Multiple Transmission Pathways and Disease Dynamics in a Waterborne Pathogen Model

A new epidemic model with indirect transmission

We consider an epidemic model in which all disease transmission is through shedding of virus by infectives and acquisition by susceptibles, rather than by direct contact. This leads to an susceptible-infectious-virus-removed (SIVR) model for which we can determine the basic reproduction number and the final size relation. We extend the model to an age of infection model with virus shedding a function of the age of infection.

Reproduction numbers of infectious disease models

This primer article focuses on the basic reproduction number, ℛ 0 , for infectious diseases, and other reproduction numbers related to ℛ 0 that are useful in guiding control strategies. Beginning with a simple population model, the concept is developed for a threshold value of ℛ 0 determining whether or not the disease dies out. The next generation matrix method of calculating ℛ 0 in a compartmental model is described and illustrated. To address control strategies, type and target reproduction numbers are defined, as well as sensitivity and elasticity indices. These theoretical ideas are then applied to models that are formulated for West Nile virus in birds (a vector-borne disease), cholera in humans (a disease with two transmission pathways), anthrax in animals (a disease that can be spread by dead carcasses and spores), and Zika in humans (spread by mosquitoes and sexual contacts). Some parameter values from literature data are used to illustrate the results. Finally, references for other ways to calculate ℛ 0 are given. These are useful for more complicated models that, for example, take account of variations in environmental fluctuation or stochasticity.

Impact of Awareness Programs on Cholera Dynamics: Two Modeling Approaches

We propose two differential equation-based models to investigate the impact of awareness programs on cholera dynamics. The first model represents the disease transmission rates as decreasing functions of the number of awareness programs, whereas the second model divides the susceptible individuals into two distinct classes depending on their awareness/unawareness of the risk of infection. We study the essential dynamical properties of each model, using both analytical and numerical approaches. We find that the two models, though closely related, exhibit significantly different dynamical behaviors. Namely, the first model follows regular threshold dynamics while rich dynamical behaviors such as backward bifurcation may arise from the second one. Our results highlight the importance of validating key modeling assumptions in the development and selection of mathematical models toward practical application.

Disease ecology, health and the environment: a framework to account for ecological and socio-economic drivers in the control of neglected tropical diseases

Reducing the burden of neglected tropical diseases (NTDs) is one of the key strategic targets advanced by the Sustainable Development Goals. Despite the unprecedented effort deployed for NTD elimination in the past decade, their control, mainly through drug administration, remains particularly challenging: persistent poverty and repeated exposure to pathogens embedded in the environment limit the efficacy of strategies focused exclusively on human treatment or medical care. Here, we present a simple modelling framework to illustrate the relative role of ecological and socio-economic drivers of environmentally transmitted parasites and pathogens. Through the analysis of system dynamics, we show that periodic drug treatments that lead to the elimination of directly transmitted diseases may fail to do so in the case of human pathogens with an environmental reservoir. Control of environmentally transmitted diseases can be more effective when human treatment is complemented with interventions targeting the environmental reservoir of the pathogen. We present mechanisms through which the environment can influence the dynamics of poverty via disease feedbacks. For illustration, we present the case studies of Buruli ulcer and schistosomiasis, two devastating waterborne NTDs for which control is particularly challenging.This article is part of the themed issue 'Conservation, biodiversity and infectious disease: scientific evidence and policy implications'.

Challenges in developing methods for quantifying the effects of weather and climate on water-associated diseases: A systematic review

