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cholera.tex
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cholera.tex
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\begin{fullwidth}
\chapter{A modelling oriented primer on cholera}
\end{fullwidth}
Cholera is an acute intestinal infection causing severe diarrhea that may lead to dehydration, and sometimes death. The global burden of cholera is difficult to estimate as the majority of cases are not reported. It is estimated that 3 million cases and 95'000 deaths occur every year, with millions more at risk in endemic areas (around 50 countries)\cite[-1\baselineskip]{Ali:UpdatedGlobalBurden:2015}. Despite its household name, cholera belongs to the neglected tropical diseases group: our understanding of many important aspects of cholera clinical course and transmission is still limited.
A political will to eliminate this ancient disease has recently arisen. The World Health Organization (WHO) initiated the Global Task Force for Cholera Control (GTFCC), which provides a concrete path towards the elimination of cholera by 2030\footnote[][-3\baselineskip]{Elimination being defined as a 90\% reduction of cholera deaths per year, see \url{gtfcc.org}.}. The consensus is that to reach this goal in endemic countries, there must be substantial long-term improvements in safe water distribution systems, adequate sanitation, and accessible hygiene education. Moreover, in the event of a cholera outbreak, timely interventions such as vaccination campaigns are crucial to limit the spread of the disease, and access to proper treatments reduces the toll of the disease on communities. %With limited resources, public health officials face a number of challenging decisions, and a data driven decision support to guide the rational deployment of cholera control strategies is needed. Furthermore, on the road toward elimination, the need to setup context-specific tailored approaches appears.
%The poorest of the poor
%most infected individuals are asymptomatic, i.e., they do not present symptoms, while other experience mild or severe symptoms. %Since their mobility is not hindered, they become a vector of the infection. If not properly treated, cholera can kill children and adults within hours.
\vspace{.7cm}
\section{History and epidemiology}
\begin{marginfigure}[-5\baselineskip]
%\centering
\includegraphics[width=\textwidth]{fig/snow-cholera-map_edit}
\margincaption[Map of the clusters of cholera cases in London, 1854]{Original map by John Snow. Stacked rectangles represent cholera cases of the 1854 Broad Street outbreak (London). The work of John Snow convinced the authorities to close the contaminated water pump (circled in red), leading to a decrease in mortality. Lithography by Charles Cheffins, in \fullcite[p. 54]{Snow:ModeCommunicationCholera:1855}.}\label{johnsnow}
\end{marginfigure}
Humanity and cholera share a long history, with supposed mentions as early as the 5th century \textsc{bce}. The disease became more widely known in the modern era. From 1817 to 1923, six successive pandemics occurred, all originated in the delta of the Ganges river but taking different paths across the world. Cholera spread around the world owing to the nascent mobility, leaving 10s of millions dead across countries and continents. Scientific developments sped up during the third cholera pandemic (1846-60). In 1854, a cholera outbreak in Broad Street (Soho, London) was studied by physician John Snow who led an impressive early work in spatial epidemiology applied to public health. Analyzing the contamination pattern among residents (fig. \ref{johnsnow}), Snow postulated that cholera spreads through water contaminated by an infectious agent, instead of foul air\sidenote[][-4\baselineskip]{At the time, the accepted mode of contamination for cholera and many other diseases was through miasma, \textit{bad air} contaminated by organic matter. Chasing odor justified urban planning along streets and river banks in Paris and London. But also in Lausanne, where rivers Flon and Louve were covered in 1832 in response to a cholera outbreak. Cholera is the name of a savory dish from Valais, a testimony of the strong impression cholera left on the Swiss.}. Simultaneously, Italian microbiologist Filippo Pacini isolated the bacterium in Florence\cite{Pacini:OsservazioniMicroscopicheDeduzioni:1854}. Thirty years after Pacini, German scientist Robert Koch independently rediscovered the cholera pathogen after investigations in Egypt and India. He first posited \textit{Vibrios} causative relationship with the disease. Finally, a hundred years later, Indian researcher Sambhu Nath De discovered the cholera toxin in 1959\cite{De:ExperimentalStudyAction:1951}.
Throughout a seventh pandemic from 1961 onward, cholera spread in several waves, through Asia in the 1960s, reaching Africa and the Middle East in the 1970s and the Americas in 1991\cite{Mutreja:EvidenceSeveralWaves:2011}. Improvement in sanitation and hygiene spared higher-income countries from the disease, and cholera became a burden of the poorest communities of the Global South. Series of outbreaks (\eg Zimbabwe 2008, Haïti 2010, Yemen 2016), continuous transmission in endemic countries, and millions at risk keep this ancient disease an ongoing public health issue.
