diff --git a/04-model-description.Rmd b/04-model-description.Rmd index ccb159d..a8585fd 100644 --- a/04-model-description.Rmd +++ b/04-model-description.Rmd @@ -188,7 +188,7 @@ Covariates will include both historical climate variables and those predicted un The rate at which infected individuals shed *V. cholerae* into the environment ($\zeta$) is a critical factor influencing cholera transmission. Shedding rates can vary widely depending on the severity of the infection, the immune response of the individual, and environmental factors. According to [Fung 2014](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3926264/), the shedding rate is estimated to range from 0.01 to 10 cells per mL per person per day. -Further studies support these findings, indicating that shedding rates can indeed fluctuate significantly. For instance, [Nelson et al (2009)](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3842031/) note that during the, depending on the phase of infection, individuals can shed $10^3$ (asymptomatic cases) to $10^12$ (severe cases) *V. cholerae* cells per gram of stool. Future version of the model may attempt to capture the nuances of shedding dynamics, but here we make the simplifying assumption that shedding is constant across infected individuals and has a wide range of variability with no prior distributional assumptions: +Further studies support these findings, indicating that shedding rates can indeed fluctuate significantly. For instance, [Nelson et al (2009)](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3842031/) note that during the, depending on the phase of infection, individuals can shed $10^3$ (asymptomatic cases) to $10^{12}$ (severe cases) *V. cholerae* cells per gram of stool. Future version of the model may attempt to capture the nuances of shedding dynamics, but here we make the simplifying assumption that shedding is constant across infected individuals and has a wide range of variability with no prior distributional assumptions: $$ \zeta \sim \text{Uniform}(0.01, 10). @@ -219,7 +219,7 @@ knitr::kable( knitr::include_graphics("figures/wash_incidence_correlation.png") ``` -```{r wash-country, echo=FALSE, fig.align='center', out.width="100%", fig.cap="The optimized weighted mean of WASH variables for AFRO countries"} +```{r wash-country, echo=FALSE, fig.align='center', out.width="100%", fig.cap="The optimized weighted mean of WASH variables for AFRO countries. Countries labeled in orange denote countries with an imputed weighted mean WASH variable. Imputed values are the weighted mean from the 3 most similar countries."} knitr::include_graphics("figures/wash_index_by_country.png") ``` diff --git a/MOSAIC-docs.aux b/MOSAIC-docs.aux index e91830b..1ef3191 100644 --- a/MOSAIC-docs.aux +++ b/MOSAIC-docs.aux @@ -98,18 +98,18 @@ \newlabel{immunity-from-vaccination}{{4.4.1}{27}{Immunity from vaccination}{subsection.4.4.1}{}} \@writefile{lof}{\contentsline {figure}{\numberline {4.8}{\ignorespaces Relationship between WASH variables and cholera incidences.}}{28}{figure.4.8}\protected@file@percent } \newlabel{fig:wash-incidence}{{4.8}{28}{Relationship between WASH variables and cholera incidences}{figure.4.8}{}} -\@writefile{lof}{\contentsline {figure}{\numberline {4.9}{\ignorespaces The optimized weighted mean of WASH variables for AFRO countries}}{29}{figure.4.9}\protected@file@percent } -\newlabel{fig:wash-country}{{4.9}{29}{The optimized weighted mean of WASH variables for AFRO countries}{figure.4.9}{}} -\@writefile{lot}{\contentsline {table}{\numberline {4.3}{\ignorespaces Summary of Effectiveness Data}}{29}{table.4.3}\protected@file@percent } -\newlabel{tab:effectiveness-papers}{{4.3}{29}{Summary of Effectiveness Data}{table.4.3}{}} +\@writefile{lof}{\contentsline {figure}{\numberline {4.9}{\ignorespaces The optimized weighted mean of WASH variables for AFRO countries. Countries labeled in orange denote countries with an imputed weighted mean WASH variable. Imputed values are the weighted mean from the 3 most similar countries.}}{29}{figure.4.9}\protected@file@percent } +\newlabel{fig:wash-country}{{4.9}{29}{The optimized weighted mean of WASH variables for AFRO countries. Countries labeled in orange denote countries with an imputed weighted mean WASH variable. Imputed values are the weighted mean from the 3 most similar countries}{figure.4.9}{}} +\@writefile{lot}{\contentsline {table}{\numberline {4.3}{\ignorespaces Summary of Effectiveness Data}}{30}{table.4.3}\protected@file@percent } +\newlabel{tab:effectiveness-papers}{{4.3}{30}{Summary of Effectiveness Data}{table.4.3}{}} \newlabel{eq:omega}{{4.8}{30}{Immunity from vaccination}{equation.4.4.8}{}} -\newlabel{eq:effectiveness}{{4.9}{30}{Immunity from vaccination}{equation.4.4.9}{}} -\@writefile{toc}{\contentsline {subsection}{\numberline {4.4.2}Immunity from natural infection}{30}{subsection.4.4.2}\protected@file@percent } -\newlabel{immunity-from-natural-infection}{{4.4.2}{30}{Immunity from natural infection}{subsection.4.4.2}{}} +\newlabel{eq:effectiveness}{{4.9}{31}{Immunity from vaccination}{equation.4.4.9}{}} \@writefile{lof}{\contentsline {figure}{\numberline {4.10}{\ignorespaces This is vaccine effectiveness}}{31}{figure.4.10}\protected@file@percent } \newlabel{fig:effectiveness}{{4.10}{31}{This is vaccine effectiveness}{figure.4.10}{}} -\@writefile{lot}{\contentsline {table}{\numberline {4.4}{\ignorespaces Sources for the duration of immunity fro natural infection.}}{31}{table.4.4}\protected@file@percent } -\newlabel{tab:immunity-sources}{{4.4}{31}{Sources for the duration of immunity fro natural infection}{table.4.4}{}} +\@writefile{toc}{\contentsline {subsection}{\numberline {4.4.2}Immunity from natural infection}{31}{subsection.4.4.2}\protected@file@percent } +\newlabel{immunity-from-natural-infection}{{4.4.2}{31}{Immunity from natural infection}{subsection.4.4.2}{}} +\@writefile{lot}{\contentsline {table}{\numberline {4.4}{\ignorespaces Sources for the duration of immunity fro natural infection.}}{32}{table.4.4}\protected@file@percent } +\newlabel{tab:immunity-sources}{{4.4}{32}{Sources for the duration of immunity fro natural infection}{table.4.4}{}} \@writefile{lof}{\contentsline {figure}{\numberline {4.11}{\ignorespaces The duration of immunity after natural infection with *V. cholerae*.}}{32}{figure.4.11}\protected@file@percent } \newlabel{fig:immune-decay}{{4.11}{32}{The duration of immunity after natural infection with *V. cholerae*}{figure.4.11}{}} \@writefile{toc}{\contentsline {section}{\numberline {4.5}Spatial dynamics}{32}{section.4.5}\protected@file@percent } @@ -133,25 +133,25 @@ \newlabel{the-spatial-hazard}{{4.5.5}{36}{The spatial hazard}{subsection.4.5.5}{}} \newlabel{eq:hazard}{{4.13}{36}{The spatial hazard}{equation.4.5.13}{}} \newlabel{eq:waiting}{{4.14}{36}{The spatial hazard}{equation.4.5.14}{}} -\@writefile{toc}{\contentsline {subsection}{\numberline {4.5.6}Coupling among locations}{36}{subsection.4.5.6}\protected@file@percent } -\newlabel{coupling-among-locations}{{4.5.6}{36}{Coupling among locations}{subsection.4.5.6}{}} \@writefile{lof}{\contentsline {figure}{\numberline {4.14}{\ignorespaces The estimated weekly probability of travel outside of each origin location $\mittau _i$ and 95\% confidence intervals is shown in panel A with the population mean indicated as a red dashed line. Panel B shows the estimated total number of travelers leaving origin $i$ each week.}}{37}{figure.4.14}\protected@file@percent } \newlabel{fig:mobility-departure}{{4.14}{37}{The estimated weekly probability of travel outside of each origin location $\tau _i$ and 95\% confidence intervals is shown in panel A with the population mean indicated as a red dashed line. Panel B shows the estimated total number of travelers leaving origin $i$ each week}{figure.4.14}{}} \@writefile{lof}{\contentsline {figure}{\numberline {4.15}{\ignorespaces The diffusion process $\mitpi _{ij}$ which gives the estimated probability of travel from origin $i$ to destination $j$ given that travel outside of origin $i$ has occurred.}}{38}{figure.4.15}\protected@file@percent } \newlabel{fig:mobility-diffusion}{{4.15}{38}{The diffusion process $\pi _{ij}$ which gives the estimated probability of travel from origin $i$ to destination $j$ given that travel outside of origin $i$ has occurred}{figure.4.15}{}} +\@writefile{toc}{\contentsline {subsection}{\numberline {4.5.6}Coupling among locations}{39}{subsection.4.5.6}\protected@file@percent } +\newlabel{coupling-among-locations}{{4.5.6}{39}{Coupling among locations}{subsection.4.5.6}{}} \newlabel{eq:correlation}{{4.15}{39}{Coupling among locations}{equation.4.5.15}{}} \@writefile{toc}{\contentsline {section}{\numberline {4.6}The observation process}{39}{section.4.6}\protected@file@percent } \newlabel{the-observation-process}{{4.6}{39}{The observation process}{section.4.6}{}} \@writefile{toc}{\contentsline {subsection}{\numberline {4.6.1}Rate of symptomatic infection}{39}{subsection.4.6.1}\protected@file@percent } \newlabel{rate-of-symptomatic-infection}{{4.6.1}{39}{Rate of symptomatic infection}{subsection.4.6.1}{}} -\@writefile{lof}{\contentsline {figure}{\numberline {4.16}{\ignorespaces Proportion of infections that are symptomatic.}}{40}{figure.4.16}\protected@file@percent } -\newlabel{fig:symptomatic-fig}{{4.16}{40}{Proportion of infections that are symptomatic}{figure.4.16}{}} -\@writefile{lot}{\contentsline {table}{\numberline {4.5}{\ignorespaces Summary of Studies on Cholera Immunity}}{41}{table.4.5}\protected@file@percent } -\newlabel{tab:symptomatic-table}{{4.5}{41}{Summary of Studies on Cholera Immunity}{table.4.5}{}} -\@writefile{toc}{\contentsline {subsection}{\numberline {4.6.2}Suspected cases}{41}{subsection.4.6.2}\protected@file@percent } -\newlabel{suspected-cases}{{4.6.2}{41}{Suspected cases}{subsection.4.6.2}{}} -\@writefile{toc}{\contentsline {subsection}{\numberline {4.6.3}Case fatality rate}{41}{subsection.4.6.3}\protected@file@percent } -\newlabel{case-fatality-rate}{{4.6.3}{41}{Case fatality rate}{subsection.4.6.3}{}} +\@writefile{lot}{\contentsline {table}{\numberline {4.5}{\ignorespaces Summary of Studies on Cholera Immunity}}{40}{table.4.5}\protected@file@percent } +\newlabel{tab:symptomatic-table}{{4.5}{40}{Summary of Studies on Cholera Immunity}{table.4.5}{}} +\@writefile{toc}{\contentsline {subsection}{\numberline {4.6.2}Suspected cases}{40}{subsection.4.6.2}\protected@file@percent } +\newlabel{suspected-cases}{{4.6.2}{40}{Suspected cases}{subsection.4.6.2}{}} +\@writefile{toc}{\contentsline {subsection}{\numberline {4.6.3}Case fatality rate}{40}{subsection.4.6.3}\protected@file@percent } +\newlabel{case-fatality-rate}{{4.6.3}{40}{Case fatality rate}{subsection.4.6.3}{}} +\@writefile{lof}{\contentsline {figure}{\numberline {4.16}{\ignorespaces Proportion of infections that are symptomatic.}}{41}{figure.4.16}\protected@file@percent } +\newlabel{fig:symptomatic-fig}{{4.16}{41}{Proportion of infections that are symptomatic}{figure.4.16}{}} \@writefile{lof}{\contentsline {figure}{\numberline {4.17}{\ignorespaces Proportion of suspected cholera cases that are true infections. Panel A shows the 'low' assumption which estimates across all settings: $\mitrho \sim \text {Beta}(5.43, 5.01)$. Panel B shows the 'high' assumption where the estimate reflects high-quality studies during outbreaks: $\mitrho \sim \text {Beta}(4.79, 1.53)$}}{42}{figure.4.17}\protected@file@percent } \newlabel{fig:rho}{{4.17}{42}{Proportion of suspected cholera cases that are true infections. Panel A shows the 'low' assumption which estimates across all settings: $\rho \sim \text {Beta}(5.43, 5.01)$. Panel B shows the 'high' assumption where the estimate reflects high-quality studies during outbreaks: $\rho \sim \text {Beta}(4.79, 1.53)$}{figure.4.17}{}} \@writefile{toc}{\contentsline {section}{\numberline {4.7}Demographics}{43}{section.4.7}\protected@file@percent } @@ -166,20 +166,20 @@ \newlabel{fig:cfr-beta}{{4.19}{46}{Beta distributions of the overall Case Fatality Rate (CFR) from 2014 to 2024. Examples show the overall CFR for the AFRO region (2\%) in black, Congo with the highest CFR (7\%) in red, and South Sudan with the lowest CFR (0.1\%) in blue}{figure.4.19}{}} \@writefile{lot}{\contentsline {table}{\numberline {4.7}{\ignorespaces Demographic for AFRO countries in 2023. Data include: total population as of January 1, 2023, daily birth rate, and daily death rate. Values are calculate from crude birth and death rates from UN World Population Prospects 2024.}}{47}{table.4.7}\protected@file@percent } \newlabel{tab:demographics}{{4.7}{47}{Demographic for AFRO countries in 2023. Data include: total population as of January 1, 2023, daily birth rate, and daily death rate. Values are calculate from crude birth and death rates from UN World Population Prospects 2024}{table.4.7}{}} +\@writefile{lot}{\contentsline {table}{\numberline {4.8}{\ignorespaces Generation Time in Weeks}}{48}{table.4.8}\protected@file@percent } +\newlabel{tab:unnamed-chunk-3}{{4.8}{48}{Generation Time in Weeks}{table.4.8}{}} \newlabel{eq:R}{{4.19}{48}{The reproductive number}{equation.4.8.19}{}} \@writefile{toc}{\contentsline {subsection}{\numberline {4.8.1}The generation time distribution}{48}{subsection.4.8.1}\protected@file@percent } \newlabel{the-generation-time-distribution}{{4.8.1}{48}{The generation time distribution}{subsection.4.8.1}{}} \newlabel{eq:generation-time}{{4.20}{48}{The generation time distribution}{equation.4.8.20}{}} -\@writefile{lof}{\contentsline {figure}{\numberline {4.20}{\ignorespaces This is generation time}}{48}{figure.4.20}\protected@file@percent } -\newlabel{fig:generation}{{4.20}{48}{This is generation time}{figure.4.20}{}} -\@writefile{lot}{\contentsline {table}{\numberline {4.8}{\ignorespaces Generation Time in Weeks}}{49}{table.4.8}\protected@file@percent } -\newlabel{tab:unnamed-chunk-3}{{4.8}{49}{Generation Time in Weeks}{table.4.8}{}} -\@writefile{toc}{\contentsline {section}{\numberline {4.9}Initial conditions}{49}{section.4.9}\protected@file@percent } -\newlabel{initial-conditions}{{4.9}{49}{Initial conditions}{section.4.9}{}} -\@writefile{toc}{\contentsline {section}{\numberline {4.10}Model calibration}{49}{section.4.10}\protected@file@percent } -\newlabel{model-calibration}{{4.10}{49}{Model calibration}{section.4.10}{}} -\@writefile{toc}{\contentsline {section}{\numberline {4.11}Caveats}{49}{section.4.11}\protected@file@percent } -\newlabel{caveats}{{4.11}{49}{Caveats}{section.4.11}{}} +\@writefile{toc}{\contentsline {section}{\numberline {4.9}Initial conditions}{48}{section.4.9}\protected@file@percent } +\newlabel{initial-conditions}{{4.9}{48}{Initial conditions}{section.4.9}{}} +\@writefile{lof}{\contentsline {figure}{\numberline {4.20}{\ignorespaces This is generation time}}{49}{figure.4.20}\protected@file@percent } +\newlabel{fig:generation}{{4.20}{49}{This is generation time}{figure.4.20}{}} +\@writefile{toc}{\contentsline {section}{\numberline {4.10}Model calibration}{50}{section.4.10}\protected@file@percent } +\newlabel{model-calibration}{{4.10}{50}{Model calibration}{section.4.10}{}} +\@writefile{toc}{\contentsline {section}{\numberline {4.11}Caveats}{50}{section.4.11}\protected@file@percent } +\newlabel{caveats}{{4.11}{50}{Caveats}{section.4.11}{}} \@writefile{toc}{\contentsline {section}{\numberline {4.12}Table of parameters}{50}{section.4.12}\protected@file@percent } \newlabel{table-of-parameters}{{4.12}{50}{Table of parameters}{section.4.12}{}} \@writefile{toc}{\contentsline {section}{\numberline {4.13}References}{50}{section.4.13}\protected@file@percent } diff --git a/MOSAIC-docs.log b/MOSAIC-docs.log index 2dc54ec..5beb7a0 100644 --- a/MOSAIC-docs.log +++ b/MOSAIC-docs.log @@ -1,4 +1,4 @@ -This is XeTeX, Version 3.141592653-2.6-0.999996 (TeX Live 2024) (preloaded format=xelatex 2024.8.31) 25 SEP 2024 13:28 +This is XeTeX, Version 3.141592653-2.6-0.999996 (TeX Live 2024) (preloaded format=xelatex 2024.8.31) 25 SEP 2024 22:27 entering extended mode restricted \write18 enabled. %&-line parsing enabled. @@ -982,13 +982,15 @@ File: figures/wash_index_by_country.png Graphic file (type bmp) [27] [28] + +[29] Overfull \hbox (143.90997pt too wide) in paragraph at lines 548--563 [][] [] -[29] +[30] File: figures/vaccine_effectiveness.png Graphic file (type bmp) @@ -1003,8 +1005,6 @@ Overfull \hbox (281.69998pt too wide) in paragraph at lines 603--614 -[30] - [31] File: figures/immune_decay.png Graphic file (type bmp) @@ -1037,6 +1037,9 @@ Overfull \hbox (562.07999pt too wide) in paragraph at lines 791--814 [][] [] + + +[39] File: figures/proportion_symptomatic.png Graphic file (type bmp) @@ -1049,15 +1052,12 @@ Underfull \hbox (badness 10000) in paragraph at lines 820--821 [] - - -[39] - -[40] File: figures/suspected_cases.png Graphic file (type bmp) +[40] + [41] [42] @@ -1149,6 +1149,6 @@ Here is how much of TeX's memory you used: 45609 multiletter control sequences out of 15000+600000 567563 words of font info for 114 fonts, out of 8000000 for 9000 36 hyphenation exceptions out of 8191 - 90i,12n,123p,2001b,415s stack positions out of 10000i,1000n,20000p,200000b,200000s + 90i,12n,123p,2001b,405s stack positions out of 10000i,1000n,20000p,200000b,200000s Output written on MOSAIC-docs.pdf (59 pages). diff --git a/MOSAIC-docs.toc b/MOSAIC-docs.toc index 9f65004..872c9e3 100644 --- a/MOSAIC-docs.toc +++ b/MOSAIC-docs.toc @@ -28,24 +28,24 @@ \contentsline {subsection}{\numberline {4.3.4}WAter, Sanitation, and Hygiene (WASH)}{26}{subsection.4.3.4}% \contentsline {section}{\numberline {4.4}Immune dynamics}{27}{section.4.4}% \contentsline {subsection}{\numberline {4.4.1}Immunity from vaccination}{27}{subsection.4.4.1}% -\contentsline {subsection}{\numberline {4.4.2}Immunity from natural infection}{30}{subsection.4.4.2}% +\contentsline {subsection}{\numberline {4.4.2}Immunity from natural infection}{31}{subsection.4.4.2}% \contentsline {section}{\numberline {4.5}Spatial dynamics}{32}{section.4.5}% \contentsline {subsection}{\numberline {4.5.1}Human mobility model}{33}{subsection.4.5.1}% \contentsline {subsection}{\numberline {4.5.2}Estimating the departure process}{35}{subsection.4.5.2}% \contentsline {subsection}{\numberline {4.5.3}Estimating the diffusion process}{35}{subsection.4.5.3}% \contentsline {subsection}{\numberline {4.5.4}The probability of spatial transmission}{36}{subsection.4.5.4}% \contentsline {subsection}{\numberline {4.5.5}The spatial hazard}{36}{subsection.4.5.5}% -\contentsline {subsection}{\numberline {4.5.6}Coupling among locations}{36}{subsection.4.5.6}% +\contentsline {subsection}{\numberline {4.5.6}Coupling among locations}{39}{subsection.4.5.6}% \contentsline {section}{\numberline {4.6}The observation process}{39}{section.4.6}% \contentsline {subsection}{\numberline {4.6.1}Rate of symptomatic infection}{39}{subsection.4.6.1}% -\contentsline {subsection}{\numberline {4.6.2}Suspected cases}{41}{subsection.4.6.2}% -\contentsline {subsection}{\numberline {4.6.3}Case fatality rate}{41}{subsection.4.6.3}% +\contentsline {subsection}{\numberline {4.6.2}Suspected cases}{40}{subsection.4.6.2}% +\contentsline {subsection}{\numberline {4.6.3}Case fatality rate}{40}{subsection.4.6.3}% \contentsline {section}{\numberline {4.7}Demographics}{43}{section.4.7}% \contentsline {section}{\numberline {4.8}The reproductive number}{43}{section.4.8}% \contentsline {subsection}{\numberline {4.8.1}The generation time distribution}{48}{subsection.4.8.1}% -\contentsline {section}{\numberline {4.9}Initial conditions}{49}{section.4.9}% -\contentsline {section}{\numberline {4.10}Model calibration}{49}{section.4.10}% -\contentsline {section}{\numberline {4.11}Caveats}{49}{section.4.11}% +\contentsline {section}{\numberline {4.9}Initial conditions}{48}{section.4.9}% +\contentsline {section}{\numberline {4.10}Model calibration}{50}{section.4.10}% +\contentsline {section}{\numberline {4.11}Caveats}{50}{section.4.11}% \contentsline {section}{\numberline {4.12}Table of parameters}{50}{section.4.12}% \contentsline {section}{\numberline {4.13}References}{50}{section.4.13}% \contentsline {chapter}{\numberline {5}Scenarios}{53}{chapter.5}% diff --git a/docs/04-model-description.md b/docs/04-model-description.md index e67dc0e..c4632dd 100644 --- a/docs/04-model-description.md +++ b/docs/04-model-description.md @@ -283,7 +283,7 @@ Covariates will include both historical climate variables and those predicted un The rate at which infected individuals shed *V. cholerae* into the environment ($\zeta$) is a critical factor influencing cholera transmission. Shedding rates can vary widely depending on the severity of the infection, the immune response of the individual, and environmental factors. According to [Fung 2014](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3926264/), the shedding rate is estimated to range from 0.01 to 10 cells per mL per person per day. -Further studies support these findings, indicating that shedding rates can indeed fluctuate significantly. For instance, [Nelson et al (2009)](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3842031/) note that during the, depending on the phase of infection, individuals can shed $10^3$ (asymptomatic cases) to $10^12$ (severe cases) *V. cholerae* cells per gram of stool. Future version of the model may attempt to capture the nuances of shedding dynamics, but here we make the simplifying assumption that shedding is constant across infected individuals and has a wide range of variability with no prior distributional assumptions: +Further studies support these findings, indicating that shedding rates can indeed fluctuate significantly. For instance, [Nelson et al (2009)](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3842031/) note that during the, depending on the phase of infection, individuals can shed $10^3$ (asymptomatic cases) to $10^{12}$ (severe cases) *V. cholerae* cells per gram of stool. Future version of the model may attempt to capture the nuances of shedding dynamics, but here we make the simplifying assumption that shedding is constant across infected individuals and has a wide range of variability with no prior distributional assumptions: $$ \zeta \sim \text{Uniform}(0.01, 10). @@ -348,8 +348,8 @@ To parameterize $\theta_j$, we calculated a weighted mean of the 8 WASH variable
-The optimized weighted mean of WASH variables for AFRO countries -