Infectious diseases attributable to unsafe water supply, sanitation and hygiene (e.g. Cholera, Leptospirosis, Giardiasis) remain an important cause of morbidity and mortality, especially in low-income countries. Climate and weather factors are known to affect the transmission and distribution of infectious diseases and statistical and mathematical modelling are continuously developing to investigate the impact of weather and climate on water-associated diseases. There have been little critical analyses of the methodological approaches. Our objective is to review and summarize statistical and modelling methods used to investigate the effects of weather and climate on infectious diseases associated with water, in order to identify limitations and knowledge gaps in developing of new methods. We conducted a systematic review of English-language papers published from 2000 to 2015. Search terms included concepts related to water-associated diseases, weather and climate, statistical, epidemiological and modelling methods. We found 102 full text papers that met our criteria and were included in the analysis. The most commonly used methods were grouped in two clusters: process-based models (PBM) and time series and spatial epidemiology (TS-SE). In general, PBM methods were employed when the bio-physical mechanism of the pathogen under study was relatively well known (e.g. Vibrio cholerae); TS-SE tended to be used when the specific environmental mechanisms were unclear (e.g. Campylobacter). Important data and methodological challenges emerged, with implications for surveillance and control of water-associated infections. The most common limitations comprised: non-inclusion of key factors (e.g. biological mechanism, demographic heterogeneity, human behavior), reporting bias, poor data quality, and collinearity in exposures. Furthermore, the methods often did not distinguish among the multiple sources of time-lags (e.g. patient physiology, reporting bias, healthcare access) between environmental drivers/exposures and disease detection. Key areas of future research include: disentangling the complex effects of weather/climate on each exposure-health outcome pathway (e.g. person-to-person vs environment-to-person), and linking weather data to individual cases longitudinally.

Modeling Key Drivers of Cholera Transmission Dynamics Provides New Perspectives for Parasitology

Hydroclimatological and anthropogenic factors are key drivers of waterborne disease transmission. Information on human settlements and host mobility on waterways along which pathogens and hosts disperse, and relevant hydroclimatological processes, can be acquired remotely and included in spatially explicit mathematical models of disease transmission. In the case of epidemic cholera, such models allowed the description of complex disease patterns and provided insight into the course of ongoing epidemics. The inclusion of spatial information in models of disease transmission can aid in emergency management and the assessment of alternative interventions. Here, we review the study of drivers of transmission via spatially explicit approaches and argue that, because many parasitic waterborne diseases share the same drivers as cholera, similar principles may apply.

Who acquires infection from whom and how? Disentangling multi-host and multi-mode transmission dynamics in the 'elimination' era

Multi-host infectious agents challenge our abilities to understand, predict and manage disease dynamics. Within this, many infectious agents are also able to use, simultaneously or sequentially, multiple modes of transmission. Furthermore, the relative importance of different host species and modes can itself be dynamic, with potential for switches and shifts in host range and/or transmission mode in response to changing selective pressures, such as those imposed by disease control interventions. The epidemiology of such multi-host, multi-mode infectious agents thereby can involve a multi-faceted community of definitive and intermediate/secondary hosts or vectors, often together with infectious stages in the environment, all of which may represent potential targets, as well as specific challenges, particularly where disease elimination is proposed. Here, we explore, focusing on examples from both human and animal pathogen systems, why and how we should aim to disentangle and quantify the relative importance of multi-host multi-mode infectious agent transmission dynamics under contrasting conditions, and ultimately, how this can be used to help achieve efficient and effective disease control.This article is part of the themed issue 'Opening the black box: re-examining the ecology and evolution of parasite transmission'.

Epidemic characteristics of two classic SIS models with disease-induced death

The epidemic characteristics of two classic SIS epidemic models, including the epidemic size, peak and turning point, are investigated. The two SIS models are with bilinear and standard incidences, respectively. For the SIS models, the susceptible individuals generally can be divided into two classes. One consists of the individuals who had not been infected by the infection, the other are individuals who have been infected and recovered from the infection. Based on this fact, the classic SIS epidemic models need to be reformulated in order to analyze the turning points of the epidemic for various cumulative cases in detail. The obtained results illustrate how to determine the epidemic characteristics of the two models, and demonstrate their dependence on the initial conditions and the relative parameters of the models.