The span of this thesis was marked by a cholera outbreak in Yemen (2016--2021) -- a humanitarian crisis with 2.5\textsc{m} suspected cases and nearly 4'000 deaths -- an outbreak in Zimbabwe, a flare-up in Algeria, in addition to many seasonal outbreaks and sustained endemic cholera transmission in Asia and sub-Saharan Africa. But also the last confirmed cholera case of Haiti, in early 2019, bringing an end to a 9 years epidemic and the hope of cholera elimination from its last foothold in the Americas.
While the number of cholera cases doubled from 2018 to 2019 to nearly a million, reported cholera deaths decreased to less than 2'000 in 2019, with Africa reporting its lowest numbers since the 2000s. Efforts towards the improvement of sanitary conditions and reactive vaccinations campaigns (24\textsc{m} doses of cholera vaccines were distributed in 2019) aim at further reducing the toll of cholera\footnote{see \fullcite{WHO:Cholera2019:2020} and previous \textit{Weekly Epidemiological Records} about cholera.}.
The following sections highlight some relevant aspects of cholera transmission, while further details about the biological and medical features of cholera are outside the scope of this thesis.
\section{Pathogen}
\begin{marginfigure}[6\baselineskip]
\centering
\includegraphics{fig/vibrio}
\margincaption[Image of the Vibrio cholerae bacteria]{Scanning electron microscope image of \textit{Vibrio cholerae}, a gram-negative rod-shaped bacteria (Public domain image by Ronald Taylor, Tom Kirn, Louisa Howard).}
\label{fig:bacteria}
\end{marginfigure}
\paragraph{Pathogen} Cholera is an infection caused by a waterborne bacteria: the \emph{Vibrio cholerae} (fig. \ref{fig:bacteria}). While many serogroups of \emph{V. cholerae} can secrete the cholera toxin responsible for massive watery diarrhea, only serogroups O139 and O1 are responsible for disease epidemics. O1 is causing most of the recent epidemics and is divided into two biotypes: Classical O1 and El Tor, which are both divided into three serotypes: Ogawa, Inaba, and the rare Hikojima\cite{Kaper:Cholera:1995}. Cholera classification has its importance as it affects many epidemiological characteristics \eg, El Tor survives longer in water and has a higher asymptomatic/symptomatic ratio\cite{WHO:CholeraVaccinesWHO:2017}. The current cholera pandemic is mainly caused by El Tor, while the Classical derivative caused the previous fifth and sixth pandemics, and its transmission is now limited to the Ganges delta\cite{Nair:CholeraDueAltered:2006, Domman:DefiningEndemicCholera:2018}.
\paragraph{Environmental reservoir} \textit{V. cholerae} has for natural habitat some aquatic ecosystems, in particular brackish waters and estuaries. There is no marine host, but complex ecological association processes take place in the aquatic medium, and natural genetic transformation is enabled by chitin, the polymer constituting the crustacean exoskeleton\cite{Reidl:VibrioCholeraeCholera:2002,Meibom:ChitinInducesNatural:2005}. There is no consensus on how long \textit{V. cholerae} remains infectious in water and under which conditions it can reproduce\cite{Mavian:ToxigenicVibrioCholerae:2020}. In cholera epidemics, it is difficult to isolate the importance of the natural bacteria reservoir compared to the freshly introduced \textit{Vibrios} excreted by infected persons, though the latter is suspected to drive transmission during outbreaks.
\paragraph{Climatic drivers} Cholera epidemics express marked seasonal patterns that might have different shapes in different countries. A complex and unclear association between precipitation and cholera infections has been put forward in many research works.
%The early suggestion of the importance of the rainfall driver in Haiti is proposed by \textcite{Gaudart:SpatioTemporalDynamicsCholera:2013}.