(\#fig:wash-country)The optimized weighted mean of WASH variables for AFRO countries

+The optimized weighted mean of WASH variables for AFRO countries. Countries labeled in orange denote countries with an imputed weighted mean WASH variable. Imputed values are the weighted mean from the 3 most similar countries. +

(\#fig:wash-country)The optimized weighted mean of WASH variables for AFRO countries. Countries labeled in orange denote countries with an imputed weighted mean WASH variable. Imputed values are the weighted mean from the 3 most similar countries.

diff --git a/docs/MOSAIC-docs.epub b/docs/MOSAIC-docs.epub index 68902c4..d12fbc7 100644 Binary files a/docs/MOSAIC-docs.epub and b/docs/MOSAIC-docs.epub differ diff --git a/docs/MOSAIC-docs.pdf b/docs/MOSAIC-docs.pdf index c539575..c480a43 100644 Binary files a/docs/MOSAIC-docs.pdf and b/docs/MOSAIC-docs.pdf differ diff --git a/docs/MOSAIC-docs.tex b/docs/MOSAIC-docs.tex index bc8827e..a1a527d 100644 --- a/docs/MOSAIC-docs.tex +++ b/docs/MOSAIC-docs.tex @@ -114,7 +114,7 @@ \chapter*{}\label{section} \hfill\break {\emph{ -Website under development. Last compiled on 2024-09-25 at 01:28 PM PDT. +Website under development. Last compiled on 2024-09-25 at 10:27 PM PDT. }} \section*{Welcome}\label{welcome} @@ -470,7 +470,7 @@ \subsection{Shedding}\label{shedding} The rate at which infected individuals shed \emph{V. cholerae} into the environment (\(\zeta\)) is a critical factor influencing cholera transmission. Shedding rates can vary widely depending on the severity of the infection, the immune response of the individual, and environmental factors. According to \href{https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3926264/}{Fung 2014}, the shedding rate is estimated to range from 0.01 to 10 cells per mL per person per day. -Further studies support these findings, indicating that shedding rates can indeed fluctuate significantly. For instance, \href{https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3842031/}{Nelson et al (2009)} note that during the, depending on the phase of infection, individuals can shed \(10^3\) (asymptomatic cases) to \(10^12\) (severe cases) \emph{V. cholerae} cells per gram of stool. Future version of the model may attempt to capture the nuances of shedding dynamics, but here we make the simplifying assumption that shedding is constant across infected individuals and has a wide range of variability with no prior distributional assumptions: +Further studies support these findings, indicating that shedding rates can indeed fluctuate significantly. For instance, \href{https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3842031/}{Nelson et al (2009)} note that during the, depending on the phase of infection, individuals can shed \(10^3\) (asymptomatic cases) to \(10^{12}\) (severe cases) \emph{V. cholerae} cells per gram of stool. Future version of the model may attempt to capture the nuances of shedding dynamics, but here we make the simplifying assumption that shedding is constant across infected individuals and has a wide range of variability with no prior distributional assumptions: \[ \zeta \sim \text{Uniform}(0.01, 10). @@ -524,7 +524,7 @@ \subsection{WAter, Sanitation, and Hygiene (WASH)}\label{water-sanitation-and-hy } -\caption{The optimized weighted mean of WASH variables for AFRO countries}\label{fig:wash-country} +\caption{The optimized weighted mean of WASH variables for AFRO countries. Countries labeled in orange denote countries with an imputed weighted mean WASH variable. Imputed values are the weighted mean from the 3 most similar countries.}\label{fig:wash-country} \end{figure} \section{Immune dynamics}\label{immune-dynamics} diff --git a/docs/figures/wash_incidence_correlation.png b/docs/figures/wash_incidence_correlation.png index ae98b6c..89e1421 100644 Binary files a/docs/figures/wash_incidence_correlation.png and b/docs/figures/wash_incidence_correlation.png differ diff --git a/docs/figures/wash_index_by_country.png b/docs/figures/wash_index_by_country.png index dbbea05..d99ee2b 100644 Binary files a/docs/figures/wash_index_by_country.png and b/docs/figures/wash_index_by_country.png differ diff --git a/docs/index.html b/docs/index.html index 688b209..d36c2f0 100644 --- a/docs/index.html +++ b/docs/index.html @@ -97,7 +97,7 @@


-Website under development. Last compiled on 2024-09-25 at 01:27 PM PDT. +Website under development. Last compiled on 2024-09-25 at 10:27 PM PDT.
diff --git a/docs/index.md b/docs/index.md index 80d5583..e1212b0 100644 --- a/docs/index.md +++ b/docs/index.md @@ -33,7 +33,7 @@ nocite: '@*' \
* -Website under development. Last compiled on 2024-09-25 at 01:28 PM PDT. +Website under development. Last compiled on 2024-09-25 at 10:27 PM PDT. *
## Welcome {-} diff --git a/docs/model-description.html b/docs/model-description.html index b9cd0ee..22a9b4d 100644 --- a/docs/model-description.html +++ b/docs/model-description.html @@ -512,7 +512,7 @@

4.3.3 Shedding

The rate at which infected individuals shed V. cholerae into the environment (\(\zeta\)) is a critical factor influencing cholera transmission. Shedding rates can vary widely depending on the severity of the infection, the immune response of the individual, and environmental factors. According to Fung 2014, the shedding rate is estimated to range from 0.01 to 10 cells per mL per person per day.

-

Further studies support these findings, indicating that shedding rates can indeed fluctuate significantly. For instance, Nelson et al (2009) note that during the, depending on the phase of infection, individuals can shed \(10^3\) (asymptomatic cases) to \(10^12\) (severe cases) V. cholerae cells per gram of stool. Future version of the model may attempt to capture the nuances of shedding dynamics, but here we make the simplifying assumption that shedding is constant across infected individuals and has a wide range of variability with no prior distributional assumptions:

+

Further studies support these findings, indicating that shedding rates can indeed fluctuate significantly. For instance, Nelson et al (2009) note that during the, depending on the phase of infection, individuals can shed \(10^3\) (asymptomatic cases) to \(10^{12}\) (severe cases) V. cholerae cells per gram of stool. Future version of the model may attempt to capture the nuances of shedding dynamics, but here we make the simplifying assumption that shedding is constant across infected individuals and has a wide range of variability with no prior distributional assumptions:

\[ \zeta \sim \text{Uniform}(0.01, 10). \]

@@ -610,8 +610,8 @@

-The optimized weighted mean of WASH variables for AFRO countries

-Figure 4.9: The optimized weighted mean of WASH variables for AFRO countries +The optimized weighted mean of WASH variables for AFRO countries. Countries labeled in orange denote countries with an imputed weighted mean WASH variable. Imputed values are the weighted mean from the 3 most similar countries.