Dose-response relationships for environmentally mediated infectious disease transmission models

Environmentally mediated infectious disease transmission models provide a mechanistic approach to examining environmental interventions for outbreaks, such as water treatment or surface decontamination. The shift from the classical SIR framework to one incorporating the environment requires codifying the relationship between exposure to environmental pathogens and infection, i.e. the dose-response relationship. Much of the work characterizing the functional forms of dose-response relationships has used statistical fit to experimental data. However, there has been little research examining the consequences of the choice of functional form in the context of transmission dynamics. To this end, we identify four properties of dose-response functions that should be considered when selecting a functional form: low-dose linearity, scalability, concavity, and whether it is a single-hit model. We find that i) middle- and high-dose data do not constrain the low-dose response, and different dose-response forms that are equally plausible given the data can lead to significant differences in simulated outbreak dynamics; ii) the choice of how to aggregate continuous exposure into discrete doses can impact the modeled force of infection; iii) low-dose linear, concave functions allow the basic reproduction number to control global dynamics; and iv) identifiability analysis offers a way to manage multiple sources of uncertainty and leverage environmental monitoring to make inference about infectivity. By applying an environmentally mediated infectious disease model to the 1993 Milwaukee Cryptosporidium outbreak, we demonstrate that environmental monitoring allows for inference regarding the infectivity of the pathogen and thus improves our ability to identify outbreak characteristics such as pathogen strain.

Global stability and uniform persistence of the reaction-convection-diffusion cholera epidemic model

We study the global stability issue of the reaction-convection-diffusion cholera epidemic PDE model and show that the basic reproduction number serves as a threshold parameter that predicts whether cholera will persist or become globally extinct. Specifically, when the basic reproduction number is beneath one, we show that the disease-free-equilibrium is globally attractive. On the other hand, when the basic reproduction number exceeds one, if the infectious hosts or the concentration of bacteria in the contaminated water are not initially identically zero, we prove the uniform persistence result and that there exists at least one positive steady state.

Disease dynamics in a coupled cholera model linking within-host and between-host interactions

A new modelling framework is proposed to study the within-host and between-host dynamics of cholera, a severe intestinal infection caused by the bacterium Vibrio cholerae. The within-host dynamics are characterized by the growth of highly infectious vibrios inside the human body. These vibrios shed from humans contribute to the environmental bacterial growth and the transmission of the disease among humans, providing a link from the within-host dynamics at the individual level to the between-host dynamics at the population and environmental level. A fast-slow analysis is conducted based on the two different time scales in our model. In particular, a bifurcation study is performed, and sufficient and necessary conditions are derived that lead to a backward bifurcation in cholera epidemics. Our result regarding the backward bifurcation highlights the challenges in the prevention and control of cholera.

Model distinguishability and inference robustness in mechanisms of cholera transmission and loss of immunity

Mathematical models of cholera and waterborne disease vary widely in their structures, in terms of transmission pathways, loss of immunity, and a range of other features. These differences can affect model dynamics, with different models potentially yielding different predictions and parameter estimates from the same data. Given the increasing use of mathematical models to inform public health decision-making, it is important to assess model distinguishability (whether models can be distinguished based on fit to data) and inference robustness (whether inferences from the model are robust to realistic variations in model structure). In this paper, we examined the effects of uncertainty in model structure in the context of epidemic cholera, testing a range of models with differences in transmission and loss of immunity structure, based on known features of cholera epidemiology. We fit these models to simulated epidemic and long-term data, as well as data from the 2006 Angola epidemic. We evaluated model distinguishability based on fit to data, and whether the parameter values, model behavior, and forecasting ability can accurately be inferred from incidence data. In general, all models were able to successfully fit to all data sets, both real and simulated, regardless of whether the model generating the simulated data matched the fitted model. However, in the long-term data, the best model fits were achieved when the loss of immunity structures matched those of the model that simulated the data. Two parameters, one representing person-to-person transmission and the other representing the reporting rate, were accurately estimated across all models, while the remaining parameters showed broad variation across the different models and data sets. The basic reproduction number (R₀) was often poorly estimated even using the correct model, due to practical unidentifiability issues in the waterborne transmission pathway which were consistent across all models. Forecasting efforts using noisy data were not successful early in the outbreaks, but once the epidemic peak had been achieved, most models were able to capture the downward incidence trajectory with similar accuracy. Forecasting from noise-free data was generally successful for all outbreak stages using any model. Our results suggest that we are unlikely to be able to infer mechanistic details from epidemic case data alone, underscoring the need for broader data collection, such as immunity/serology status, pathogen dose response curves, and environmental pathogen data. Nonetheless, with sufficient data, conclusions from forecasting and some parameter estimates were robust to variations in the model structure, and comparative modeling can help to determine how realistic variations in model structure may affect the conclusions drawn from models and data.