After Hurricane Matthew and heavy rainfall in October 2016, cholera flared up after a low transmission episode in Haiti\cite{Pasetto:RealtimeForecastingCholera:2018}. Similarly, cholera in many African countries follows a seasonal trend, with the higher transmission during the rainy season\cite{Baracchini:SeasonalityCholeraDynamics:2017}. Rainfall might play a major role in water contamination, for instance through the washout of open-air defecation and raw sewage circulation in the environment. A literature review about studies investigating the relationship between cholera and rainfall is provided in \textsc{Chapter~2}. Temperature, water acidity, sunlight, and other environmental factors have also been shown to affect the survival and reproduction of \textit{V. cholerae} in water bodies. Hence, macroclimate phenomena such as the El Niño Southern Oscillation have been associated with changes in transmission, even if no causal link could be established\cite[-2\baselineskip]{Pascual:CholeraDynamicsNinoSouthern:2000}\footnote{Marc Lipsitch and Cécile Viboud beautifully describe the difficulty evaluating the environmental factors in disease transmission, writing for influenza: ``This potpourri of possible mechanisms places us in a kind of Popperian purgatory, in which data in support of every hypothesis exist, yet none of the hypotheses has been subjected to tests that are rigorous enough to reject it.''. Quote from \fullcite{Lipsitch:InfluenzaSeasonalityLifting:2009}}.
\section{Cholera and the human}
\paragraph{Disease} A human host becomes infected through the ingestion of a critical dose of \emph{V. cholerae}\cite{Kaper:Cholera:1995}\cite{Nelson:CholeraTransmissionHost:2009}. The susceptibility depends on many factors such as gastric acidity and age, with children under 5 much more likely to become infected\cite{Sack:Cholera:2004}. \textit{V. cholerae} colonizes the small intestine for an incubation period lasting 12 hours to 5 days\cite{Azman:IncubationPeriodCholera:2013} before symptoms. Then, a wide range of outcomes are possible. Most of the time the infection is asymptomatic. On the other end of the spectrum severe infection (or \emph{cholera gravis}), characterized by vomiting and profuse rice water diarrhea, occurs in 1\% to 15\% of the cases. Many stages of mild infections lie in between these extremes. Unless tested in a laboratory, symptoms are indistinguishable from those of numerous other infections causing diarrhea\footnotemark[17]$^{,}$\footnotemark[18]$^{,}$\footnote{\fullcite{King:InapparentInfectionsCholera:2008};% \fullciteshortb{Kaper:Cholera:1995, Nelson:CholeraTransmissionHost:2009};
\fullcite{vandeLinde:ObservationsSpreadCholera:1965,Mccormack:CommunityStudyInapparent:1969}}. The severity of the symptoms correlates with the quantity of \textit{V. cholerae} ingested\cite{Brouwer:DoseresponseRelationshipsEnvironmentally:2017}, and depends on the cholera strain and personal characteristics, age, immunity, pregnancy, blood type\cite{WHO:CholeraVaccinesWHO:2017,Azman:IncubationPeriodCholera:2013}, ...%Table 1 for a review of individual data studies + say unclear how contagious asmptomatic
Treatment is crucial: severely infected individuals might lose up to 20 liters of water through diarrhea in a day and may die within hours. If left untreated, severe cases mortality reaches up to 60\% but proper therapy lowers it below 1\%\cite{Luquero:MortalityRatesCholera:2016}.
\paragraph{Shedding and transmission} The number of bacteria shed by an infectious individual varies with the intensity of the infection. It is estimated to range from $10^3$ to $10^{9}$ \textit{Vibrios} per gram of stool for asymptomatic infected and severely infected individual respectively\footnotemark[18]. Similarly, the duration of the shedding period typically ranges from a day up to two weeks\footnotemark[17]$^{,}$\footnotemark[18].
Transmission occurs along the fecal-oral route (consumption of contaminated water or food, contact with fomites), but contamination is also possible from the aquatic reservoirs, through ingestion of water or seafood. The principle mechanism of transmission is the intake of water contaminated by the untreated feces of other infectious individuals. In fact, freshly shed \textit{Vibrios} may be in a hyper-infectious state, which could be of great importance in driving epidemic transmission\cite{Butler:CholeraStoolBacteria:2006}.
Asymptomatic individuals remain mobile and are thought to transmit cholera, thus may be important vectors for cholera dissemination.
%TODO: Two times Piarroux
\paragraph{Human mobility and hydrological transport} Spatial spread of cholera outbreaks may occur through two networks. \textit{V. cholerae} may be transported through the river network. An example is the start of the 2010 Haiti epidemic along the Artibonite river\cite{Piarroux:UnderstandingCholeraEpidemic:2011}. Human mobility also plays a major role in the spreading of the infections possibly due to the large proportion of asymptomatic that transport and transmit cholera across regions. Indeed, also in Haiti cholera was brought into the state by infected United Nations peacekeepers\cite{Piarroux:UnderstandingCholeraEpidemic:2011}. %However, mobility, especially in humanitarian crisis situations often associated with cholera, remains difficult to predict\cite{Lu:PredictabilityPopulationDisplacement:2012,Riley:LargeScaleSpatialTransmissionModels:2007,Bengtsson:ImprovedResponseDisasters:2011,Rebaudet:DrySeasonHaiti:2013}.