+Figure 4.9: The optimized weighted mean of WASH variables for AFRO countries. Countries labeled in orange denote countries with an imputed weighted mean WASH variable. Imputed values are the weighted mean from the 3 most similar countries.

diff --git a/docs/search.json b/docs/search.json index 6312e6e..41ee78c 100644 --- a/docs/search.json +++ b/docs/search.json @@ -1 +1 @@ -[{"path":"index.html","id":"section","chapter":"","heading":"","text":"","code":""},{"path":"index.html","id":"welcome","chapter":"","heading":"Welcome","text":"Welcome Metapopulation Outbreak Simulation Agent-based Implementation Cholera (MOSAIC). MOSAIC framework simulates transmission dynamics cholera Sub-Saharan Africa (SSA) provides tools understand impact interventions, vaccination, well large-scale drivers like climate change. MOSAIC built using Light-agent Spatial Model ERadication (LASER) platform, site serves documentation model’s methods associated analyses. Please note MOSAIC currently development, content may change regularly. sharing increase visibility welcome feedback aspect model.","code":""},{"path":"index.html","id":"contact","chapter":"","heading":"Contact","text":"MOSAIC developed team researchers Institute Disease Modeling (IDM) dedicated developing modeling methods software tools help decision-makers understand respond infectious disease outbreaks. website currently maintained John Giles (@gilesjohnr). general questions, contact John Giles (john.giles@gatesfoundation.org), Jillian Gauld (jillian.gauld@gatesfoundation.org), /Rajiv Sodhi (rajiv.sodhi@gatesfoundation.org).","code":""},{"path":"index.html","id":"funding","chapter":"","heading":"Funding","text":"work developed Institute Disease Modeling support funded research grants made Bill & Melinda Gates Foundation.","code":""},{"path":"introduction.html","id":"introduction","chapter":"1 Introduction","heading":"1 Introduction","text":"","code":""},{"path":"introduction.html","id":"history","chapter":"1 Introduction","heading":"1.1 History","text":"Vibrio cholerae introduced African continent Asia 1970s since become endemic many countries.","code":""},{"path":"introduction.html","id":"recent-surge","chapter":"1 Introduction","heading":"1.2 Recent Surge","text":"Although sporadic cholera outbreaks past five decades, significant surge cases since 2021. increase likely due factors climate change disruptions municipal services COVID-19 pandemic.","code":""},{"path":"introduction.html","id":"gtfcc-goals","chapter":"1 Introduction","heading":"1.3 GTFCC Goals","text":"Global Task Force Cholera Control (GTFCC) aims reduce cholera deaths 90% 2030.","code":""},{"path":"introduction.html","id":"ocv-stockpiles","chapter":"1 Introduction","heading":"1.4 OCV Stockpiles","text":"major concern recent surge cases depletion oral cholera vaccine (OCV) stockpiles. response, officials shifted single-dose OCV strategies. Efforts underway replenish stockpiles, key question best allocate reduce transmission regionally support GTFCC’s goal.","code":""},{"path":"introduction.html","id":"climate-change","chapter":"1 Introduction","heading":"1.5 Climate Change","text":"Environmental factors play significant role cholera outbreaks, warmer wetter conditions creating favorable environment Vibrio cholerae. conditions likely exacerbate already challenging endemic outbreak settings. Models incorporate climatic forcing can provide insights future cholera dynamics due climate change aid achieving GTFCC goal.","code":""},{"path":"rationale.html","id":"rationale","chapter":"2 Rationale","heading":"2 Rationale","text":"significant challenge controlling cholera transmission Sub-Saharan Africa (SSA) lack comprehensive datasets dynamic models designed support ongoing policy-making. persistent endemic nature cholera SSA presents complex quantitative challenge, requiring sophisticated models produce meaningful inferences. Models incorporate necessary natural history disease dynamics, operate adequate spatial temporal scales, crucial providing timely actionable information address ongoing future cholera outbreaks.Although developing data models scales challenging, goal iteratively create landscape-scale transmission model cholera SSA can provide weekly predictions key epidemiological metrics. modeling methods leverage wide array --date data sources, including incidence mortality reports, patterns human movement, vaccination history schedules, environmental factors.Key questions address include administer limited supply oral cholera vaccine (OCV) severe weather events climate change impact future outbreaks. landscape-scale model accounts endemic transmission patterns valuable tool addressing questions.","code":""},{"path":"data.html","id":"data","chapter":"3 Data","heading":"3 Data","text":"MOSAIC model requires diverse set data sources, directly used define model parameters (e.g., birth death rates), others help fit models priori provide informative priors transmission model. additional data sources become available, future versions model adapt incorporate . now, following data sources represent minimum requirements initiate viable first model.","code":""},{"path":"data.html","id":"historical-incidence-and-deaths","chapter":"3 Data","heading":"3.1 Historical Incidence and Deaths","text":"Data historical cholera incidence deaths crucial establishing baseline transmission patterns. compiled annual total reported cases deaths AFRO region countries January 1970 August 2024. data comes several sources include:World Data (1970-2021): Number Reported Cases Cholera (1949-2021) Number Reported Deaths Cholera (1949-2021). World Data group compiled data previously published annual reports.Annual Report 2022: data manually extracted World Health Organization’s Weekly Epidemiological Record 38, 2023, 98, 431–452.Global Cholera Acute Watery Diarrhea Dashboard (2023-2024): Unofficial tallies reported cases deaths 2023 part 2024 available Global Cholera AWD Dashboard.","code":""},{"path":"data.html","id":"recent-incidence-and-deaths","chapter":"3 Data","heading":"3.2 Recent Incidence and Deaths","text":"capture recent cholera trends, retrieved reported cases deaths data Global Cholera Acute Watery Diarrhea Dashboard REST API. data provide weekly incidence deaths January 2023 August 2024 provides --date counts country level.","code":""},{"path":"data.html","id":"vaccinations","chapter":"3 Data","heading":"3.3 Vaccinations","text":"Accurate data oral cholera vaccine (OCV) campaigns vaccination history vital understanding impact vaccination efforts. data come :Cholera Vaccine Dashboard: resource (link) provides detailed information OCV distribution vaccination campaigns 2016 2024.GTFCC OCV Dashboard: Managed Médecins Sans Frontières, dashboard (link) tracks OCV deployments globally, offering granular insights vaccination efforts 2013 2024.","code":""},{"path":"data.html","id":"human-mobility-data","chapter":"3 Data","heading":"3.4 Human Mobility Data","text":"Human mobility patterns significantly influence cholera transmission. Relevant data include:OAG Passenger Booking Data: dataset (link) offers insights air passenger movements, can used model spread cholera across regions.Namibia Call Data Records: additional source Giles et al. (2020) (link) provides detailed mobility data based mobile phone records, useful localized modeling.","code":""},{"path":"data.html","id":"climate-data","chapter":"3 Data","heading":"3.5 Climate Data","text":"Climate conditions, including temperature, precipitation, extreme weather events, play critical role cholera dynamics. captured :OpenMeteo Historical Weather Data API: API (link) offers access historical climate data, essential modeling environmental factors influencing cholera outbreaks.","code":""},{"path":"data.html","id":"storms-and-floods","chapter":"3 Data","heading":"3.5.1 Storms and Floods","text":"Data extreme weather events, specifically storms floods, obtained :EM-DAT International Disaster Database: Maintained Centre Research Epidemiology Disasters (CRED) UCLouvain, database (link) provides comprehensive records disasters 2000 present, including affecting African countries.","code":""},{"path":"data.html","id":"wash-water-sanitation-and-hygiene","chapter":"3 Data","heading":"3.6 WASH (Water, Sanitation, and Hygiene)","text":"Data water, sanitation, hygiene (WASH) critical understanding environmental infrastructural factors influence cholera transmission. data sourced :UNICEF Joint Monitoring Program (JMP) Database: resource (link) offers detailed information household-level access clean water sanitation, integral cholera prevention efforts.","code":""},{"path":"data.html","id":"demographics","chapter":"3 Data","heading":"3.7 Demographics","text":"Demographic data, including population size, birth rates, death rates, foundational accurate disease modeling. data sourced :UN World Population Prospects 2024: database (link) provides probabilistic projections key demographic metrics, essential estimating population-level impacts cholera.","code":""},{"path":"model-description.html","id":"model-description","chapter":"4 Model description","heading":"4 Model description","text":"describe methods MOSAIC beta version 0.1. model version provides starting point understanding cholera transmission Sub-Saharan Africa, incorporating important drivers disease dynamics human mobility, environmental conditions, vaccination schedules. MOSAIC continues evolve, future iterations refine model components based available data improved model mechanisms, hope increase applicability real-world scenarios.model operates weekly time steps January 2023 August 2024 includes 46 countries Sub-Saharan Africa (SSA) shown Figure 4.1.\nFigure 4.1: map Sub-Saharan Africa countries experienced cholera outbreak past 5 10 years highlighted green.\n","code":""},{"path":"model-description.html","id":"transmission-dynamics","chapter":"4 Model description","heading":"4.1 Transmission dynamics","text":"model metapopulation structure familiar compartments Susceptible, Infected, Recovered individuals SIRS dynamics. model also contains compartments vaccinated individuals (V) Water & environment based transmission (W) refer SVIWRS.\nFigure 4.2: diagram SVIWRS (Susceptible-Vaccinated-Infected-Water/environmental-Recovered-Susceptible) model shows model compartments circles rate parameters displayed. primary data sources model fit shown square nodes (Vaccination data, reported cases deaths).\nSVIWRS metapopulation model, shown Figure 4.2, governed following difference equations:\\[\\begin{equation}\n\\begin{aligned}\nS_{j,t+1} &= b_j N_{jt} + S_{jt} - \\phi \\nu_{jt} S_{jt} + \\omega V_{jt} - \\Lambda_{j,t+1} - \\Psi_{j,t+1} + \\varepsilon R_{jt} - d_j S_{jt}\\\\[11pt]\nV_{j,t+1} &= V_{jt} + \\phi \\nu_{jt} S_{jt} - \\omega V_{jt} - d_j V_{jt}\\\\[11pt]\nI_{j,t+1} &= I_{jt} + \\Lambda_{j,t+1} + \\Psi_{j,t+1} - \\gamma I_{jt} - \\mu \\sigma I_{jt} - d_j I_{jt}\\\\[11pt]\nW_{j,t+1} &= W_{jt} + \\zeta I_{jt} - \\delta_{jt} W_{jt}\\\\[11pt]\nR_{j,t+1} &= R_{jt} + \\gamma I_{jt} - \\varepsilon R_{jt} - d_j R_{jt}\\\\[11pt]\n\\end{aligned}\n\\tag{4.1}\n\\end{equation}\\]descriptions parameters Equation (4.1), see Table (4.14). Transmission dynamics driven two force infection terms, \\(\\Lambda_{jt}\\) \\(\\Psi_{jt}\\). force infection due human--human (\\(\\Lambda_{jt}\\)) :\\[\\begin{equation}\n\\begin{aligned}\n\\Lambda_{j,t+1} &= \\frac{\n\\beta_{jt}^{\\text{hum}} \\Big(\\big(S_{jt}(1-\\tau_{j})\\big) \\big(I_{jt}(1-\\tau_{j}) + \\sum_{\\forall \\= j} (\\pi_{ij}\\tau_iI_{}) \\big)\\Big)^\\alpha}{N_{jt}}.\\\\[11pt]\n\\end{aligned}\n\\tag{4.2}\n\\end{equation}\\]\\(\\beta_{jt}^{\\text{hum}}\\) rate human--human transmission. Movement within among metapopulations governed \\(\\tau_i\\), probability departing origin location \\(\\), \\(\\pi_{ij}\\) relative probability travel origin \\(\\) destination \\(j\\) (see section spatial dynamics). include environmental effects, force infection due environment--human transmission (\\(\\Psi_{jt}\\)) defined :\\[\\begin{equation}\n\\begin{aligned}\n\\Psi_{j,t+1} &= \\frac{\\beta_{jt}^{\\text{env}} \\big(S_{jt}(1-\\tau_{j})\\big) (1-\\theta_j) W_{jt}}{\\kappa+W_{jt}},\\\\[11pt]\n\\end{aligned}\n\\tag{4.3}\n\\end{equation}\\]\\(\\beta_{jt}^{\\text{env}}\\) rate environment--human transmission \\(\\theta_j\\) proportion population location \\(j\\) least basic access Water, Sanitation, Hygiene (WASH). environmental compartment model also scaled concentration (cells per mL) V. cholerae required 50% probability infection Fung 2014. See section environmental transmission water/environment compartment climatic drivers transmission.Note model processes stochastic. Transition rates converted probabilities commonly used formula \\(p(t) = 1-e^{-rt}\\) (see Ross 2007), integer quantities moved model compartments time step according binomial process like example recovery infected individuals (\\(\\gamma I_{jt}\\)):\\[\\begin{equation}\n\\frac{\\partial R}{\\partial t} \\sim \\text{Binom}(I_{jt}, 1-\\exp(-\\gamma))\n\\tag{4.4}\n\\end{equation}\\]","code":""},{"path":"model-description.html","id":"seasonality","chapter":"4 Model description","heading":"4.2 Seasonality","text":"Cholera transmission seasonal typically associated rainy season, transmission rate terms \\(\\beta_{jt}^{\\text{*}}\\) temporally forced. human--human transmission used truncated sine-cosine form Fourier series two harmonic features flexibility capture seasonal transmission dynamics driven extended rainy seasons /biannual trends:\\[\\begin{equation}\n\\beta_{jt}^{\\text{hum}} = \\beta_{j0}^{\\text{hum}} + a_1 \\cos\\left(\\frac{2\\pi t}{p}\\right) + b_1 \\sin\\left(\\frac{2\\pi t}{p}\\right) + a_2 \\cos\\left(\\frac{4\\pi t}{p}\\right) + b_2 \\sin\\left(\\frac{4\\pi t}{p}\\right)\n\\tag{4.5}\n\\end{equation}\\], \\(\\beta_{j0}^{\\text{hum}}\\) mean human--human transmission rate location \\(j\\) time steps. Seasonal dynamics determined parameters \\(a_1\\), \\(b_1\\) \\(a_2\\), \\(b_2\\) gives amplitude first second waves respectively. periodic cycle \\(p\\) 52, function controls temporal variation \\(\\beta_{jt}^{\\text{hum}}\\) 52 weeks year.\nFigure 4.3: example temporal distribution human--human transmission rate across 52 weeks year given cosine wave function. wave function fitted country designed align rainy season indicated shaded region figure.\nestimated parameters Fourier series (\\(a_1\\), \\(b_1\\), \\(a_2\\), \\(b_2\\)) using Levenberg–Marquardt algorithm minpack.lm R library. Given lack reported cholera case data many countries SSA association cholera transmission rainy season, leveraged seasonal precipitation data help fit Fourier wave function countries. first gathered weekly precipitation values 1994 2024 30 uniformly distributed points within country Open-Meteo Historical Weather Data API. fit Fourier series weekly precipitation data used parameters starting values fitting model sparse cholera case data.\nFigure 4.4: Example grid 30 uniformly distributed points within Mozambique (). scatterplot shows weekly summed precipitation values 30 grid points cholera cases plotted scale Z-Score shows variance around mean terms standard deviation. Fitted Fourier series fucntions shown blue (fit precipitation data) red (fit cholera case data) lines.\ncountries reported case data, inferred seasonal dynamics using fitted wave function neighboring country available case data. selected neighbor chosen cluster countries (grouped hierarchically four clusters based precipitation seasonality using Ward’s method; see Figure 4.5) highest correlation seasonal precipitation country lacking case data. rare event country reported case data found within seasonal cluster, expanded search 10 nearest neighbors continued expanding adding next nearest neighbor match found.\nFigure 4.5: ) Map showing clustering African countries based seasonal precipitation patterns (1994-2024). Countries colored according cluster assignments, identified using hierarchical clustering. B) Fourier series fitted weekly precipitation country. line plot shows seasonal pattern countries within given cluster. Clusteres used infer seasonal transmission dynamics countries reported cholera cases.\nUsing model fitting methods described , cluster-based approach inferring seasonal Fourier series pattern countries without reported cholera cases, modeled seasonal dynamics 41 countries MOSAIC framework. dynamics visualized Figure 4.6, corresponding Fourier model coefficients presented Table 4.1.