Optimal allocation of the limited oral cholera vaccine supply between endemic and epidemic settings

The World Health Organization (WHO) recently established a global stockpile of oral cholera vaccine (OCV) to be preferentially used in epidemic response (reactive campaigns) with any vaccine remaining after 1 year allocated to endemic settings. Hence, the number of cholera cases or deaths prevented in an endemic setting represents the minimum utility of these doses, and the optimal risk-averse response to any reactive vaccination request (i.e. the minimax strategy) is one that allocates the remaining doses between the requested epidemic response and endemic use in order to ensure that at least this minimum utility is achieved. Using mathematical models, we find that the best minimax strategy is to allocate the majority of doses to reactive campaigns, unless the request came late in the targeted epidemic. As vaccine supplies dwindle, the case for reactive use of the remaining doses grows stronger. Our analysis provides a lower bound for the amount of OCV to keep in reserve when responding to any request. These results provide a strategic context for the fulfilment of requests to the stockpile, and define allocation strategies that minimize the number of OCV doses that are allocated to suboptimal situations.

The potential for sexual transmission to compromise control of Ebola virus outbreaks

Recent evidence suggests that sexual contact may give rise to transmission of Ebola virus long after infection has been cleared from blood. We develop a simple mathematical model that incorporates contact transmission and sexual transmission parametrized from data relating to the 2013-2015 West African Ebola epidemic. The model explores scenarios where contact transmission is reduced following infection events, capturing behaviour change, and quantifies how these actions reducing transmission may be compromised by sexual transmission in terms of increasing likelihood, size and duration of outbreaks. We characterize the extent to which sexual transmission operates in terms of the probability of initial infection resolving to sexual infectiousness and the sexual transmission rate, and relate these parameters to the overall case burden. We find that sexual transmission can have large effects on epidemic dynamics (increasing attack ratios from 25% in scenarios without sexual transmission but with contact-transmission-reducing behaviour, up to 80% in equivalent scenarios with sexual transmission).

Lethal exposure: An integrated approach to pathogen transmission via environmental reservoirs

To mitigate the effects of zoonotic diseases on human and animal populations, it is critical to understand what factors alter transmission dynamics. Here we assess the risk of exposure to lethal concentrations of the anthrax bacterium, Bacillus anthracis, for grazing animals in a natural system over time through different transmission mechanisms. We follow pathogen concentrations at anthrax carcass sites and waterholes for five years and estimate infection risk as a function of grass, soil or water intake, age of carcass sites, and the exposure required for a lethal infection. Grazing, not drinking, seems the dominant transmission route, and transmission is more probable from grazing at carcass sites 1-2 years of age. Unlike most studies of virulent pathogens that are conducted under controlled conditions for extrapolation to real situations, we evaluate exposure risk under field conditions to estimate the probability of a lethal dose, showing that not all reservoirs with detectable pathogens are significant transmission pathways.

PREDICTIVE MODELING OF CHOLERA OUTBREAKS IN BANGLADESH

Despite seasonal cholera outbreaks in Bangladesh, little is known about the relationship between environmental conditions and cholera cases. We seek to develop a predictive model for cholera outbreaks in Bangladesh based on environmental predictors. To do this, we estimate the contribution of environmental variables, such as water depth and water temperature, to cholera outbreaks in the context of a disease transmission model. We implement a method which simultaneously accounts for disease dynamics and environmental variables in a Susceptible-Infected-Recovered-Susceptible (SIRS) model. The entire system is treated as a continuous-time hidden Markov model, where the hidden Markov states are the numbers of people who are susceptible, infected, or recovered at each time point, and the observed states are the numbers of cholera cases reported. We use a Bayesian framework to fit this hidden SIRS model, implementing particle Markov chain Monte Carlo methods to sample from the posterior distribution of the environmental and transmission parameters given the observed data. We test this method using both simulation and data from Mathbaria, Bangladesh. Parameter estimates are used to make short-term predictions that capture the formation and decline of epidemic peaks. We demonstrate that our model can successfully predict an increase in the number of infected individuals in the population weeks before the observed number of cholera cases increases, which could allow for early notification of an epidemic and timely allocation of resources.