\paragraph{Immunity} Infected individuals that recover from the infection are immunized against \textit{V. cholerae} of the same serogroup. The duration of acquired immunity is difficult to estimate and depends on many factors. Acquired immunity has been reported to range from few months to several years, with most recent estimates ranging from 2 to 10 years, possibly depending on the virulence of the infection\cite{Levine:DurationInfectionDerivedImmunity:1981,Kaper:Cholera:1995,Woodward:CholeraReinfectionMan:1971,Glass:SeroepidemiologicalStudiesEI:1985,Clemens:BiotypeDeterminantNatural:1991,Leung:ProtectionAffordedPrevious:2021}.
%TODO Kaper and Nelson, here are done with footnotemarks here
\subsection{Interventions against cholera}
Interventions against cholera may be preventive or concern the treatment of infected individuals. Case management and treatment plays an important role in reducing the transmission and the toll of the epidemic and mainly consists of:
\begin{description}
\item[Oral (or Intravenous) Rehydratation Therapy] The main treatment for cholera consists of replacing fluids as fast as they are lost. Despite its simplicity, it is very effective in reducing mortality. Fluids with the same electrolyte composition must be administered\cite{Kuhn:GlucoseNotRiceBased:2014}. Re-hydration is usually done in treatment centers but may take place at the patient home. This differentiation might determine if stools contribute to the infection cycle or are properly disposed of.
\item[Antibiotics] reduce the severity and the duration of the infection. WHO recommends their use only for the most severe cases as antibiotic resistance of \emph{V. cholerae} is raising worldwide\cite{Sack:GettingSeriousCholera:2006}.
\end{description}
Prevention measures may be carried out before and during the outbreak, and are described below.
\paragraph{Surveillance and reporting} During an outbreak, surveillance consists of the timely detection and reporting of new cases. In many countries where outbreaks occur annually during the rainy season, the observation of past epidemics provides insight on the severity and timing of the new infections, that can be used for preparation\cite{Baracchini:SeasonalityCholeraDynamics:2017}. While possible, environmental monitoring for \textit{V. Cholerae} in the water has never succeeded to warn about an upcoming epidemic.
Since without laboratory equipment it is impossible to distinguish cholera from another pathogen in a patient with acute watery diarrhea, there has been an effort to standardize the clinical definition of a suspected cholera case. %, which varies between countries and during outbreaks.
A suspected case combines acute watery diarrhea and severe dehydration, the latter condition being dropped in case of an outbreak. This diagnosis may be precised using rapid diagnosis tests (RTDs), with a pretty high sensibility but low sensitivity. Precise identification is obtained through culture, the current gold standard\footnote{\fullcite{Camacho:CholeraEpidemicYemen:2018} and \fullcite{CDC:DiagnosisDetectionCholera:2018}}. RTDs and culture are not always available, especially during an outbreak, and over-reporting due to the misclassification of other diarrheal diseases is possible. Conversely, under-reporting might also occur as transmission settings might be isolated or plagued with conflicts or natural disasters. WHO guidelines recommend that, when a patient enters a treatment center, his name, address, sex, age (over or below 5), and symptoms are recorded\cite{WHO:FirstStepsManaging:2010}. % https://apps.who.int/iris/bitstream/handle/10665/334241/WER9537-eng-fre.pdf?ua=1 end, and https://www.cdc.gov/cholera/diagnosis.html
\paragraph{WaSH} Water, Sanitation and Hygiene (WaSH) is a broad term that encompasses many intervention strategies which are key to the long-term control of cholera. Improved sanitary conditions brought cholera elimination in first-world countries. WaSH is divided into short and long-term measures. Short term strategies involve sterilization, decontamination, hand washing, education sessions, and water purification and filtering (chlorination, ...)\cite[-3\baselineskip]{Rebaudet:NationalAlertresponseStrategy:2018,Fewtrell:WaterSanitationHygiene:2005}. Long-term sanitation strategies involve the construction of infrastructures for fecal sludge management, sewage systems, toilets, and access to safe water sources. From a modeling point of view, WaSH reduces exposure (water purification, sari filtration\cite{Colwell:ReductionCholeraBangladeshi:2003}) and shedding (sewage and fecal sludge management). By its nature and the possible long-term effects, WaSH improvement is difficult to quantify in modeling frameworks.