\nFigure 4.6: Seasonal transmission patterns countries modeled MOSAIC modeled truncated Fourier series Equation (4.5). Blues lines give Fourier series model fits precipitation (1994-2024) red lines give models fits reported cholera cases (2023-2024). countries reported case data available, Fourier model inferred nearest country similar seasonal precipitation patterns determined hierarchical clustering. Countries inferred case data neighboring locations annotated red. X-axis represents weeks year (1-52), Y-axis shows Z-score weekly precipitation cholera cases.\n\nTable 4.1: Table 4.2: Estimated coefficients truncated Fourier model Equation (4.5) fit countries reported cholera cases. Model fits shown Figure 4.6.\n","code":""},{"path":"model-description.html","id":"environmental-transmission","chapter":"4 Model description","heading":"4.3 Environmental transmission","text":"Environmental transmission critical factor cholera spread consists several key components: rate infected individuals shed V. cholerae environment, pathogen’s survival rate environmental conditions, overall suitability environment sustaining bacteria time.","code":""},{"path":"model-description.html","id":"climate-driven-transmission","chapter":"4 Model description","heading":"4.3.1 Climate-driven transmission","text":"capture impacts climate-drivers cholera transmission, included parameter \\(\\psi_{jt}\\), represents current state environmental suitability respect : ) survival time V. cholerae environment , ii) rate environment--human transmission contributes overall force infection.\\[\\begin{equation}\n\\beta_{jt}^{\\text{env}} = \\beta_{j0}^{\\text{env}} \\Bigg(1 + \\frac{\\psi_{jt}-\\bar\\psi_j}{\\bar\\psi_j} \\Bigg) \\quad \\text{} \\quad \\bar\\psi_j = \\frac{1}{T} \\sum_{t=1}^{T} \\psi_{jt}\n\\tag{4.6}\n\\end{equation}\\]formulation effectively scales base environmental transmission rate \\(\\beta_{jt}^{\\text{env}}\\) varies time according climatically driven model suitability. Note , unlike cosine wave function \\(\\beta_{jt}^{\\text{hum}}\\), temporal term can increase decrease time following multi-annual cycles.[Fig: Example temporal forcing environment--human transmission]Environmental suitability (\\(\\psi_{jt}\\)) also impacts survival rate V. cholerae environment (\\(\\delta_{jt}\\)) form:\\[\\begin{equation}\n\\delta_{jt} = \\delta_{\\text{min}} + \\psi_{jt} \\times (\\delta_{\\text{max}} - \\delta_{\\text{min}})\n\\tag{4.7}\n\\end{equation}\\]normalizes variance suitability parameter bounded within minimum (\\(\\delta_{\\text{min}}\\)) maximum (\\(\\delta_{\\text{max}}\\)) survival times V. cholerae.\nFigure 4.7: Relationship environmental suitability (\\(\\psi_{jt}\\)) rate V. cholerae decay environment (\\(\\delta_j\\)). green line shows mildest penalty V. cholerae survival, survival environment \\(1/\\delta_{\\text{min}}\\) = 3 days suitability = 0 \\(1/\\delta_{\\text{max}}\\) = 90 days suitability = 1.\n","code":""},{"path":"model-description.html","id":"modeling-suitability","chapter":"4 Model description","heading":"4.3.2 Modeling suitability","text":"environmental suitability (\\(\\psi_{jt}\\)) V. cholerae modeled time series location, using covariates include environmental factors, past present climate measures, severe weather events, large-scale regional climate drivers. factors influenced climate change, source data projects covariate future different climate change scenarios. Environmental suitability, \\(\\psi_{jt}\\), generally defined :\\[\n\\psi_{jt} = f(\\text{temperature, precipitation, humidity, wind speed, soil moisture})\n\\]function \\(f(\\cdot)\\) can modeled using various approaches, including Generalized Linear Models (GLMs), Generalized Additive Models (GAMs), Boosted Regression Trees (BRTs), machine learning methods Recurrent Neural Networks (RNNs) Long Short-Term Memory Networks (LSTMs). simpler approach might involve Bayesian variable selection using BAS R package. model fitted available data, projections suitability location. Implementing rolling-window validation across time series help assess model performance. model can directly fitted reported case counts converted binary threshold, depending analysis needs. primary goal explain portion variance reported case counts proxy environmental suitability.Covariates include historical climate variables predicted climate change scenarios. example, MRI-AGCM3-2-S EC_Earth3P_HR models provide weather variables ~20km resolution, including temperature, relative humidity, wind, precipitation, solar radiation, cloud cover, soil moisture. covariates time-lagged short-term cumulative versions. initial version model likely use data OpenMeteo Historical Weather Data API. Additional data sources integrated subsequent versions suitability model. instance, flood cyclone data incorporated later, though initial version model. also seek data ENSO (El Niño, Neutral, La Niña) Indian Ocean sea surface temperature index. Open-source projections variables near future (months year two) likely available.","code":""},{"path":"model-description.html","id":"shedding","chapter":"4 Model description","heading":"4.3.3 Shedding","text":"rate infected individuals shed V. cholerae environment (\\(\\zeta\\)) critical factor influencing cholera transmission. Shedding rates can vary widely depending severity infection, immune response individual, environmental factors. According Fung 2014, shedding rate estimated range 0.01 10 cells per mL per person per day.studies support findings, indicating shedding rates can indeed fluctuate significantly. instance, Nelson et al (2009) note , depending phase infection, individuals can shed \\(10^3\\) (asymptomatic cases) \\(10^12\\) (severe cases) V. cholerae cells per gram stool. Future version model may attempt capture nuances shedding dynamics, make simplifying assumption shedding constant across infected individuals wide range variability prior distributional assumptions:\\[\n\\zeta \\sim \\text{Uniform}(0.01, 10).\n\\]","code":""},{"path":"model-description.html","id":"water-sanitation-and-hygiene-wash","chapter":"4 Model description","heading":"4.3.4 WAter, Sanitation, and Hygiene (WASH)","text":"Since V. cholerae transmitted fecal contamination water consumables, level exposure contaminated substrates significantly impacts transmission rates. Interventions involving Water, Sanitation, Hygiene (WASH) long first line defense reducing cholera transmission, context, WASH variables can serve proxy rate contact environmental risk factors. MOSAIC model, WASH variables incorporated mechanistically, allowing intervention scenarios include changes WASH. However, necessary distill available WASH variables single parameter represents WASH-determined contact rate contaminated substrates location \\(j\\), define \\(\\theta_j\\).parameterize \\(\\theta_j\\), calculated weighted mean 8 WASH variables Sikder et al 2023 originally modeled Local Burden Disease WaSH Collaborators 2020. 8 WASH variables (listed Table 4.3) provide population-weighted measures proportion population either: ) access WASH resources (e.g., piped water, septic sewer sanitation), ii) exposed risk factors (e.g. surface water, open defecation). risk associated WASH variables, used complement (\\(1-\\text{value}\\)) give proportion population exposed risk factor. used optim function R L-BFGS-B algorithm estimate set optimal weights (Table 4.3) maximize correlation weighted mean 8 WASH variables reported cholera incidence per 1000 population across 40 SSA countries 2000 2016. optimal weighted mean correlation coefficient \\(r =\\) -0.33 (-0.51 -0.09 95% CI) higher basic mean correlations provided individual WASH variables (see Figure 4.8). weighted mean provides single variable 0 1 represents overall proportion population access WASH /exposed environmental risk factors. Thus, WASH-mediated contact rate sources environmental transmission represented (\\(1-\\theta_j\\)) environment--human force infection (\\(\\Psi_{jt}\\)). Values \\(\\theta_j\\) countries shown Figure 4.9.\nTable 4.3: Table 4.4: Table optimized weights used calculate single mean WASH index countries.\n\nFigure 4.8: Relationship WASH variables cholera incidences.\n\nFigure 4.9: optimized weighted mean WASH variables AFRO countries\n","code":""},{"path":"model-description.html","id":"immune-dynamics","chapter":"4 Model description","heading":"4.4 Immune dynamics","text":"","code":""},{"path":"model-description.html","id":"immunity-from-vaccination","chapter":"4 Model description","heading":"4.4.1 Immunity from vaccination","text":"impacts Oral Cholera Vaccine (OCV) campaigns incorporated model Vaccinated compartment (V). rate individuals effectively vaccinated defined \\(\\phi\\nu_tS_{jt}\\), \\(S_{jt}\\) available number susceptible individuals location \\(j\\) time \\(t\\), \\(\\nu_t\\) number OCV doses administered time \\(t\\) \\(\\phi\\) estimated vaccine effectiveness. Note just one vaccinated compartment time, though future model versions may include \\(V_1\\) \\(V_2\\) compartments explore two dose vaccination strategies emulate complex waning patterns.vaccination rate \\(\\nu_t\\) estimated quantity. Rather, directly defined reported number OCV doses administered OCV dashboard : https://www..int/groups/icg/cholera.\\[\n\\nu_t := \\text{Reported rate OCV administration} \n\\]evidence waning immunity comes 4 cohort studies (Table 4.5) Bangladesh (Qadri et al 2016 2018), South Sudan (Azman et al 2016), Democratic Republic Congo (Malembaka et al 2024).Table 4.5: Summary Effectiveness DataWe estimated vaccine effectiveness waning immunity fitting exponential decay model reported effectiveness one dose OCV studies using following formulation:\\[\\begin{equation}\n\\text{Proportion immune}\\ t \\ \\text{days vaccination} = \\phi \\times (1 - \\omega) ^ {t-t_{\\text{vaccination}}}\n\\tag{4.8}\n\\end{equation}\\]\\(\\phi\\) effectiveness one dose OCV, based specification, also initial proportion immune directly vaccination. decay rate parameter \\(\\omega\\) rate initial vaccine derived immunity decays per day post vaccination, \\(t\\) \\(t_{\\text{vaccination}}\\) time (days) function evaluated time vaccination respectively. fitted model data cohort studies shown Table (4.5) found \\(\\omega = 0.00057\\) (\\(0-0.0019\\) 95% CI), gives mean estimate 4.8 years vaccine derived immune duration unreasonably large confidence intervals (1.4 years infinite immunity). However, point estimate 4.8 years consistent anecdotes one dose OCV effective least 3 years.wide confidence intervals likely due wide range reported estimates proportion immune short duration 7–90 days range (Azman et al 2016 Qadri et al 2016). Therefore, chose use point estimate \\(\\omega\\) incorporate uncertainty based initial proportion immune (.e. vaccine effectiveness \\(\\phi\\)) shortly vaccination. Using decay model Equation (4.8) estimated \\(\\phi\\) \\(0.64\\) (\\(0.32-0.96\\) 95% CI). fit Beta distribution quantiles \\(\\phi\\) minimizing sums squares using Nelder-Mead optimization algorithm render following distribution (shown Figure 4.10B):\\[\\begin{equation}\n\\phi \\sim \\text{Beta}(4.57, 2.41).\n\\tag{4.9}\n\\end{equation}\\]\nFigure 4.10: vaccine effectiveness\n","code":""},{"path":"model-description.html","id":"immunity-from-natural-infection","chapter":"4 Model description","heading":"4.4.2 Immunity from natural infection","text":"duration immunity natural infection likely longer lasting vaccination OCV (especially given current one dose strategy). SIR-type models, rate individuals leave Recovered compartment governed immune decay parameter \\(\\varepsilon\\). estimated durability immunity natural infection based two cohort studies fit following exponential decay model estimate rate immunity decay time:\\[\n\\text{Proportion immune}\\ t \\ \\text{days infection} = 0.99 \\times (1 - \\varepsilon) ^ {t-t_{\\text{infection}}}\n\\]\nmake necessary simplifying assumption within 0–90 days natural infection V. cholerae, individuals 95–99% immune. fit model reported data Ali et al (2011) Clemens et al (1991) (see Table 4.6).Table 4.6: Sources duration immunity fro natural infection.estimated mean immune decay \\(\\bar\\varepsilon = 3.9 \\times 10^{-4}\\) (\\(1.7 \\times 10^{-4}-1.03 \\times 10^{-3}\\) 95% CI) equivalent immune duration \\(7.21\\) years (\\(2.66-16.1\\) years 95% CI) shown Figure 4.11A. slightly longer previous modeling work estimating duration immunity ~5 years (King et al 2008). Uncertainty around \\(\\varepsilon\\) model represented Log-Normal distribution shown Figure 4.11B:\\[\n\\varepsilon \\sim \\text{Lognormal}(\\bar\\varepsilon+\\frac{\\sigma^2}{2}, 0.25)\n\\]\nFigure 4.11: duration immunity natural infection V. cholerae.\n","code":""},{"path":"model-description.html","id":"spatial-dynamics","chapter":"4 Model description","heading":"4.5 Spatial dynamics","text":"parameters model diagram Figure 4.2 \\(jt\\) subscript denote spatial structure model. country modeled independent metapopulation connected others via spatial force infection \\(\\Lambda_{jt}\\) moves contagion among metapopulations according connectivity provided parameters \\(\\tau_i\\) (probability departure) \\(\\pi_{ij}\\) (probability diffusion destination \\(j\\)). parameters estimated using departure-diffusion model fitted average weekly air traffic volume 41 countries included MOSAIC framework (Figure 4.12).\nFigure 4.12: average number air passengers per week 2017 among countries.\n\nFigure 4.13: network map showing average number air passengers per week 2017.\n","code":""},{"path":"model-description.html","id":"human-mobility-model","chapter":"4 Model description","heading":"4.5.1 Human mobility model","text":"departure-diffusion model estimates diagonal -diagonal elements mobility matrix (\\(M\\)) separately combines using conditional probability rules. model first estimates probability travel outside origin location \\(\\)—departure process—distribution travel origin location \\(\\) normalizing connectivity values across \\(j\\) destinations—diffusion process. values \\(\\pi_{ij}\\) sum unity along row, diagonal included, indicating relative quantity. say, \\(\\pi_{ij}\\) gives probability going \\(\\) \\(j\\) given travel outside origin \\(\\) occurs. Therefore, can use basic conditional probability rules define travel routes diagonal elements (trips made within origin \\(\\)) \n\\[\n\\Pr( \\neg \\text{depart}_i ) = 1 - \\tau_i\n\\]\n-diagonal elements (trips made outside origin \\(\\)) \n\\[\n\\Pr( \\text{depart}_i, \\text{diffuse}_{\\rightarrow j}) = \\Pr( \\text{diffuse}_{\\rightarrow j} \\mid \\text{depart}_i ) \\Pr(\\text{depart}_i ) = \\pi_{ij} \\tau_i.\n\\]\nexpected mean number trips route \\(\\rightarrow j\\) :\\[\\begin{equation}\nM_{ij} =\n\\begin{cases}\n\\theta N_i (1-\\tau_i) \\ & \\text{} \\ = j \\\\\n\\theta N_i \\tau_i \\pi_{ij} \\ & \\text{} \\ \\ne j.\n\\end{cases}\n\\tag{4.10}\n\\end{equation}\\], \\(\\theta\\) proportionality constant representing overall number trips per person origin population size \\(N_i\\), \\(\\tau_i\\) probability leaving origin \\(\\), \\(\\pi_{ij}\\) probability travel destination \\(j\\) given travel outside origin \\(\\) occurs.","code":""},{"path":"model-description.html","id":"estimating-the-departure-process","chapter":"4 Model description","heading":"4.5.2 Estimating the departure process","text":"probability travel outside origin estimated location \\(\\) give location-specific departure probability \\(\\tau_i\\).\n\\[\n\\tau_i \\sim \\text{Beta}(1+s, 1+r)\n\\]\nBinomial probabilities origin \\(\\tau_i\\) drawn Beta distributed prior shape (\\(s\\)) rate (\\(r\\)) parameters.\n\\[\n\\begin{aligned}\ns &\\sim \\text{Gamma}(0.01, 0.01)\\\\\nr &\\sim \\text{Gamma}(0.01, 0.01)\n\\end{aligned}\n\\]","code":""},{"path":"model-description.html","id":"estimating-the-diffusion-process","chapter":"4 Model description","heading":"4.5.3 Estimating the diffusion process","text":"use normalized formulation power law gravity model defined diffusion process, probability travelling destination \\(j\\) given travel outside origin \\(\\) (\\(\\pi_{ij}\\)) defined :\\[\\begin{equation}\n\\pi_{ij} = \\frac{\nN_j^\\omega d_{ij}^{-\\gamma}\n}{\n\\sum\\limits_{\\forall j \\ne } N_j^\\omega d_{ij}^{-\\gamma}\n}\n\\tag{4.11}\n\\end{equation}\\], \\(\\omega\\) scales attractive force \\(j\\) destination based population size \\(N_j\\). kernel function \\(d_{ij}^{-\\gamma}\\) serves penalty proportion travel \\(\\) \\(j\\) based distance. Prior distributions diffusion model parameters defined :\n\\[\n\\begin{aligned}\n\\omega &\\sim \\text{Gamma}(1, 1)\\\\\n\\gamma &\\sim \\text{Gamma}(1, 1)\n\\end{aligned}\n\\]models \\(\\tau_i\\) \\(\\pi_{ij}\\) fitted air traffic data OAG using mobility R package (Giles 2020). Estimates mobility model parameters shown Figures 4.14 4.15.