Eradication of Ebola Based on Dynamic Programming

This paper mainly studies the eradication of the Ebola virus, proposing a scientific system, including three modules for the eradication of Ebola virus. Firstly, we build a basic model combined with nonlinear incidence rate and maximum treatment capacity. Secondly, we use the dynamic programming method and the Dijkstra Algorithm to set up M-S (storage) and several delivery locations in West Africa. Finally, we apply the previous results to calculate the total cost, production cost, storage cost, and shortage cost.

The impact of spatial arrangements on epidemic disease dynamics and intervention strategies

The role of spatial arrangements on the spread and management strategies of a cholera epidemic is investigated. We consider the effect of human and pathogen movement on optimal vaccination strategies. A metapopulation model is used, incorporating a susceptible-infected-recovered system of differential equations coupled with an equation modelling the concentration of Vibrio cholerae in an aquatic reservoir. The model compared spatial arrangements and varying scenarios to draw conclusions on how to effectively manage outbreaks. The work is motivated by the 2010 cholera outbreak in Haiti. Results give guidance for vaccination strategies in response to an outbreak.

A multigroup model for cholera dynamics and control

We propose a general multigroup model for cholera dynamics that involves both direct and indirect transmission pathways and that incorporates spatial heterogeneity. Under biologically feasible conditions, we show that the basic reproduction number R0 remains a sharp threshold for cholera dynamics in multigroup settings. We verify the analysis by numerical simulation results. We also perform an optimal control study to explore optimal vaccination strategy for cholera outbreaks.

Analysis of a waterborne disease model with socioeconomic classes

Waterborne diseases such as cholera continue to pose serious public health problems in the world today. Transmission parameters can vary greatly with socioeconomic class (SEC) and the availability of clean water. We formulate a multi-patch waterborne disease model such that each patch represents a particular SEC with its own water source, allowing individuals to move between SECs. For a 2-SEC model, we investigate the conditions under which each SEC is responsible for driving a cholera outbreak. We determine the effect of SECs on disease transmission dynamics by comparing the basic reproduction number of the 2-SEC model to that of a homogeneous model that does not take SECs into account. We conclude by extending several results of the 2-SEC model to an n-SEC model.

Influence of human behavior on cholera dynamics

This paper is devoted to studying the impact of human behavior on cholera infection. We start with a cholera ordinary differential equation (ODE) model that incorporates human behavior via modeling disease prevalence dependent contact rates for direct and indirect transmissions and infectious host shedding. Local and global dynamics of the model are analyzed with respect to the basic reproduction number. We then extend the ODE model to a reaction-convection-diffusion partial differential equation (PDE) model that accounts for the movement of both human hosts and bacteria. Particularly, we investigate the cholera spreading speed by analyzing the traveling wave solutions of the PDE model, and disease threshold dynamics by numerically evaluating the basic reproduction number of the PDE model. Our results show that human behavior can reduce (a) the endemic and epidemic levels, (b) cholera spreading speeds and (c) the risk of infection (characterized by the basic reproduction number).

MODELING THE EFFECT OF VACCINES ON CHOLERA TRANSMISSION

Cholera is a diarrhoeal disease that is caused by an intestinal bacterium, Vibrio cholerae. Recently an outbreak of cholera in Haiti brought public attention to this deadly disease. In this work, the goal of our differential equation model is to find an effective optimal vaccination strategy to minimize the disease related mortality and to reduce the associated costs. The effect of seasonality in pathogen transmission on vaccination strategies was investigated under several types of disease scenarios, including an endemic case and a new outbreak case. This model is an extension of a general water-borne pathogen model. This work involves the optimal control problem formulation, analysis and numerical simulations.