\paragraph{Vaccination} is a safe and effective way to protect individuals from cholera and to reduce the propagation of the epidemic. It can be used in a preventive or reactive way. Several vaccines exist for cholera, with different characteristics. As of today, two main oral cholera vaccines (OCVs) are used in vaccination campaigns around the world: WC-rBS and BivWC\footnote[][-3\baselineskip]{Another vaccine, Vaxchora, was recently approved by the FDA, mostly for travelers.}. The main characteristics of these two vaccines are shown in tab.~\ref{tab:vacc}.
Vaccines can either be administered in a targeted fashion or to the whole population in mass vaccination campaigns, either preventively or reactively during outbreaks\footnote[][-5.5\baselineskip]{See \href{twitter.com/TheFerrariLab/status/1340681737316294659}{this interesting thread} on the surprisingly recent history of reactive vaccination.}. Despite the efforts to build a worldwide vaccine stockpile, the demand for cholera vaccine vastly exceeds the supply\cite[-5\baselineskip]{Parker:AdaptingGlobalShortage:2017a,Seidlein:PreventingCholeraOutbreaks:2018}.
\begin{table}[h]
\centering\small
\label{tab:prior}
\begin{tabular}{lp{40mm}p{40mm}}
\toprule
Generic Name & BiWC & WC-rBS\\
\midrule
Commercial name & mORCVAX, Shanchol, Euvichol, Cholvax & Dukoral \\
Target strain O1 & yes (classical, El Tor, Ogawa, Inaba)& yes (classical, El Tor, Ogawa, Inaba), also target a cholera toxin \\
Target strain O139 & yes & no \\
Doses & 2 doses, 2 weeks apart & 2 doses (3 for children) 1--6 weeks apart \\
%Vaccine Efficacy & 58\% & \\
Field effectiveness & between 37\% and 87\% for two years & 78\% protection 1--6 months after vaccination\\
Age & $>$ 1 year & $>$ 2 year \\
Usage & Mass vaccination, Global OCV stockpile, 25\textsc{m} doses administered & Mainly for travelers ($>$ 1M doses administered)\\
Protection length & 3 years (1 dose: short term protection) & 2 years\\
Constraints & -- & needs buffer solution\\
Price per dose & 1.85\$ & 5.25\$ \\
Usage & Since 1998 in most recent outbreaks & Between 1997 and 2009 in Uganda, Tanzania, Indonesia,~... \\
\bottomrule
\end{tabular}
\caption[Characteristic of existing oral cholera vaccines][-2\baselineskip]{Characteristic of existing oral cholera vaccines. Field effectiveness and vaccine efficacy are difficult to evaluate and hence omitted from this table. See \fullcite{WHO:CholeraVaccinesWHO:2017}, \fullcite{Bi:ProtectionCholeraKilled:2017}(and \fullciteshortb{WHO:BackgroundPaperWholeCell:2017,Azman:PopulationLevelEffectCholera:2016,Luquero:FirstOutbreakResponse:2013,WHO:BackgroundPaperIntegration:2009,Luquero:UseVibrioCholerae:2014,Qadri:EfficacySingledoseRegimen:2018,Azman:ImpactOneDoseTwoDose:2015,Tohme:OralCholeraVaccine:2015}).}%TODO citation here
\label{tab:vacc}
\end{table}
\section{ECHO's cholera model}
Cholera has been the subject of modeling studies since the first attempt by Capasso and Paveri-Fontana\cite[-4\baselineskip]{Capasso:MathematicalModel1973:1979}\footnote[][-1\baselineskip]{For an overview of the history of cholera modeling, from SI to SIR to SIRB, the reader is referred to \fullcite{Rinaldo:RiverNetworksEcological:2020a}.}, and cholera modeling has received renewed attention during the 2010 outbreak in Haiti. Most of recent cholera modeling literature focuses on phenomenological (or statistical) models with different degrees of mechanistic processes\shortcite{Azman:UrbanCholeraTransmission:2012,Finger:PotentialImpactCasearea:2018,Camacho:CholeraEpidemicYemen:2018,Lessler:MappingBurdenCholera:2018,Koelle:DisentanglingExtrinsicIntrinsic:2004}, but numerous mechanistic cholera models are proposed, differing in the way they account for the epidemiological processes\shortcite{Kirpich:ControllingCholeraOuest:2017,Tuite:CholeraEpidemicHaiti:2011,Chao:VaccinationStrategiesEpidemic:2011,Kirpich:CholeraTransmissionOuest:2015}. %Most cholera models are spatially implicit, however there have been a number of attempts to describe the spatial spread of the epidemic. For example, Andrew and Basu used an approach with isolated nodes and independent transmission parameters in each node\shortcite{Andrews:TransmissionDynamicsControl:2011}.