\nFigure 4.14: estimated weekly probability travel outside origin location \\(\\tau_i\\) 95% confidence intervals shown panel population mean indicated red dashed line. Panel B shows estimated total number travelers leaving origin \\(\\) week.\n\nFigure 4.15: diffusion process \\(\\pi_{ij}\\) gives estimated probability travel origin \\(\\) destination \\(j\\) given travel outside origin \\(\\) occurred.\n","code":""},{"path":"model-description.html","id":"the-probability-of-spatial-transmission","chapter":"4 Model description","heading":"4.5.4 The probability of spatial transmission","text":"likelihood introductions cholera disparate locations major concern cholera outbreaks. However, can difficult characterize given endemic dynamics patterns human movement. include measures spatial heterogeneity first simple importation probability based connectivity possibility incoming infections. basic probability transmission origin \\(\\) particular destination \\(j\\) time \\(t\\) defined :\\[\\begin{equation}\np(,j,t) = 1 - e^{-\\beta_{jt}^{\\text{hum}} (((1-\\tau_j)S_{jt})/N_{jt}) \\pi_{ij}\\tau_iI_{}}\n\\tag{4.12}\n\\end{equation}\\]","code":""},{"path":"model-description.html","id":"the-spatial-hazard","chapter":"4 Model description","heading":"4.5.5 The spatial hazard","text":"Although concerned endemic dynamics , likely periods time early rainy season cholera cases rate transmission low enough spatial spread resemble epidemic dynamics time. times periods, can estimate arrival time contagion location cases yet reported. estimating spatial hazard transmission:\\[\\begin{equation}\nh(j,t) = \\frac{\n\\beta_{jt}^{\\text{hum}} \\Big(1 - \\exp\\big(-((1-\\tau_j)S_{jt}/N_{jt}) \\sum_{\\forall \\= j} \\pi_{ij}\\tau_i (I_{}/N_{}) \\big) \\Big)\n}{\n1/\\big(1 + \\beta_{jt}^{\\text{hum}} (1-\\tau_j)S_{jt}\\big)\n}.\n\\tag{4.13}\n\\end{equation}\\]normalizing give waiting time distribution locations:\\[\\begin{equation}\nw(j,t) = h(j,T) \\prod_{t=1}^{T-1}1-h(j,t).\n\\tag{4.14}\n\\end{equation}\\]","code":""},{"path":"model-description.html","id":"coupling-among-locations","chapter":"4 Model description","heading":"4.5.6 Coupling among locations","text":"Another measure spatial heterogeneity quantify coupling disease dynamics among metapopulations using correlation coefficient. , use definition spatial correlation locations \\(\\) \\(j\\) \\(C_{ij}\\) described Keeling Rohani (2002), gives measure similar infection dynamics locations.\\[\\begin{equation}\nC_{ij} = \\frac{\n( y_{} - \\bar{y}_i )( y_{jt} - \\bar{y}_j )\n}{\n\\sqrt{\\text{var}(y_i) \\text{var}(y_j)}\n}\n\\tag{4.15}\n\\end{equation}\\]\n\\(y_{} = I_{}/N_i\\) \\(y_{jt} = I_{jt}/N_j\\). Mean prevalence location \\(\\bar{y_i} = \\frac{1}{T} \\sum_{t=1}^{T} y_{}\\) \\(\\bar{y_j} = \\frac{1}{T} \\sum_{t=1}^{T} y_{jt}\\).","code":""},{"path":"model-description.html","id":"the-observation-process","chapter":"4 Model description","heading":"4.6 The observation process","text":"","code":""},{"path":"model-description.html","id":"rate-of-symptomatic-infection","chapter":"4 Model description","heading":"4.6.1 Rate of symptomatic infection","text":"presentation infection V. cholerae can extremely variable. severity infection depends many factors amount infectious dose, age host, level immunity host either vaccination previous infection, naivety particular strain V. cholerae. Additional circumstantial factors nutritional status overall pathogen burden may also impact infection severity. population level, observed proportion infections symptomatic also dependent endemicity cholera region. Highly endemic areas (e.g. parts Bangladesh; Hegde et al 2024) may low proportion symptomatic infections due many previous exposures. Inversely, populations largely naive V. cholerae exhibit relatively higher proportion symptomatic infections (e.g. Haiti; Finger et al 2024).Accounting nuances first version model possible, can past studies contain information can help set sensible bounds definition proportion infections symptomatic (\\(\\sigma\\)). compiled short list studies done sero-surveys cohort studies assess likelihood symptomatic infections different locations displayed results Table (4.7).provide reasonably informed prior proportion infections symptomatic, calculated combine mean confidence intervals studies Table 4.7 fit Beta distribution corresponds quantiles using least-squares Nelder-Mead algorithm. resulting prior distribution symptomatic proportion \\(\\sigma\\) :\\[\\begin{equation}\n\\sigma \\sim \\text{Beta}(4.30, 13.51)\n\\end{equation}\\]Table 4.7: Summary Studies Cholera ImmunityThe prior distribution \\(\\sigma\\) plotted Figure 4.16A reported values proportion symptomatic previous studies shown 4.16B.\nFigure 4.16: Proportion infections symptomatic.\n","code":""},{"path":"model-description.html","id":"suspected-cases","chapter":"4 Model description","heading":"4.6.2 Suspected cases","text":"clinical presentation diarrheal diseases often similar across various pathogens, can lead systematic biases reported number cholera cases. anticipated number suspected cholera cases related actual number infections factor \\(1/\\rho\\), \\(\\rho\\) represents proportion suspected cases true infections. adjust bias, use estimates meta-analysis Weins et al. (2023), suggests suspected cholera cases outnumber true infections approximately 2 1, mean across studies indicating 52% (24-80% 95% CI) suspected cases actual cholera infections. higher estimate reported ourbreak settings (78%, 40-99% 95% CI). account variability estimate, fit Beta distribution reported quantiles using least squares approach Nelder-Mead algorithm, resulting prior distribution shown Figure 4.17B:\\[\\begin{equation}\n\\rho \\sim \\text{Beta}(4.79, 1.53).\n\\end{equation}\\]\nFigure 4.17: Proportion suspected cholera cases true infections. Panel shows ‘low’ assumption estimates across settings: \\(\\rho \\sim \\text{Beta}(5.43, 5.01)\\). Panel B shows ‘high’ assumption estimate reflects high-quality studies outbreaks: \\(\\rho \\sim \\text{Beta}(4.79, 1.53)\\)\n","code":""},{"path":"model-description.html","id":"case-fatality-rate","chapter":"4 Model description","heading":"4.6.3 Case fatality rate","text":"Case Fatality Rate (CFR) among symptomatic infections calculated using reported cases deaths data January 2021 August 2024. data collated various issues Weekly Epidemiological Record Global Cholera Acute Watery Diarrhea (AWD) Dashboard (see Data section) provide annual aggregations reported cholera cases deaths. used Binomial exact test (binom.test R) calculate mean probability number deaths (successes) given number reported cases (sample size), Clopper-Pearson method calculating binomial confidence intervals. fit Beta distributions mean CFR 95% confidence intervals calculated country using least squares Nelder-Mead algorithm give distributional uncertainty around CFR estimate country (\\(\\mu_j\\)).\\[\n\\mu_j \\sim \\text{Beta}(s_{1,j}, s_{2,j})\n\\]\\(s_{1,}\\) \\(s_{2,j}\\) two positive shape parameters Beta distribution estimated destination \\(j\\). definition \\(\\mu_j\\) CFR reported cases subset total number infections. Therefore, infer total number deaths attributable cholera infection, assume CFR observed cases proportionally equivalent CFR cases calculate total deaths \\(D\\) follows:\\[\\begin{equation}\n\\begin{aligned}\n\\text{CFR}_{\\text{observed}} &= \\text{CFR}_{\\text{total}}\\\\\n\\\\[3pt]\n\\frac{[\\text{observed deaths}]}{[\\text{observed cases}]} &=\n\\frac{[\\text{total deaths}]}{[\\text{infections}]}\\\\\n\\\\[3pt]\n\\text{total deaths} &= \\frac{[\\text{observed deaths}] \\times [\\text{true infections}]}{[\\text{observed cases}]}\\\\\n\\\\[3pt]\nD_{jt} &= \\frac{ [\\sigma\\rho\\mu_j I_{jt}] \\times [I_{jt}] }{ [\\sigma\\rho I_{jt}] }\n\\end{aligned}\n\\end{equation}\\]\nTable 4.8: Table 4.9: CFR Values Beta Shape Parameters AFRO Countries\n\nFigure 4.18: Case Fatality Rate (CFR) Total Cases Country AFRO Region 2014 2024. Panel : Case Fatality Ratio (CFR) 95% confidence intervals. Panel B: total number cholera cases. AFRO Region highlighted black, countries less 3/0.2 = 150 total reported cases assigned mean CFR AFRO.\n\nFigure 4.19: Beta distributions overall Case Fatality Rate (CFR) 2014 2024. Examples show overall CFR AFRO region (2%) black, Congo highest CFR (7%) red, South Sudan lowest CFR (0.1%) blue.\n","code":""},{"path":"model-description.html","id":"demographics-1","chapter":"4 Model description","heading":"4.7 Demographics","text":"model includes basic demographic change using reported birth death rates \\(j\\) countries, \\(b_j\\) \\(d_j\\) respectively. rates static defined United Nations Department Economic Social Affairs Population Division World Population Prospects 2024. Values \\(b_j\\) \\(d_j\\) derived crude rates converted birth rate per day death rate per day (shown Table 4.10).\nTable 4.10: Table 4.11: Demographic AFRO countries 2023. Data include: total population January 1, 2023, daily birth rate, daily death rate. Values calculate crude birth death rates UN World Population Prospects 2024.\n","code":""},{"path":"model-description.html","id":"the-reproductive-number","chapter":"4 Model description","heading":"4.8 The reproductive number","text":"reproductive number common metric epidemic growth represents average number secondary cases generated primary case specific time epidemic. track \\(R\\) changes time estimating instantaneous reproductive number \\(R_t\\) described Cori et al 2013. track \\(R_t\\) across metapopulations model give \\(R_{jt}\\) using following formula:\\[\\begin{equation}\nR_{jt} = \\frac{I_{jt}}{\\sum_{\\Delta t=1}^{t} g(\\Delta t) I_{j,t-\\Delta t}}\n\\tag{4.16}\n\\end{equation}\\]\\(I_{jt}\\) number new infections destination \\(j\\) time \\(t\\), \n\\(g(\\Delta t)\\) represents probability value generation time distribution cholera. accomplished using weighed sum denominator highly influenced generation time distribution.","code":""},{"path":"model-description.html","id":"the-generation-time-distribution","chapter":"4 Model description","heading":"4.8.1 The generation time distribution","text":"generation time distribution gives time individual infected infect subsequent individuals. parameterized quantity using Gamma distribution mean 5 days:\\[\\begin{equation}\ng(\\cdot) \\sim \\text{Gamma}(0.5, 0.1).\n\\tag{4.17}\n\\end{equation}\\], shape=0.5, rate=0.1, mean given shape/rate. Previous studies use mean 5 days (Kahn et al 2020 Azman 2016), however mean 3, 5, 7, 10 days may admissible (Azman 2012).\nFigure 4.20: generation time\n\nTable 4.12: Table 4.13: Generation Time Weeks\n","code":""},{"path":"model-description.html","id":"initial-conditions","chapter":"4 Model description","heading":"4.9 Initial conditions","text":"Since first version model begin Jan 2023 (take advantage available weekly data), initial conditions surrounding population immunity must estimated. set initial conditions, use historical data find total number reported cases location previous X years, multiply \\(1/\\sigma\\) estimate total infections symptomatic cases reported, adjust based waning immunity. also sum total number vaccinations past X years adjust vaccine efficacy \\(\\phi\\) waning immunity vaccination \\(\\omega\\).total number infected? reported cases… back symptomatic asymptomatictotal number infected? reported cases… back symptomatic asymptomaticTotal number immune due natural infections past X yearsTotal number immune due natural infections past X yearstotal number immune due past vaccinations X yearstotal number immune due past vaccinations X yearsUse deconvolution based immune decay estimated vaccine section","code":""},{"path":"model-description.html","id":"model-calibration","chapter":"4 Model description","heading":"4.10 Model calibration","text":"model calibrated using Latin hypercube sampling hyper-parameters model likelihoods fit incidence deaths.model calibrated using Latin hypercube sampling hyper-parameters model likelihoods fit incidence deaths.important challenge flexibly fitting data often missing available aggregated forms.important challenge flexibly fitting data often missing available aggregated forms.[Fig: different spatial temporal scales available data]","code":""},{"path":"model-description.html","id":"caveats","chapter":"4 Model description","heading":"4.11 Caveats","text":"Simplest model start. Easier initial spatial structure minimum additional compartments calibrate available data (vaccination, cases, deaths).Country level aggregations. First generation data 2023/24…Assumes vaccinating susceptible individuals.climate, summarizing whole country.","code":""},{"path":"model-description.html","id":"table-of-parameters","chapter":"4 Model description","heading":"4.12 Table of parameters","text":"Table 4.14: Descriptions model parameters along prior distributions sources applicable.","code":""},{"path":"model-description.html","id":"references","chapter":"4 Model description","heading":"4.13 References","text":"","code":""},{"path":"scenarios.html","id":"scenarios","chapter":"5 Scenarios","heading":"5 Scenarios","text":"key aim MOSAIC model provide near-term forecasts cholera transmission Sub-Saharan Africa (SSA) using current data available. However, MOSAIC just forecasting tool; dynamic model designed explore various scenarios influence critical factors vaccination, environmental conditions, Water, Sanitation, Hygiene (WASH) interventions.","code":""},{"path":"scenarios.html","id":"vaccination","chapter":"5 Scenarios","heading":"5.1 Vaccination","text":"","code":""},{"path":"scenarios.html","id":"spatial-and-temporal-strategies","chapter":"5 Scenarios","heading":"5.1.1 Spatial and Temporal Strategies","text":"Understanding spatial temporal distribution cholera vaccination efforts crucial effective outbreak control. Key resources include:Stockpile Status: availability oral cholera vaccine emergency stockpiles can tracked UNICEF’s Emergency Stockpile Availability.OCV Dashboard: dashboard (link) provides insights deployment oral cholera vaccines (OCV) across different regions.","code":""},{"path":"scenarios.html","id":"reactive-vaccination","chapter":"5 Scenarios","heading":"5.1.2 Reactive Vaccination","text":"timing logistics reactive vaccination campaigns critical controlling ongoing outbreaks. Relevant resources include:Recommended Timing: Guidelines recommendations timing reactive OCV campaigns available (link).Requests Delay Time Distributions: Information vaccine request processes distribution delays vaccine deployment can accessed GTFCC OCV Dashboard (link).","code":""},{"path":"scenarios.html","id":"impacts-of-climate-change","chapter":"5 Scenarios","heading":"5.2 Impacts of Climate Change","text":"","code":""},{"path":"scenarios.html","id":"severe-weather-events","chapter":"5 Scenarios","heading":"5.2.1 Severe Weather Events","text":"Projections climate shocks, including frequency severity cyclones floods, essential modeling future impacts climate change cholera transmission. Key references include:Chen Chavas 2020: study cyclone season dynamics climate change scenarios (link).Sparks Toumi 2024: Research projected flood frequencies due climate change (link).Switzer et al. 2023: analysis climate shock impacts cholera outbreaks (link).","code":""},{"path":"scenarios.html","id":"long-term-trends","chapter":"5 Scenarios","heading":"5.2.2 Long-Term Trends","text":"Long-term trends weather variables various climate change scenarios can explored using following resource:Weather Variables Climate Change: OpenMeteo Climate API provides access projected weather data different climate change scenarios (link).","code":""},{"path":"usage.html","id":"usage","chapter":"6 Usage","heading":"6 Usage","text":"open-source code used run MOSAIC currently development presented future.","code":""},{"path":"news.html","id":"news","chapter":"7 News","heading":"7 News","text":"","code":""},{"path":"news.html","id":"past-versions-of-mosaic","chapter":"7 News","heading":"7.1 Past versions of MOSAIC","text":"Table 7.1: Current future planned model versions brief descriptions.","code":""},{"path":"references-1.html","id":"references-1","chapter":"8 References","heading":"8 References","text":"","code":""}] +[{"path":"index.html","id":"section","chapter":"","heading":"","text":"","code":""},{"path":"index.html","id":"welcome","chapter":"","heading":"Welcome","text":"Welcome Metapopulation Outbreak Simulation Agent-based Implementation Cholera (MOSAIC). MOSAIC framework simulates transmission dynamics cholera Sub-Saharan Africa (SSA) provides tools understand impact interventions, vaccination, well large-scale drivers like climate change. MOSAIC