Analyzing transmission dynamics of cholera with public health interventions

Cholera continues to be a serious public health concern in developing countries and the global increase in the number of reported outbreaks suggests that activities to control the diseases and surveillance programs to identify or predict the occurrence of the next outbreaks are not adequate. These outbreaks have increased in frequency, severity, duration and endemicity in recent years. Mathematical models for infectious diseases play a critical role in predicting and understanding disease mechanisms, and have long provided basic insights in the possible ways to control infectious diseases. In this paper, we present a new deterministic cholera epidemiological model with three types of control measures incorporated into a cholera epidemic setting: treatment, vaccination and sanitation. Essential dynamical properties of the model with constant intervention controls which include local and global stabilities for the equilibria are carefully analyzed. Further, using optimal control techniques, we perform a study to investigate cost-effective solutions for time-dependent public health interventions in order to curb disease transmission in epidemic settings. Our results show that the basic reproductive number (R0) remains the model's epidemic threshold despite the inclusion of a package of cholera interventions. For time-dependent controls, the results suggest that these interventions closely interplay with each other, and the costs of controls directly affect the length and strength of each control in an optimal strategy.

An evolutionary computing approach for parameter estimation investigation of a model for cholera

We consider the problem of using time-series data to inform a corresponding deterministic model and introduce the concept of genetic algorithms (GA) as a tool for parameter estimation, providing instructions for an implementation of the method that does not require access to special toolboxes or software. We give as an example a model for cholera, a disease for which there is much mechanistic uncertainty in the literature. We use GA to find parameter sets using available time-series data from the introduction of cholera in Haiti and we discuss the value of comparing multiple parameter sets with similar performances in describing the data.

From the bench to modeling--R0 at the interface between empirical and theoretical approaches in epidemiology of environmentally transmitted infectious diseases

For environmentally transmitted infectious diseases (ETIDs), which spread through the contaminated environment (such as foods, surfaces and fomites), there is a lack of consensus about the mathematical approach to derive R0, leading to inconsistent predictions about the spread and control of these infections in their host populations. The objective of this study was to explain three current, though conflicting, approaches to derive a theoretical expression for R0 for ETIDs and assess their validity through comparison with available empirical data. Salmonella Typhimurium in laboratory mice was used as a theoretical and empirical model system. The three conflicting theoretical expressions for R0 were derived using the next generation matrix approach according to three unverified hypotheses about the role of the environmental phase in the transmission of ETIDs. The hypotheses assume that the environment contaminated with a pathogen is: (H1) an extension of the host's infectious period, (H2) a reservoir for the infectious agent, or (H3) has both of these characteristics. For the parameter values describing the empirical model system, the theoretical values of R0 corresponding to the hypotheses H2 and H3 were very similar (1.38 and 1.62, respectively) and their values were approximately half of the value of R0 for hypothesis H1 (2.94). The theoretical R0 values were compared with an empirical R0 (1.58, 95% confidence interval: 1.14, 2.02) estimated using a Martingale method from published experimental data on Salmonella Typhimurium transmission in mice. The results of the comparison suggested that hypothesis H1 is unlikely to hold true but it could not be rejected with confidence because of uncertainty in the value of the pathogen growth rate in the environment. The hypotheses H2 and H3 were both equally strongly supported with the empirical data suggesting that either of them could be valid. A sensitivity analysis identified critical information gaps about the indirect transmission rate of infection and the pathogen growth rate in the environment. Moreover, we identified experimental conditions for which the theoretical R0 predictions based on the hypotheses H2 and H3 differ greatly, which would assist their discrimination and conclusive validation against future empirical studies. Once a valid theoretical R0 is identified for Salmonella Typhimurium in mice, its generalizability to other host-pathogen-environment systems should be tested. The present study may serve as a template for integrated empirical and theoretical research of R0 in the epidemiology of ETIDs.