A spatially-explicit model has been developed at the ECHO laboratory in the past 10 years\cite{Bertuzzo:SpacetimeEvolutionCholera:2008}. It has been used to studies on the dynamics of several cholera epidemics, such as in South Africa in 2000\cite{Mari:ModellingCholeraEpidemics:2012}, Senegal in 2005, Haiti from 2010 to 2019\cite{Bertuzzo:PredictionSpatialEvolution:2011,Bertuzzo:ProbabilityExtinctionHaiti:2016}, Democratic Republic of the Congo from 2004 to 2011 and many others\cite{Finger:PotentialImpactCasearea:2018}.
A complete formulation of the model is presented here, including vaccination and human and hydrologic mobility\cite{Bertuzzo:ProbabilityExtinctionHaiti:2016,Pasetto:RealtimeForecastingCholera:2018}. This model has inspired to a various degree all the other models described in this thesis.
The area potentially concerned by the epidemic is divided into $n$ regions. The sub-regions, defined by political boundaries or geomorphological features (like watersheds\cite{Bertuzzo:ProbabilityExtinctionHaiti:2016}), are represented as connected nodes. Each of the $n$ nodes represent a human community with population $H_i$, $i=1,\dots, n$.
The model is a variation of the SIR model introduced first by Kermack and McKendrick\cite{Kermack:ContributionMathematicalTheory:1927}, with additional compartments for the vaccinated individuals and the bacteria concentration in the environment. At time $t$ and for each node $i$, the individuals are divided into six compartments:
\begin{itemize}
\item $S_i(t)$: susceptible individuals have no immunity and may enter in contact with the bacteria and become infected (symptomatic or not),
\item $I_i(t)$: infected and infectious individual, that shed bacteria into the community reservoir,
\item $R_i(t)$: recovered are temporally immune, and don't participate in disease transmission,
\item $V^S_i(t)$: vaccinated susceptible (as the oral cholera vaccines do no offer total protection against infection),
\item $V^I_i(t)$: vaccinated infected,
\item $V^R_i(t)$: vaccinated recovered.
\end{itemize}
In addition, the model considers the bacterial concentration of \textit{V.~cholerae} in the water reservoir of the community, $B_i(t)$.
The $n$ nodes are connected by both human mobility and pathogen transport through water. Individuals commute from node $i$ to node $j$ with probability $Q_{ij}$. Bacteria are transported along with the river network from node $i$ to node $j$ with probability $P_{ij}$.
The cholera dynamics are described by the following set of coupled ordinary differential equations:
\begin{fullwidth}
\begingroup
\allowdisplaybreaks
\begin{eqnarray}
\frac{dI_i}{dt} &=& \sigma F_i(t) S_i - (\gamma + \mu + \alpha) I_i \label{eq:I2}\\
\frac{dR_i}{dt} &=& (1-\sigma) F_i(t) S_i + \gamma I_i - (\rho + \mu+\frac{\nu_i(t)}{S_i+R_i}) R_i \label{eq:R2}\\
\frac{dV^S_i}{dt} &=& \nu_i(t) \frac{S_i}{S_i+R_i}-\mu V^S_i \label{eq:VS2}\\
\frac{dV^I_i}{dt} &=& \sigma (1-\eta) F_i(t) V^S_i - (\gamma + \mu + \alpha) V^I_i \label{eq:VI2}\\
\frac{dV^R_i}{dt} &=& \nu_i(t) \frac{R_i}{S_i+R_i} + (1-\sigma) (1-\eta) F_i(t) V^S_i + \gamma V^I_i - (\mu+\rho_v) V^R_i \label{eq:VR2}\\
\frac{dB_i}{dt} &=& - \mu_B B_i +\frac{p}{W_i}\left[1 + \lambda J_i(t) \right] \left((1-m)(I_i +V_i^I)+m \sum_{j=1}^n Q_{ij} (I_j +V_j^I)\right)- l \left( B_i - \sum_{j=1}^n P_{ji} \frac{W_j}{W_i} B_j \right).