built using Light-agent Spatial Model ERadication (LASER) platform, site serves documentation model’s methods associated analyses. Please note MOSAIC currently development, content may change regularly. sharing increase visibility welcome feedback aspect model.","code":""},{"path":"index.html","id":"contact","chapter":"","heading":"Contact","text":"MOSAIC developed team researchers Institute Disease Modeling (IDM) dedicated developing modeling methods software tools help decision-makers understand respond infectious disease outbreaks. website currently maintained John Giles (@gilesjohnr). general questions, contact John Giles (john.giles@gatesfoundation.org), Jillian Gauld (jillian.gauld@gatesfoundation.org), /Rajiv Sodhi (rajiv.sodhi@gatesfoundation.org).","code":""},{"path":"index.html","id":"funding","chapter":"","heading":"Funding","text":"work developed Institute Disease Modeling support funded research grants made Bill & Melinda Gates Foundation.","code":""},{"path":"introduction.html","id":"introduction","chapter":"1 Introduction","heading":"1 Introduction","text":"","code":""},{"path":"introduction.html","id":"history","chapter":"1 Introduction","heading":"1.1 History","text":"Vibrio cholerae introduced African continent Asia 1970s since become endemic many countries.","code":""},{"path":"introduction.html","id":"recent-surge","chapter":"1 Introduction","heading":"1.2 Recent Surge","text":"Although sporadic cholera outbreaks past five decades, significant surge cases since 2021. increase likely due factors climate change disruptions municipal services COVID-19 pandemic.","code":""},{"path":"introduction.html","id":"gtfcc-goals","chapter":"1 Introduction","heading":"1.3 GTFCC Goals","text":"Global Task Force Cholera Control (GTFCC) aims reduce cholera deaths 90% 2030.","code":""},{"path":"introduction.html","id":"ocv-stockpiles","chapter":"1 Introduction","heading":"1.4 OCV Stockpiles","text":"major concern recent surge cases depletion oral cholera vaccine (OCV) stockpiles. response, officials shifted single-dose OCV strategies. Efforts underway replenish stockpiles, key question best allocate reduce transmission regionally support GTFCC’s goal.","code":""},{"path":"introduction.html","id":"climate-change","chapter":"1 Introduction","heading":"1.5 Climate Change","text":"Environmental factors play significant role cholera outbreaks, warmer wetter conditions creating favorable environment Vibrio cholerae. conditions likely exacerbate already challenging endemic outbreak settings. Models incorporate climatic forcing can provide insights future cholera dynamics due climate change aid achieving GTFCC goal.","code":""},{"path":"rationale.html","id":"rationale","chapter":"2 Rationale","heading":"2 Rationale","text":"significant challenge controlling cholera transmission Sub-Saharan Africa (SSA) lack comprehensive datasets dynamic models designed support ongoing policy-making. persistent endemic nature cholera SSA presents complex quantitative challenge, requiring sophisticated models produce meaningful inferences. Models incorporate necessary natural history disease dynamics, operate adequate spatial temporal scales, crucial providing timely actionable information address ongoing future cholera outbreaks.Although developing data models scales challenging, goal iteratively create landscape-scale transmission model cholera SSA can provide weekly predictions key epidemiological metrics. modeling methods leverage wide array --date data sources, including incidence mortality reports, patterns human movement, vaccination history schedules, environmental factors.Key questions address include administer limited supply oral cholera vaccine (OCV) severe weather events climate change impact future outbreaks. landscape-scale model accounts endemic transmission patterns valuable tool addressing questions.","code":""},{"path":"data.html","id":"data","chapter":"3 Data","heading":"3 Data","text":"MOSAIC model requires diverse set data sources, directly used define model parameters (e.g., birth death rates), others help fit models priori provide informative priors transmission model. additional data sources become available, future versions model adapt incorporate . now, following data sources represent minimum requirements initiate viable first model.","code":""},{"path":"data.html","id":"historical-incidence-and-deaths","chapter":"3 Data","heading":"3.1 Historical Incidence and Deaths","text":"Data historical cholera incidence deaths crucial establishing baseline transmission patterns. compiled annual total reported cases deaths AFRO region countries January 1970 August 2024. data comes several sources include:World Data (1970-2021): Number Reported Cases Cholera (1949-2021) Number Reported Deaths Cholera (1949-2021). World Data group compiled data previously published annual reports.Annual Report 2022: data manually extracted World Health Organization’s Weekly Epidemiological Record 38, 2023, 98, 431–452.Global Cholera Acute Watery Diarrhea Dashboard (2023-2024): Unofficial tallies reported cases deaths 2023 part 2024 available Global Cholera AWD Dashboard.","code":""},{"path":"data.html","id":"recent-incidence-and-deaths","chapter":"3 Data","heading":"3.2 Recent Incidence and Deaths","text":"capture recent cholera trends, retrieved reported cases deaths data Global Cholera Acute Watery Diarrhea Dashboard REST API. data provide weekly incidence deaths January 2023 August 2024 provides --date counts country level.","code":""},{"path":"data.html","id":"vaccinations","chapter":"3 Data","heading":"3.3 Vaccinations","text":"Accurate data oral cholera vaccine (OCV) campaigns vaccination history vital understanding impact vaccination efforts. data come :Cholera Vaccine Dashboard: resource (link) provides detailed information OCV distribution vaccination campaigns 2016 2024.GTFCC OCV Dashboard: Managed Médecins Sans Frontières, dashboard (link) tracks OCV deployments globally, offering granular insights vaccination efforts 2013 2024.","code":""},{"path":"data.html","id":"human-mobility-data","chapter":"3 Data","heading":"3.4 Human Mobility Data","text":"Human mobility patterns significantly influence cholera transmission. Relevant data include:OAG Passenger Booking Data: dataset (link) offers insights air passenger movements, can used model spread cholera across regions.Namibia Call Data Records: additional source Giles et al. (2020) (link) provides detailed mobility data based mobile phone records, useful localized modeling.","code":""},{"path":"data.html","id":"climate-data","chapter":"3 Data","heading":"3.5 Climate Data","text":"Climate conditions, including temperature, precipitation, extreme weather events, play critical role cholera dynamics. captured :OpenMeteo Historical Weather Data API: API (link) offers access historical climate data, essential modeling environmental factors influencing cholera outbreaks.","code":""},{"path":"data.html","id":"storms-and-floods","chapter":"3 Data","heading":"3.5.1 Storms and Floods","text":"Data extreme weather events, specifically storms floods, obtained :EM-DAT International Disaster Database: Maintained Centre Research Epidemiology Disasters (CRED) UCLouvain, database (link) provides comprehensive records disasters 2000 present, including affecting African countries.","code":""},{"path":"data.html","id":"wash-water-sanitation-and-hygiene","chapter":"3 Data","heading":"3.6 WASH (Water, Sanitation, and Hygiene)","text":"Data water, sanitation, hygiene (WASH) critical understanding environmental infrastructural factors influence cholera transmission. data sourced :UNICEF Joint Monitoring Program (JMP) Database: resource (link) offers detailed information household-level access clean water sanitation, integral cholera prevention efforts.","code":""},{"path":"data.html","id":"demographics","chapter":"3 Data","heading":"3.7 Demographics","text":"Demographic data, including population size, birth rates, death rates, foundational accurate disease modeling. data sourced :UN World Population Prospects 2024: database (link) provides probabilistic projections key demographic metrics, essential estimating population-level impacts cholera.","code":""},{"path":"model-description.html","id":"model-description","chapter":"4 Model description","heading":"4 Model description","text":"describe methods MOSAIC beta version 0.1. model version provides starting point understanding cholera transmission Sub-Saharan Africa, incorporating important drivers disease dynamics human mobility, environmental conditions, vaccination schedules. MOSAIC continues evolve, future iterations refine model components based available data improved model mechanisms, hope increase applicability real-world scenarios.model operates weekly time steps January 2023 August 2024 includes 46 countries Sub-Saharan Africa (SSA) shown Figure 4.1.\nFigure 4.1: map Sub-Saharan Africa countries experienced cholera outbreak past 5 10 years highlighted green.\n","code":""},{"path":"model-description.html","id":"transmission-dynamics","chapter":"4 Model description","heading":"4.1 Transmission dynamics","text":"model metapopulation structure familiar compartments Susceptible, Infected, Recovered individuals SIRS dynamics. model also contains compartments vaccinated individuals (V) Water & environment based transmission (W) refer SVIWRS.\nFigure 4.2: diagram SVIWRS (Susceptible-Vaccinated-Infected-Water/environmental-Recovered-Susceptible) model shows model compartments circles rate parameters displayed. primary data sources model fit shown square nodes (Vaccination data, reported cases deaths).\nSVIWRS metapopulation model, shown Figure 4.2, governed following difference equations:\\[\\begin{equation}\n\\begin{aligned}\nS_{j,t+1} &= b_j N_{jt} + S_{jt} - \\phi \\nu_{jt} S_{jt} + \\omega V_{jt} - \\Lambda_{j,t+1} - \\Psi_{j,t+1} + \\varepsilon R_{jt} - d_j S_{jt}\\\\[11pt]\nV_{j,t+1} &= V_{jt} + \\phi \\nu_{jt} S_{jt} - \\omega V_{jt} - d_j V_{jt}\\\\[11pt]\nI_{j,t+1} &= I_{jt} + \\Lambda_{j,t+1} + \\Psi_{j,t+1} - \\gamma I_{jt} - \\mu \\sigma I_{jt} - d_j I_{jt}\\\\[11pt]\nW_{j,t+1} &= W_{jt} + \\zeta I_{jt} - \\delta_{jt} W_{jt}\\\\[11pt]\nR_{j,t+1} &= R_{jt} + \\gamma I_{jt} - \\varepsilon R_{jt} - d_j R_{jt}\\\\[11pt]\n\\end{aligned}\n\\tag{4.1}\n\\end{equation}\\]descriptions parameters Equation (4.1), see Table (4.14). Transmission dynamics driven two force infection terms, \\(\\Lambda_{jt}\\) \\(\\Psi_{jt}\\). force infection due human--human (\\(\\Lambda_{jt}\\)) :\\[\\begin{equation}\n\\begin{aligned}\n\\Lambda_{j,t+1} &= \\frac{\n\\beta_{jt}^{\\text{hum}} \\Big(\\big(S_{jt}(1-\\tau_{j})\\big) \\big(I_{jt}(1-\\tau_{j}) + \\sum_{\\forall \\= j} (\\pi_{ij}\\tau_iI_{}) \\big)\\Big)^\\alpha}{N_{jt}}.\\\\[11pt]\n\\end{aligned}\n\\tag{4.2}\n\\end{equation}\\]\\(\\beta_{jt}^{\\text{hum}}\\) rate human--human transmission. Movement within among metapopulations governed \\(\\tau_i\\), probability departing origin location \\(\\), \\(\\pi_{ij}\\) relative probability travel origin \\(\\) destination \\(j\\) (see section spatial dynamics). include environmental effects, force infection due environment--human transmission (\\(\\Psi_{jt}\\)) defined :\\[\\begin{equation}\n\\begin{aligned}\n\\Psi_{j,t+1} &= \\frac{\\beta_{jt}^{\\text{env}} \\big(S_{jt}(1-\\tau_{j})\\big) (1-\\theta_j) W_{jt}}{\\kappa+W_{jt}},\\\\[11pt]\n\\end{aligned}\n\\tag{4.3}\n\\end{equation}\\]\\(\\beta_{jt}^{\\text{env}}\\) rate environment--human transmission \\(\\theta_j\\) proportion population location \\(j\\) least basic access Water, Sanitation, Hygiene (WASH). environmental compartment model also scaled concentration (cells per mL) V. cholerae required 50% probability infection Fung 2014. See section environmental transmission water/environment compartment climatic drivers transmission.Note model processes stochastic. Transition rates converted probabilities commonly used formula \\(p(t) = 1-e^{-rt}\\) (see Ross 2007), integer quantities moved model compartments time step according binomial process like example recovery infected individuals (\\(\\gamma I_{jt}\\)):\\[\\begin{equation}\n\\frac{\\partial R}{\\partial t} \\sim \\text{Binom}(I_{jt}, 1-\\exp(-\\gamma))\n\\tag{4.4}\n\\end{equation}\\]","code":""},{"path":"model-description.html","id":"seasonality","chapter":"4 Model description","heading":"4.2 Seasonality","text":"Cholera transmission seasonal typically associated rainy season, transmission rate terms \\(\\beta_{jt}^{\\text{*}}\\) temporally forced. human--human transmission used truncated sine-cosine form Fourier series two harmonic features flexibility capture seasonal transmission dynamics driven extended rainy seasons /biannual trends:\\[\\begin{equation}\n\\beta_{jt}^{\\text{hum}} = \\beta_{j0}^{\\text{hum}} + a_1 \\cos\\left(\\frac{2\\pi t}{p}\\right) + b_1 \\sin\\left(\\frac{2\\pi t}{p}\\right) + a_2 \\cos\\left(\\frac{4\\pi t}{p}\\right) + b_2 \\sin\\left(\\frac{4\\pi t}{p}\\right)\n\\tag{4.5}\n\\end{equation}\\], \\(\\beta_{j0}^{\\text{hum}}\\) mean human--human transmission rate location \\(j\\) time steps. Seasonal dynamics determined parameters \\(a_1\\), \\(b_1\\) \\(a_2\\), \\(b_2\\) gives amplitude first second waves respectively. periodic cycle \\(p\\) 52, function controls temporal variation \\(\\beta_{jt}^{\\text{hum}}\\) 52 weeks year.\nFigure 4.3: example temporal distribution human--human transmission rate across 52 weeks year given cosine wave function. wave function fitted country designed align rainy season indicated shaded region figure.\nestimated parameters Fourier series (\\(a_1\\), \\(b_1\\), \\(a_2\\), \\(b_2\\)) using Levenberg–Marquardt algorithm minpack.lm R library. Given lack reported cholera case data many countries SSA association cholera transmission rainy season, leveraged seasonal precipitation data help fit Fourier wave function countries. first gathered weekly precipitation values 1994 2024 30 uniformly distributed points within country Open-Meteo Historical Weather Data API. fit Fourier series weekly precipitation data used parameters starting values fitting model sparse cholera case data.\nFigure 4.4: Example grid 30 uniformly distributed points within Mozambique (). scatterplot shows weekly summed precipitation values 30 grid points cholera cases plotted scale Z-Score shows variance around mean terms standard deviation. Fitted Fourier series fucntions shown blue (fit precipitation data) red (fit cholera case data) lines.\ncountries reported case data, inferred seasonal dynamics using fitted wave function neighboring country available case data. selected neighbor chosen cluster countries (grouped hierarchically four clusters based precipitation seasonality using Ward’s method; see Figure 4.5) highest correlation seasonal precipitation country lacking case data. rare event country reported case data found within seasonal cluster, expanded search 10 nearest neighbors continued expanding adding next nearest neighbor match found.\nFigure 4.5: ) Map showing clustering African countries based seasonal precipitation patterns (1994-2024). Countries colored according cluster assignments, identified using hierarchical clustering. B) Fourier series fitted weekly precipitation country. line plot shows seasonal pattern countries within given cluster. Clusteres used infer seasonal transmission dynamics countries reported cholera cases.\nUsing model fitting methods described , cluster-based approach inferring seasonal Fourier series pattern countries without reported cholera cases, modeled seasonal dynamics 41 countries MOSAIC framework. dynamics visualized Figure 4.6, corresponding Fourier model coefficients presented Table 4.1.\nFigure 4.6: Seasonal transmission patterns countries modeled MOSAIC modeled truncated Fourier series Equation (4.5). Blues lines give Fourier series model fits precipitation (1994-2024) red lines give models fits reported cholera cases (2023-2024). countries reported case data available, Fourier model inferred nearest country similar seasonal precipitation patterns determined hierarchical clustering. Countries inferred case data neighboring locations annotated red. X-axis represents weeks year (1-52), Y-axis shows Z-score weekly precipitation cholera cases.\n\nTable 4.1: Table 4.2: Estimated coefficients truncated Fourier model Equation (4.5) fit countries reported cholera cases. Model fits shown Figure 4.6.