Analysis of cholera epidemics with bacterial growth and spatial movement

In this work, we propose novel epidemic models (named, susceptible-infected-recovered-susceptible-bacteria) for cholera dynamics by incorporating a general formulation of bacteria growth and spatial variation. In the first part, a generalized ordinary differential equation (ODE) model is presented and it is found that bacterial growth contributes to the increase in the basic reproduction number, [Formula: see text]. With the derived basic reproduction number, we analyse the local and global dynamics of the model. Particularly, we give a rigorous proof on the endemic global stability by employing the geometric approach. In the second part, we extend the ODE model to a partial differential equation (PDE) model with the inclusion of diffusion to capture the movement of human hosts and bacteria in a heterogeneous environment. The disease threshold of this PDE model is studied again by using the basic reproduction number. The results on the threshold dynamics of the ODE and PDE models are compared, and verified through numerical simulation. Additionally, our analysis shows that incorporating diffusive spatial spread does not produce a Turing instability when [Formula: see text] associated with the ODE model is less than the unity.

Model for disease dynamics of a waterborne pathogen on a random network

A network epidemic SIWR model for cholera and other diseases that can be transmitted via the environment is developed and analyzed. The person-to-person contacts are modeled by a random contact network, and the contagious environment is modeled by an external node that connects to every individual. The model is adapted from the Miller network SIR model, and in the homogeneous mixing limit becomes the Tien and Earn deterministic cholera model without births and deaths. The dynamics of our model shows excellent agreement with stochastic simulations. The basic reproduction number [Formula: see text] is computed, and on a Poisson network shown to be the sum of the basic reproduction numbers of the person-to-person and person-to-water-to-person transmission pathways. However, on other networks, [Formula: see text] depends nonlinearly on the transmission along the two pathways. Type reproduction numbers are computed and quantify measures to control the disease. Equations giving the final epidemic size are obtained.

Estimating the probability of an extinction or major outbreak for an environmentally transmitted infectious disease

Indirect transmission through the environment, pathogen shedding by infectious hosts, replication of free-living pathogens within the environment, and environmental decontamination are suspected to play important roles in the spread and control of environmentally transmitted infectious diseases. To account for these factors, the classic Susceptible-Infectious-Recovered-Susceptible epidemic model is modified to include a compartment representing the amount of free-living pathogen within the environment. The model accounts for host demography, direct and indirect transmission, replication of free-living pathogens in the environment, and removal of free-living pathogens by natural death or environmental decontamination. Based on the assumptions of the deterministic model, a continuous-time Markov chain model is developed. An estimate for the probability of disease extinction or a major outbreak is obtained by approximating the Markov chain with a multitype branching process. Numerical simulations illustrate important differences between the deterministic and stochastic counterparts, relevant for outbreak prevention, that depend on indirect transmission, pathogen shedding by infectious hosts, replication of free-living pathogens, and environmental decontamination. The probability of a major outbreak is computed for salmonellosis in a herd of dairy cattle as well as cholera in a human population. An explicit expression for the probability of disease extinction or a major outbreak in terms of the model parameters is obtained for systems with no direct transmission or replication of free-living pathogens.

Modelling and control of cholera on networks with a common water source

A mathematical model is formulated for the transmission and spread of cholera in a heterogeneous host population that consists of several patches of homogeneous host populations sharing a common water source. The basic reproduction number ℛ0 is derived and shown to determine whether or not cholera dies out. Explicit formulas are derived for target/type reproduction numbers that measure the control strategies required to eradicate cholera from all patches.

Traveling wave solutions for epidemic cholera model with disease-related death

Based on Codeço's cholera model (2001), an epidemic cholera model that incorporates the pathogen diffusion and disease-related death is proposed. The formula for minimal wave speed c^∗ is given. To prove the existence of traveling wave solutions, an invariant cone is constructed by upper and lower solutions and Schauder's fixed point theorem is applied. The nonexistence of traveling wave solutions is proved by two-sided Laplace transform. However, to apply two-sided Laplace transform, the prior estimate of exponential decrease of traveling wave solutions is needed. For this aim, a new method is proposed, which can be applied to reaction-diffusion systems consisting of more than three equations.