\end{eqnarray}
\endgroup
\end{fullwidth}
Here demographic equilibrium is assumed, hence the population~$H_i$ of each node is assumed to be constant, which implies that the number of susceptible individuals at time $t$ is $S_i(t) = H_i - I_i(t) - R_i(t) - V_i^S(t) - V^I_i(t) - V_i^R(t)$, but it is also possible to model $\frac{dS_i}{dt}$ dynamically.
The force of infection represents the rate at which an individual enters in contact with the disease. It is written as:
\begin{equation}
F_i(t) = \beta \left[ (1 - m) \frac{B_i}{K + B_i} + m \sum_{j=1}^n Q_{ij} \frac{B_j}{K + B_j} \right].
\label{force}
\end{equation}
The parameter~$\beta$ represents the maximum exposure rate (and is proportional to the basic reproduction number $R_0$, see \textsc{Chapter 5}), which may vary in time due to non-pharmaceutical interventions or the awareness of the population\cite{Bertuzzo:ProbabilityExtinctionHaiti:2016}. The fraction $B_{i}/(K+B_{i})$\footnote{In the classical SIR model, this fraction would be $\frac{I}{H}$, a mass action reaction on the proportion of infected in the population.} is the probability of becoming infected due to the exposure to a concentration~$B_i$ of \textit{V.~cholerae}, $K$ being the half-saturation constant\cite{Codeco:EndemicEpidemicDynamics:2001}. The force of infection in a given node depends for a fraction ($1-m$) to the local concentration of \textit{V.~cholerae} $B_i$. The remaining fraction $m$, the community-level probability that individuals travel outside their node, are exposed to the concentration~$B_j$ of the remote communities.
The human mobility is encoded in matrix $Q$, where $Q_{ij}$ represents the probability that a mobile individual living in node $i$ reaches~$j$ as a destination. Because of human mobility, a susceptible individual residing at node $i$ can be exposed to pathogens in the destination community $j$.
In the lack of detailed mobility data, the probabilities~$Q_{ij}$ can be estimated through a gravity model of human mobility\cite{Erlander:GravityModelTransportation:1990}:
\begin{equation}
Q_{ij} = \frac{H_j e^{-d_{ij}/D}}{\sum_{k \neq i}^n H_k e^{-d_{ik}/D}} \, ,
\label{eq:mob}
\end{equation}
where the attractiveness of node~$j$ depends on its population size $H_j$, while the deterrence factor is assumed to be dependent on the distance~$d_{ij}$ between the two communities via an exponential kernel (with shape factor~$D$).
Individuals are removed from the susceptible compartment $S$ at rate $F(t)$. A fraction $\sigma$ of the infected individuals develops symptoms, passing from compartment $S$ to $I$ and shed \textit{V.~cholerae} into the reservoir at rate $\theta$. The remaining fraction~$(1-\sigma)$ does not develop symptoms and does not contribute to the disease transmission\footnote{In the model presented in \textsc{Chapter~3}, asymptomatic individuals are assumed to contribute to disease transmission, and are explicitly modeled in a compartment $A$.}, and is considered temporally immune, thus passing from compartment $S$ to $R$. Symptomatic infected individuals recover at a rate~$\gamma$\footnote{Hence the mean infectious period is $\frac{1}{\gamma}$ [duration]. Correspondingly the life expectancy without cholera would be $\frac{1}{\mu}$.}, or die due to cholera or other causes at rates~$\alpha$ or $\mu$, respectively.
Recovered individuals lose their immunity at rate~$\rho$, the average rate of loss of immunity for individuals that previously had been asymptomatic or symptomatic, or die at a rate~$\mu$. In both cases, they return to susceptible (death is associated with the birth of a susceptible to keep the population constant).
The environmental concentration of \textit{V.~cholerae} $B_i$ may increase due to infected individuals shedding \textit{Vibrios} and hydrologic travel. As for the force of infection, bacterial shedding is proportional for a fraction $1-m$ on the local infected individuals and for a fraction $m$ on the infected individuals moving according to the mobility network. The increase in bacteria concentration is modeled with the rate $p/W_i$, where $p$ is the rate at which bacteria excreted by an infected individual reach and contaminate the local water reservoir of volume $W_i$ (assumed to be proportional to population size, i.e., $W_i=c H_i$\cite{Rinaldo:Reassessment20102011:2012}). Rainfall-induced runoff might cause additional pathogen loads to enter the water reservoir due to effects such as the overflow of pit latrines and washout of open-air defecation sites. The contamination rate $p$ is thus intensified by the rainfall intensity $J_i(t)$ via a coefficient $\lambda$\cite{Rinaldo:Reassessment20102011:2012,Righetto:RainfallMediationsSpreading:2013}. By introducing the dimensionless bacterial concentrations $B_i^*=B_i/K$, it is possible to group three model parameters into a single shedding intensity ratio $\theta=p/(cK)$\cite{Bertuzzo:SpacetimeEvolutionCholera:2008}. Bacteria undergo hydrologic dispersal at a rate~$l$: pathogens travel from node~$i$ to~$j$ with probability $P_{ij}$, which is assumed to be one if node~$j$ is the downstream nearest neighborhood $i$, and zero otherwise. \textit{V.~cholerae} decays at rate $\mu_B$ in the water reservoir.