\n","code":""},{"path":"model-description.html","id":"environmental-transmission","chapter":"4 Model description","heading":"4.3 Environmental transmission","text":"Environmental transmission critical factor cholera spread consists several key components: rate infected individuals shed V. cholerae environment, pathogen’s survival rate environmental conditions, overall suitability environment sustaining bacteria time.","code":""},{"path":"model-description.html","id":"climate-driven-transmission","chapter":"4 Model description","heading":"4.3.1 Climate-driven transmission","text":"capture impacts climate-drivers cholera transmission, included parameter \\(\\psi_{jt}\\), represents current state environmental suitability respect : ) survival time V. cholerae environment , ii) rate environment--human transmission contributes overall force infection.\\[\\begin{equation}\n\\beta_{jt}^{\\text{env}} = \\beta_{j0}^{\\text{env}} \\Bigg(1 + \\frac{\\psi_{jt}-\\bar\\psi_j}{\\bar\\psi_j} \\Bigg) \\quad \\text{} \\quad \\bar\\psi_j = \\frac{1}{T} \\sum_{t=1}^{T} \\psi_{jt}\n\\tag{4.6}\n\\end{equation}\\]formulation effectively scales base environmental transmission rate \\(\\beta_{jt}^{\\text{env}}\\) varies time according climatically driven model suitability. Note , unlike cosine wave function \\(\\beta_{jt}^{\\text{hum}}\\), temporal term can increase decrease time following multi-annual cycles.[Fig: Example temporal forcing environment--human transmission]Environmental suitability (\\(\\psi_{jt}\\)) also impacts survival rate V. cholerae environment (\\(\\delta_{jt}\\)) form:\\[\\begin{equation}\n\\delta_{jt} = \\delta_{\\text{min}} + \\psi_{jt} \\times (\\delta_{\\text{max}} - \\delta_{\\text{min}})\n\\tag{4.7}\n\\end{equation}\\]normalizes variance suitability parameter bounded within minimum (\\(\\delta_{\\text{min}}\\)) maximum (\\(\\delta_{\\text{max}}\\)) survival times V. cholerae.\nFigure 4.7: Relationship environmental suitability (\\(\\psi_{jt}\\)) rate V. cholerae decay environment (\\(\\delta_j\\)). green line shows mildest penalty V. cholerae survival, survival environment \\(1/\\delta_{\\text{min}}\\) = 3 days suitability = 0 \\(1/\\delta_{\\text{max}}\\) = 90 days suitability = 1.\n","code":""},{"path":"model-description.html","id":"modeling-suitability","chapter":"4 Model description","heading":"4.3.2 Modeling suitability","text":"environmental suitability (\\(\\psi_{jt}\\)) V. cholerae modeled time series location, using covariates include environmental factors, past present climate measures, severe weather events, large-scale regional climate drivers. factors influenced climate change, source data projects covariate future different climate change scenarios. Environmental suitability, \\(\\psi_{jt}\\), generally defined :\\[\n\\psi_{jt} = f(\\text{temperature, precipitation, humidity, wind speed, soil moisture})\n\\]function \\(f(\\cdot)\\) can modeled using various approaches, including Generalized Linear Models (GLMs), Generalized Additive Models (GAMs), Boosted Regression Trees (BRTs), machine learning methods Recurrent Neural Networks (RNNs) Long Short-Term Memory Networks (LSTMs). simpler approach might involve Bayesian variable selection using BAS R package. model fitted available data, projections suitability location. Implementing rolling-window validation across time series help assess model performance. model can directly fitted reported case counts converted binary threshold, depending analysis needs. primary goal explain portion variance reported case counts proxy environmental suitability.Covariates include historical climate variables predicted climate change scenarios. example, MRI-AGCM3-2-S EC_Earth3P_HR models provide weather variables ~20km resolution, including temperature, relative humidity, wind, precipitation, solar radiation, cloud cover, soil moisture. covariates time-lagged short-term cumulative versions. initial version model likely use data OpenMeteo Historical Weather Data API. Additional data sources integrated subsequent versions suitability model. instance, flood cyclone data incorporated later, though initial version model. also seek data ENSO (El Niño, Neutral, La Niña) Indian Ocean sea surface temperature index. Open-source projections variables near future (months year two) likely available.","code":""},{"path":"model-description.html","id":"shedding","chapter":"4 Model description","heading":"4.3.3 Shedding","text":"rate infected individuals shed V. cholerae environment (\\(\\zeta\\)) critical factor influencing cholera transmission. Shedding rates can vary widely depending severity infection, immune response individual, environmental factors. According Fung 2014, shedding rate estimated range 0.01 10 cells per mL per person per day.studies support findings, indicating shedding rates can indeed fluctuate significantly. instance, Nelson et al (2009) note , depending phase infection, individuals can shed \\(10^3\\) (asymptomatic cases) \\(10^{12}\\) (severe cases) V. cholerae cells per gram stool. Future version model may attempt capture nuances shedding dynamics, make simplifying assumption shedding constant across infected individuals wide range variability prior distributional assumptions:\\[\n\\zeta \\sim \\text{Uniform}(0.01, 10).\n\\]","code":""},{"path":"model-description.html","id":"water-sanitation-and-hygiene-wash","chapter":"4 Model description","heading":"4.3.4 WAter, Sanitation, and Hygiene (WASH)","text":"Since V. cholerae transmitted fecal contamination water consumables, level exposure contaminated substrates significantly impacts transmission rates. Interventions involving Water, Sanitation, Hygiene (WASH) long first line defense reducing cholera transmission, context, WASH variables can serve proxy rate contact environmental risk factors. MOSAIC model, WASH variables incorporated mechanistically, allowing intervention scenarios include changes WASH. However, necessary distill available WASH variables single parameter represents WASH-determined contact rate contaminated substrates location \\(j\\), define \\(\\theta_j\\).parameterize \\(\\theta_j\\), calculated weighted mean 8 WASH variables Sikder et al 2023 originally modeled Local Burden Disease WaSH Collaborators 2020. 8 WASH variables (listed Table 4.3) provide population-weighted measures proportion population either: ) access WASH resources (e.g., piped water, septic sewer sanitation), ii) exposed risk factors (e.g. surface water, open defecation). risk associated WASH variables, used complement (\\(1-\\text{value}\\)) give proportion population exposed risk factor. used optim function R L-BFGS-B algorithm estimate set optimal weights (Table 4.3) maximize correlation weighted mean 8 WASH variables reported cholera incidence per 1000 population across 40 SSA countries 2000 2016. optimal weighted mean correlation coefficient \\(r =\\) -0.33 (-0.51 -0.09 95% CI) higher basic mean correlations provided individual WASH variables (see Figure 4.8). weighted mean provides single variable 0 1 represents overall proportion population access WASH /exposed environmental risk factors. Thus, WASH-mediated contact rate sources environmental transmission represented (\\(1-\\theta_j\\)) environment--human force infection (\\(\\Psi_{jt}\\)). Values \\(\\theta_j\\) countries shown Figure 4.9.\nTable 4.3: Table 4.4: Table optimized weights used calculate single mean WASH index countries.\n\nFigure 4.8: Relationship WASH variables cholera incidences.\n\nFigure 4.9: optimized weighted mean WASH variables AFRO countries. Countries labeled orange denote countries imputed weighted mean WASH variable. Imputed values weighted mean 3 similar countries.\n","code":""},{"path":"model-description.html","id":"immune-dynamics","chapter":"4 Model description","heading":"4.4 Immune dynamics","text":"","code":""},{"path":"model-description.html","id":"immunity-from-vaccination","chapter":"4 Model description","heading":"4.4.1 Immunity from vaccination","text":"impacts Oral Cholera Vaccine (OCV) campaigns incorporated model Vaccinated compartment (V). rate individuals effectively vaccinated defined \\(\\phi\\nu_tS_{jt}\\), \\(S_{jt}\\) available number susceptible individuals location \\(j\\) time \\(t\\), \\(\\nu_t\\) number OCV doses administered time \\(t\\) \\(\\phi\\) estimated vaccine effectiveness. Note just one vaccinated compartment time, though future model versions may include \\(V_1\\) \\(V_2\\) compartments explore two dose vaccination strategies emulate complex waning patterns.vaccination rate \\(\\nu_t\\) estimated quantity. Rather, directly defined reported number OCV doses administered OCV dashboard : https://www..int/groups/icg/cholera.\\[\n\\nu_t := \\text{Reported rate OCV administration} \n\\]evidence waning immunity comes 4 cohort studies (Table 4.5) Bangladesh (Qadri et al 2016 2018), South Sudan (Azman et al 2016), Democratic Republic Congo (Malembaka et al 2024).Table 4.5: Summary Effectiveness DataWe estimated vaccine effectiveness waning immunity fitting exponential decay model reported effectiveness one dose OCV studies using following formulation:\\[\\begin{equation}\n\\text{Proportion immune}\\ t \\ \\text{days vaccination} = \\phi \\times (1 - \\omega) ^ {t-t_{\\text{vaccination}}}\n\\tag{4.8}\n\\end{equation}\\]\\(\\phi\\) effectiveness one dose OCV, based specification, also initial proportion immune directly vaccination. decay rate parameter \\(\\omega\\) rate initial vaccine derived immunity decays per day post vaccination, \\(t\\) \\(t_{\\text{vaccination}}\\) time (days) function evaluated time vaccination respectively. fitted model data cohort studies shown Table (4.5) found \\(\\omega = 0.00057\\) (\\(0-0.0019\\) 95% CI), gives mean estimate 4.8 years vaccine derived immune duration unreasonably large confidence intervals (1.4 years infinite immunity). However, point estimate 4.8 years consistent anecdotes one dose OCV effective least 3 years.wide confidence intervals likely due wide range reported estimates proportion immune short duration 7–90 days range (Azman et al 2016 Qadri et al 2016). Therefore, chose use point estimate \\(\\omega\\) incorporate uncertainty based initial proportion immune (.e. vaccine effectiveness \\(\\phi\\)) shortly vaccination. Using decay model Equation (4.8) estimated \\(\\phi\\) \\(0.64\\) (\\(0.32-0.96\\) 95% CI). fit Beta distribution quantiles \\(\\phi\\) minimizing sums squares using Nelder-Mead optimization algorithm render following distribution (shown Figure 4.10B):\\[\\begin{equation}\n\\phi \\sim \\text{Beta}(4.57, 2.41).\n\\tag{4.9}\n\\end{equation}\\]\nFigure 4.10: vaccine effectiveness\n","code":""},{"path":"model-description.html","id":"immunity-from-natural-infection","chapter":"4 Model description","heading":"4.4.2 Immunity from natural infection","text":"duration immunity natural infection likely longer lasting vaccination OCV (especially given current one dose strategy). SIR-type models, rate individuals leave Recovered compartment governed immune decay parameter \\(\\varepsilon\\). estimated durability immunity natural infection based two cohort studies fit following exponential decay model estimate rate immunity decay time:\\[\n\\text{Proportion immune}\\ t \\ \\text{days infection} = 0.99 \\times (1 - \\varepsilon) ^ {t-t_{\\text{infection}}}\n\\]\nmake necessary simplifying assumption within 0–90 days natural infection V. cholerae, individuals 95–99% immune. fit model reported data Ali et al (2011) Clemens et al (1991) (see Table 4.6).Table 4.6: Sources duration immunity fro natural infection.estimated mean immune decay \\(\\bar\\varepsilon = 3.9 \\times 10^{-4}\\) (\\(1.7 \\times 10^{-4}-1.03 \\times 10^{-3}\\) 95% CI) equivalent immune duration \\(7.21\\) years (\\(2.66-16.1\\) years 95% CI) shown Figure 4.11A. slightly longer previous modeling work estimating duration immunity ~5 years (King et al 2008). Uncertainty around \\(\\varepsilon\\) model represented Log-Normal distribution shown Figure 4.11B:\\[\n\\varepsilon \\sim \\text{Lognormal}(\\bar\\varepsilon+\\frac{\\sigma^2}{2}, 0.25)\n\\]\nFigure 4.11: duration immunity natural infection V. cholerae.\n","code":""},{"path":"model-description.html","id":"spatial-dynamics","chapter":"4 Model description","heading":"4.5 Spatial dynamics","text":"parameters model diagram Figure 4.2 \\(jt\\) subscript denote spatial structure model. country modeled independent metapopulation connected others via spatial force infection \\(\\Lambda_{jt}\\) moves contagion among metapopulations according connectivity provided parameters \\(\\tau_i\\) (probability departure) \\(\\pi_{ij}\\) (probability diffusion destination \\(j\\)). parameters estimated using departure-diffusion model fitted average weekly air traffic volume 41 countries included MOSAIC framework (Figure 4.12).\nFigure 4.12: average number air passengers per week 2017 among countries.\n\nFigure 4.13: network map showing average number air passengers per week 2017.\n","code":""},{"path":"model-description.html","id":"human-mobility-model","chapter":"4 Model description","heading":"4.5.1 Human mobility model","text":"departure-diffusion model estimates diagonal -diagonal elements mobility matrix (\\(M\\)) separately combines using conditional probability rules. model first estimates probability travel outside origin location \\(\\)—departure process—distribution travel origin location \\(\\) normalizing connectivity values across \\(j\\) destinations—diffusion process. values \\(\\pi_{ij}\\) sum unity along row, diagonal included, indicating relative quantity. say, \\(\\pi_{ij}\\) gives probability going \\(\\) \\(j\\) given travel outside origin \\(\\) occurs. Therefore, can use basic conditional probability rules define travel routes diagonal elements (trips made within origin \\(\\)) \n\\[\n\\Pr( \\neg \\text{depart}_i ) = 1 - \\tau_i\n\\]\n-diagonal elements (trips made outside origin \\(\\)) \n\\[\n\\Pr( \\text{depart}_i, \\text{diffuse}_{\\rightarrow j}) = \\Pr( \\text{diffuse}_{\\rightarrow j} \\mid \\text{depart}_i ) \\Pr(\\text{depart}_i ) = \\pi_{ij} \\tau_i.\n\\]\nexpected mean number trips route \\(\\rightarrow j\\) :\\[\\begin{equation}\nM_{ij} =\n\\begin{cases}\n\\theta N_i (1-\\tau_i) \\ & \\text{} \\ = j \\\\\n\\theta N_i \\tau_i \\pi_{ij} \\ & \\text{} \\ \\ne j.\n\\end{cases}\n\\tag{4.10}\n\\end{equation}\\], \\(\\theta\\) proportionality constant representing overall number trips per person origin population size \\(N_i\\), \\(\\tau_i\\) probability leaving origin \\(\\), \\(\\pi_{ij}\\) probability travel destination \\(j\\) given travel outside origin \\(\\) occurs.","code":""},{"path":"model-description.html","id":"estimating-the-departure-process","chapter":"4 Model description","heading":"4.5.2 Estimating the departure process","text":"probability travel outside origin estimated location \\(\\) give location-specific departure probability \\(\\tau_i\\).\n\\[\n\\tau_i \\sim \\text{Beta}(1+s, 1+r)\n\\]\nBinomial probabilities origin \\(\\tau_i\\) drawn Beta distributed prior shape (\\(s\\)) rate (\\(r\\)) parameters.\n\\[\n\\begin{aligned}\ns &\\sim \\text{Gamma}(0.01, 0.01)\\\\\nr &\\sim \\text{Gamma}(0.01, 0.01)\n\\end{aligned}\n\\]","code":""},{"path":"model-description.html","id":"estimating-the-diffusion-process","chapter":"4 Model description","heading":"4.5.3 Estimating the diffusion process","text":"use normalized formulation power law gravity model defined diffusion process, probability travelling destination \\(j\\) given travel outside origin \\(\\) (\\(\\pi_{ij}\\)) defined :\\[\\begin{equation}\n\\pi_{ij} = \\frac{\nN_j^\\omega d_{ij}^{-\\gamma}\n}{\n\\sum\\limits_{\\forall j \\ne } N_j^\\omega d_{ij}^{-\\gamma}\n}\n\\tag{4.11}\n\\end{equation}\\], \\(\\omega\\) scales attractive force \\(j\\) destination based population size \\(N_j\\). kernel function \\(d_{ij}^{-\\gamma}\\) serves penalty proportion travel \\(\\) \\(j\\) based distance. Prior distributions diffusion model parameters defined :\n\\[\n\\begin{aligned}\n\\omega &\\sim \\text{Gamma}(1, 1)\\\\\n\\gamma &\\sim \\text{Gamma}(1, 1)\n\\end{aligned}\n\\]models \\(\\tau_i\\) \\(\\pi_{ij}\\) fitted air traffic data OAG using mobility R package (Giles 2020). Estimates mobility model parameters shown Figures 4.14 4.15.\nFigure 4.14: estimated weekly probability travel outside origin location \\(\\tau_i\\) 95% confidence intervals shown panel population mean indicated red dashed line. Panel B shows estimated total number travelers leaving origin \\(\\) week.\n\nFigure 4.15: diffusion process \\(\\pi_{ij}\\) gives estimated probability travel origin \\(\\) destination \\(j\\) given travel outside origin \\(\\) occurred.