The estimation of the local incidence is computed integrating over the new symptomatic individuals,
\begin{equation}
\frac{d C_i}{dt} \ = \ \sigma F_i S_i \, , \label{eq:C}
\end{equation}
During the vaccination campaign, oral cholera vaccines doses are distributed at rate $\nu_i(t)$ to susceptible and recovered individuals, which enter the compartments $V^S$ and $V^R$. As the OCV provides a partial immunity having efficacy $\eta$, $0\leq \eta \leq 1$, vaccinated susceptibles ($V^S$) can become infected ($V^I$) through a decreased force of infection of a factor $(1-\eta)$ with respect to non-vaccinated individuals. Vaccinated infected individuals behave exactly like infected ones, but are placed in a different compartment to exclude them from future vaccination campaigns. After recovering at rate $\gamma$, they lose their vaccine protection at rate $\rho_{v}$.
Derivations of this model are used in the next 2 chapters. Hydrological transport is always neglected ($P_{ij}=0, \forall i,j$), and other differences are:
\begin{itemize}
\item in \textsc{Chapter~2}: the model is very similar, with only one spatial node. Compartment for the incubating individuals $E$ and $V_E$ are added because of the higher temporal resolution of the date. Moreover, a precise non-linear formulation of the effect of rainfall is developed. The force of infection is further generalized to encompass non-linear rainfall effects and human-to-human transmission. Finally, alongside the present ordinary differential equations formulation, a version of the model with stochastic transitions is developed to better deal with low incidence data.
\item in \textsc{Chapter~3}: the model is also a very similar stochastic translation of the present model, with mobility (but no hydrological transport) implemented as human-to-human transmission. Shedding (infectious) asymptomatic are modeled in an additional compartment $A$. The effect of vaccination is described very precisely with vaccine efficacy waning over time. As the effect of synchronized immunity is sought after, the time spent recovered is more accurately represented as an erlang distribution with parameter 3. The rainfall uses the same non-linear effect introduced in \textsc{Chapter~2}.
\end{itemize}
The \textsc{covid}-19 models proposed later are very different but share some features \eg the mobility formulation is close in \textsc{Chapter~4} and \textsc{Chapter~6}, and more generally the general structure is similar across all compartmental models.
%ffective targeted interventions
%could eliminate 50\% of the region’s cholera by covering 35·3 million people (95% CrI 26·3 million to 62·0 million),
%which is less than 4\% of the total population\cite{lessler_mapping_2018} + hotspot vs optimal strategies
%hotspot\cite{azman_micro-hotspots_2018}
%The Dry Season in Haiti: a Window of Opportunity to Eliminate Cholera\cite{rebaudet_dry_2013}
%\paragraph{Interventions design} The planning of the interventions described above is difficult to establish, as many factors enter into consideration including logistical and political constraints. Expert opinion is a valuable resource, however its use during outbreaks is not necessarily feasible and a consensus on strategy choice does not always emerges\cite{Cyranoski:CholeraVaccinePlan:2011}. Moreover, the global vaccine shortage\cite{Parker:AdaptingGlobalShortage:2017a,Seidlein:PreventingCholeraOutbreaks:2018} calls for an optimal use of current resources. Mass vaccination campaigns should be conducted only when strictly necessary. Case-area targeted interventions (CATIs) are an effective way of mitigating an outbreak while saving scarce resources. However, the optimal allocation in time and space of such interventions is strongly context-dependent which hinders the definition of general guidelines\cite{Eubank:ModellingDiseaseOutbreaks:2004,Finger:PotentialImpactCasearea:2018,Seidlein:PreventingCholeraOutbreaks:2018,Azman:MicrohotspotsRiskUrban:2018,Lessler:MappingBurdenCholera:2018,Rebaudet:DrySeasonHaiti:2013}.
%This thesis will explore the possibility to apply optimal control for both short- and long-term interventions across all scales.