\n","code":""},{"path":"model-description.html","id":"the-probability-of-spatial-transmission","chapter":"4 Model description","heading":"4.5.4 The probability of spatial transmission","text":"likelihood introductions cholera disparate locations major concern cholera outbreaks. However, can difficult characterize given endemic dynamics patterns human movement. include measures spatial heterogeneity first simple importation probability based connectivity possibility incoming infections. basic probability transmission origin \\(\\) particular destination \\(j\\) time \\(t\\) defined :\\[\\begin{equation}\np(,j,t) = 1 - e^{-\\beta_{jt}^{\\text{hum}} (((1-\\tau_j)S_{jt})/N_{jt}) \\pi_{ij}\\tau_iI_{}}\n\\tag{4.12}\n\\end{equation}\\]","code":""},{"path":"model-description.html","id":"the-spatial-hazard","chapter":"4 Model description","heading":"4.5.5 The spatial hazard","text":"Although concerned endemic dynamics , likely periods time early rainy season cholera cases rate transmission low enough spatial spread resemble epidemic dynamics time. times periods, can estimate arrival time contagion location cases yet reported. estimating spatial hazard transmission:\\[\\begin{equation}\nh(j,t) = \\frac{\n\\beta_{jt}^{\\text{hum}} \\Big(1 - \\exp\\big(-((1-\\tau_j)S_{jt}/N_{jt}) \\sum_{\\forall \\= j} \\pi_{ij}\\tau_i (I_{}/N_{}) \\big) \\Big)\n}{\n1/\\big(1 + \\beta_{jt}^{\\text{hum}} (1-\\tau_j)S_{jt}\\big)\n}.\n\\tag{4.13}\n\\end{equation}\\]normalizing give waiting time distribution locations:\\[\\begin{equation}\nw(j,t) = h(j,T) \\prod_{t=1}^{T-1}1-h(j,t).\n\\tag{4.14}\n\\end{equation}\\]","code":""},{"path":"model-description.html","id":"coupling-among-locations","chapter":"4 Model description","heading":"4.5.6 Coupling among locations","text":"Another measure spatial heterogeneity quantify coupling disease dynamics among metapopulations using correlation coefficient. , use definition spatial correlation locations \\(\\) \\(j\\) \\(C_{ij}\\) described Keeling Rohani (2002), gives measure similar infection dynamics locations.\\[\\begin{equation}\nC_{ij} = \\frac{\n( y_{} - \\bar{y}_i )( y_{jt} - \\bar{y}_j )\n}{\n\\sqrt{\\text{var}(y_i) \\text{var}(y_j)}\n}\n\\tag{4.15}\n\\end{equation}\\]\n\\(y_{} = I_{}/N_i\\) \\(y_{jt} = I_{jt}/N_j\\). Mean prevalence location \\(\\bar{y_i} = \\frac{1}{T} \\sum_{t=1}^{T} y_{}\\) \\(\\bar{y_j} = \\frac{1}{T} \\sum_{t=1}^{T} y_{jt}\\).","code":""},{"path":"model-description.html","id":"the-observation-process","chapter":"4 Model description","heading":"4.6 The observation process","text":"","code":""},{"path":"model-description.html","id":"rate-of-symptomatic-infection","chapter":"4 Model description","heading":"4.6.1 Rate of symptomatic infection","text":"presentation infection V. cholerae can extremely variable. severity infection depends many factors amount infectious dose, age host, level immunity host either vaccination previous infection, naivety particular strain V. cholerae. Additional circumstantial factors nutritional status overall pathogen burden may also impact infection severity. population level, observed proportion infections symptomatic also dependent endemicity cholera region. Highly endemic areas (e.g. parts Bangladesh; Hegde et al 2024) may low proportion symptomatic infections due many previous exposures. Inversely, populations largely naive V. cholerae exhibit relatively higher proportion symptomatic infections (e.g. Haiti; Finger et al 2024).Accounting nuances first version model possible, can past studies contain information can help set sensible bounds definition proportion infections symptomatic (\\(\\sigma\\)). compiled short list studies done sero-surveys cohort studies assess likelihood symptomatic infections different locations displayed results Table (4.7).provide reasonably informed prior proportion infections symptomatic, calculated combine mean confidence intervals studies Table 4.7 fit Beta distribution corresponds quantiles using least-squares Nelder-Mead algorithm. resulting prior distribution symptomatic proportion \\(\\sigma\\) :\\[\\begin{equation}\n\\sigma \\sim \\text{Beta}(4.30, 13.51)\n\\end{equation}\\]Table 4.7: Summary Studies Cholera ImmunityThe prior distribution \\(\\sigma\\) plotted Figure 4.16A reported values proportion symptomatic previous studies shown 4.16B.\nFigure 4.16: Proportion infections symptomatic.\n","code":""},{"path":"model-description.html","id":"suspected-cases","chapter":"4 Model description","heading":"4.6.2 Suspected cases","text":"clinical presentation diarrheal diseases often similar across various pathogens, can lead systematic biases reported number cholera cases. anticipated number suspected cholera cases related actual number infections factor \\(1/\\rho\\), \\(\\rho\\) represents proportion suspected cases true infections. adjust bias, use estimates meta-analysis Weins et al. (2023), suggests suspected cholera cases outnumber true infections approximately 2 1, mean across studies indicating 52% (24-80% 95% CI) suspected cases actual cholera infections. higher estimate reported ourbreak settings (78%, 40-99% 95% CI). account variability estimate, fit Beta distribution reported quantiles using least squares approach Nelder-Mead algorithm, resulting prior distribution shown Figure 4.17B:\\[\\begin{equation}\n\\rho \\sim \\text{Beta}(4.79, 1.53).\n\\end{equation}\\]\nFigure 4.17: Proportion suspected cholera cases true infections. Panel shows ‘low’ assumption estimates across settings: \\(\\rho \\sim \\text{Beta}(5.43, 5.01)\\). Panel B shows ‘high’ assumption estimate reflects high-quality studies outbreaks: \\(\\rho \\sim \\text{Beta}(4.79, 1.53)\\)\n","code":""},{"path":"model-description.html","id":"case-fatality-rate","chapter":"4 Model description","heading":"4.6.3 Case fatality rate","text":"Case Fatality Rate (CFR) among symptomatic infections calculated using reported cases deaths data January 2021 August 2024. data collated various issues Weekly Epidemiological Record Global Cholera Acute Watery Diarrhea (AWD) Dashboard (see Data section) provide annual aggregations reported cholera cases deaths. used Binomial exact test (binom.test R) calculate mean probability number deaths (successes) given number reported cases (sample size), Clopper-Pearson method calculating binomial confidence intervals. fit Beta distributions mean CFR 95% confidence intervals calculated country using least squares Nelder-Mead algorithm give distributional uncertainty around CFR estimate country (\\(\\mu_j\\)).\\[\n\\mu_j \\sim \\text{Beta}(s_{1,j}, s_{2,j})\n\\]\\(s_{1,}\\) \\(s_{2,j}\\) two positive shape parameters Beta distribution estimated destination \\(j\\). definition \\(\\mu_j\\) CFR reported cases subset total number infections. Therefore, infer total number deaths attributable cholera infection, assume CFR observed cases proportionally equivalent CFR cases calculate total deaths \\(D\\) follows:\\[\\begin{equation}\n\\begin{aligned}\n\\text{CFR}_{\\text{observed}} &= \\text{CFR}_{\\text{total}}\\\\\n\\\\[3pt]\n\\frac{[\\text{observed deaths}]}{[\\text{observed cases}]} &=\n\\frac{[\\text{total deaths}]}{[\\text{infections}]}\\\\\n\\\\[3pt]\n\\text{total deaths} &= \\frac{[\\text{observed deaths}] \\times [\\text{true infections}]}{[\\text{observed cases}]}\\\\\n\\\\[3pt]\nD_{jt} &= \\frac{ [\\sigma\\rho\\mu_j I_{jt}] \\times [I_{jt}] }{ [\\sigma\\rho I_{jt}] }\n\\end{aligned}\n\\end{equation}\\]\nTable 4.8: Table 4.9: CFR Values Beta Shape Parameters AFRO Countries\n\nFigure 4.18: Case Fatality Rate (CFR) Total Cases Country AFRO Region 2014 2024. Panel : Case Fatality Ratio (CFR) 95% confidence intervals. Panel B: total number cholera cases. AFRO Region highlighted black, countries less 3/0.2 = 150 total reported cases assigned mean CFR AFRO.\n\nFigure 4.19: Beta distributions overall Case Fatality Rate (CFR) 2014 2024. Examples show overall CFR AFRO region (2%) black, Congo highest CFR (7%) red, South Sudan lowest CFR (0.1%) blue.\n","code":""},{"path":"model-description.html","id":"demographics-1","chapter":"4 Model description","heading":"4.7 Demographics","text":"model includes basic demographic change using reported birth death rates \\(j\\) countries, \\(b_j\\) \\(d_j\\) respectively. rates static defined United Nations Department Economic Social Affairs Population Division World Population Prospects 2024. Values \\(b_j\\) \\(d_j\\) derived crude rates converted birth rate per day death rate per day (shown Table 4.10).\nTable 4.10: Table 4.11: Demographic AFRO countries 2023. Data include: total population January 1, 2023, daily birth rate, daily death rate. Values calculate crude birth death rates UN World Population Prospects 2024.\n","code":""},{"path":"model-description.html","id":"the-reproductive-number","chapter":"4 Model description","heading":"4.8 The reproductive number","text":"reproductive number common metric epidemic growth represents average number secondary cases generated primary case specific time epidemic. track \\(R\\) changes time estimating instantaneous reproductive number \\(R_t\\) described Cori et al 2013. track \\(R_t\\) across metapopulations model give \\(R_{jt}\\) using following formula:\\[\\begin{equation}\nR_{jt} = \\frac{I_{jt}}{\\sum_{\\Delta t=1}^{t} g(\\Delta t) I_{j,t-\\Delta t}}\n\\tag{4.16}\n\\end{equation}\\]\\(I_{jt}\\) number new infections destination \\(j\\) time \\(t\\), \n\\(g(\\Delta t)\\) represents probability value generation time distribution cholera. accomplished using weighed sum denominator highly influenced generation time distribution.","code":""},{"path":"model-description.html","id":"the-generation-time-distribution","chapter":"4 Model description","heading":"4.8.1 The generation time distribution","text":"generation time distribution gives time individual infected infect subsequent individuals. parameterized quantity using Gamma distribution mean 5 days:\\[\\begin{equation}\ng(\\cdot) \\sim \\text{Gamma}(0.5, 0.1).\n\\tag{4.17}\n\\end{equation}\\], shape=0.5, rate=0.1, mean given shape/rate. Previous studies use mean 5 days (Kahn et al 2020 Azman 2016), however mean 3, 5, 7, 10 days may admissible (Azman 2012).\nFigure 4.20: generation time\n\nTable 4.12: Table 4.13: Generation Time Weeks\n","code":""},{"path":"model-description.html","id":"initial-conditions","chapter":"4 Model description","heading":"4.9 Initial conditions","text":"Since first version model begin Jan 2023 (take advantage available weekly data), initial conditions surrounding population immunity must estimated. set initial conditions, use historical data find total number reported cases location previous X years, multiply \\(1/\\sigma\\) estimate total infections symptomatic cases reported, adjust based waning immunity. also sum total number vaccinations past X years adjust vaccine efficacy \\(\\phi\\) waning immunity vaccination \\(\\omega\\).total number infected? reported cases… back symptomatic asymptomatictotal number infected? reported cases… back symptomatic asymptomaticTotal number immune due natural infections past X yearsTotal number immune due natural infections past X yearstotal number immune due past vaccinations X yearstotal number immune due past vaccinations X yearsUse deconvolution based immune decay estimated vaccine section","code":""},{"path":"model-description.html","id":"model-calibration","chapter":"4 Model description","heading":"4.10 Model calibration","text":"model calibrated using Latin hypercube sampling hyper-parameters model likelihoods fit incidence deaths.model calibrated using Latin hypercube sampling hyper-parameters model likelihoods fit incidence deaths.important challenge flexibly fitting data often missing available aggregated forms.important challenge flexibly fitting data often missing available aggregated forms.[Fig: different spatial temporal scales available data]","code":""},{"path":"model-description.html","id":"caveats","chapter":"4 Model description","heading":"4.11 Caveats","text":"Simplest model start. Easier initial spatial structure minimum additional compartments calibrate available data (vaccination, cases, deaths).Country level aggregations. First generation data 2023/24…Assumes vaccinating susceptible individuals.climate, summarizing whole country.","code":""},{"path":"model-description.html","id":"table-of-parameters","chapter":"4 Model description","heading":"4.12 Table of parameters","text":"Table 4.14: Descriptions model parameters along prior distributions sources applicable.","code":""},{"path":"model-description.html","id":"references","chapter":"4 Model description","heading":"4.13 References","text":"","code":""},{"path":"scenarios.html","id":"scenarios","chapter":"5 Scenarios","heading":"5 Scenarios","text":"key aim MOSAIC model provide near-term forecasts cholera transmission Sub-Saharan Africa (SSA) using current data available. However, MOSAIC just forecasting tool; dynamic model designed explore various scenarios influence critical factors vaccination, environmental conditions, Water, Sanitation, Hygiene (WASH) interventions.","code":""},{"path":"scenarios.html","id":"vaccination","chapter":"5 Scenarios","heading":"5.1 Vaccination","text":"","code":""},{"path":"scenarios.html","id":"spatial-and-temporal-strategies","chapter":"5 Scenarios","heading":"5.1.1 Spatial and Temporal Strategies","text":"Understanding spatial temporal distribution cholera vaccination efforts crucial effective outbreak control. Key resources include:Stockpile Status: availability oral cholera vaccine emergency stockpiles can tracked UNICEF’s Emergency Stockpile Availability.OCV Dashboard: dashboard (link) provides insights deployment oral cholera vaccines (OCV) across different regions.","code":""},{"path":"scenarios.html","id":"reactive-vaccination","chapter":"5 Scenarios","heading":"5.1.2 Reactive Vaccination","text":"timing logistics reactive vaccination campaigns critical controlling ongoing outbreaks. Relevant resources include:Recommended Timing: Guidelines recommendations timing reactive OCV campaigns available (link).Requests Delay Time Distributions: Information vaccine request processes distribution delays vaccine deployment can accessed GTFCC OCV Dashboard (link).","code":""},{"path":"scenarios.html","id":"impacts-of-climate-change","chapter":"5 Scenarios","heading":"5.2 Impacts of Climate Change","text":"","code":""},{"path":"scenarios.html","id":"severe-weather-events","chapter":"5 Scenarios","heading":"5.2.1 Severe Weather Events","text":"Projections climate shocks, including frequency severity cyclones floods, essential modeling future impacts climate change cholera transmission. Key references include:Chen Chavas 2020: study cyclone season dynamics climate change scenarios (link).Sparks Toumi 2024: Research projected flood frequencies due climate change (link).Switzer et al. 2023: analysis climate shock impacts cholera outbreaks (link).","code":""},{"path":"scenarios.html","id":"long-term-trends","chapter":"5 Scenarios","heading":"5.2.2 Long-Term Trends","text":"Long-term trends weather variables various climate change scenarios can explored using following resource:Weather Variables Climate Change: OpenMeteo Climate API provides access projected weather data different climate change scenarios (link).","code":""},{"path":"usage.html","id":"usage","chapter":"6 Usage","heading":"6 Usage","text":"open-source code used run MOSAIC currently development presented future.","code":""},{"path":"news.html","id":"news","chapter":"7 News","heading":"7 News","text":"","code":""},{"path":"news.html","id":"past-versions-of-mosaic","chapter":"7 News","heading":"7.1 Past versions of MOSAIC","text":"Table 7.1: Current future planned model versions brief descriptions.","code":""},{"path":"references-1.html","id":"references-1","chapter":"8 References","heading":"8 References","text":"","code":""}] diff --git a/figures/wash_incidence_correlation.png b/figures/wash_incidence_correlation.png index ae98b6c..89e1421 100644 Binary files a/figures/wash_incidence_correlation.png and b/figures/wash_incidence_correlation.png differ diff --git a/figures/wash_index_by_country.png b/figures/wash_index_by_country.png index dbbea05..d99ee2b 100644 Binary files a/figures/wash_index_by_country.png and b/figures/wash_index_by_country.png differ diff --git a/tables/WASH_weighted_mean_theta.csv b/tables/WASH_weighted_mean_theta.csv index 7fc6a0a..356c58a 100644 --- a/tables/WASH_weighted_mean_theta.csv +++ b/tables/WASH_weighted_mean_theta.csv @@ -39,3 +39,13 @@ "South Africa","ZAF",0.913309900436136 "Zambia","ZMB",0.627881103961712 "Zimbabwe","ZWE",0.662058780328589 +"Central African Republic","CAF",0.762547227258778 +"Gabon","GAB",0.631520091542042 +"Ghana","GHA",0.669666481983608 +"Guinea","GIN",0.779348244741159 +"Gambia","GMB",0.687360184773828 +"Guinea-Bissau","GNB",0.778921138655779 +"Niger","NER",0.803807835653375 +"Nigeria","NGA",0.760084782053426 +"Rwanda","RWA",0.670273659221199 +"South Sudan","SSD",0.812776283418475