Incorporating heterogeneity into the transmission dynamics of a waterborne disease model

We formulate a mathematical model that captures the essential dynamics of waterborne disease transmission to study the effects of heterogeneity on the spread of the disease. The effects of heterogeneity on some important mathematical features of the model such as the basic reproduction number, type reproduction number and final outbreak size are analysed accordingly. We conduct a real-world application of this model by using it to investigate the heterogeneity in transmission in the recent cholera outbreak in Haiti. By evaluating the measure of heterogeneity between the administrative departments in Haiti, we discover a significant difference in the dynamics of the cholera outbreak between the departments.

Investigation of key interventions for shigellosis outbreak control in China

Shigellosis is a major public health concern in China, where waterborne disease outbreaks are common. Shigellosis-containing strategies, mostly single or multiple interventions, are implemented by primary-level health departments. Systematic assessment of the effectiveness of these measures is scarce. To estimate the efficacy of commonly used intervention strategies, we developed a Susceptible–Exposed–Infectious/Asymptomatic–Recovered–Water model. No intervention was predicted to result in a total attack rate (TAR) of 90% of the affected population (95% confidence interval [CI]: 86.65–92.80) and duration of outbreak (DO) of 89 days, and the use of single-intervention strategies can be futile or even counter-productive. Prophylactics and water disinfection did not improve TAR or DO. School closure for up to 3 weeks did not help but only increased DO. Isolation alone significantly increased DO. Only antibiotics treatment could shorten the DO to 35 days with TAR unaffected. We observed that these intervention effects were additive when in combined usage under most circumstances. Combined intervention “Isolation+antibiotics+prophylactics+water disinfection” was predicted to result in the lowest TAR (41.9%, 95%CI: 36.97–47.04%) and shortest DO (28 days). Our actual Shigellosis control implementation that also included school closure for 1 week, attained comparable results and the modeling produced an epidemic curve of Shigellosis highly similar to our actual outbreak data. This lends a strong support to the reality of our model that provides a possible reference for public health professionals to evaluate their strategies towards Shigellosis control.

Cholera transmission dynamic models for public health practitioners

Great progress has been made in mathematical models of cholera transmission dynamics in recent years. However, little impact, if any, has been made by models upon public health decision-making and day-to-day routine of epidemiologists. This paper provides a brief introduction to the basics of ordinary differential equation models of cholera transmission dynamics. We discuss a basic model adapted from Codeço (2001), and how it can be modified to incorporate different hypotheses, including the importance of asymptomatic or inapparent infections, and hyperinfectious V. cholerae and human-to-human transmission. We highlight three important challenges of cholera models: (1) model misspecification and parameter uncertainty, (2) modeling the impact of water, sanitation and hygiene interventions and (3) model structure. We use published models, especially those related to the 2010 Haitian outbreak as examples. We emphasize that the choice of models should be dictated by the research questions in mind. More collaboration is needed between policy-makers, epidemiologists and modelers in public health.

Modelling cholera in periodic environments

We propose a deterministic compartmental model for cholera dynamics in periodic environments. The model incorporates seasonal variation into a general formulation for the incidence (or, force of infection) and the pathogen concentration. The basic reproduction number of the periodic model is derived, based on which a careful analysis is conducted on the epidemic and endemic dynamics of cholera. Several specific examples are presented to demonstrate this general model, and numerical simulation results are used to validate the analytical prediction.

Modeling cholera outbreaks

Mathematical modeling can be a valuable tool for studying infectious disease outbreak dynamics and simulating the effects of possible interventions. Here, we describe approaches to modeling cholera outbreaks and how models have been applied to explore intervention strategies, particularly in Haiti. Mathematical models can play an important role in formulating and evaluating complex cholera outbreak response options. Major challenges to cholera modeling are insufficient data for calibrating models and the need to tailor models for different outbreak scenarios.

Examining rainfall and cholera dynamics in Haiti using statistical and dynamic modeling approaches

Haiti has been in the midst of a cholera epidemic since October 2010. Rainfall is thought to be associated with cholera here, but this relationship has only begun to be quantitatively examined. In this paper, we quantitatively examine the link between rainfall and cholera in Haiti for several different settings (including urban, rural, and displaced person camps) and spatial scales, using a combination of statistical and dynamic models. Statistical analysis of the lagged relationship between rainfall and cholera incidence was conducted using case crossover analysis and distributed lag nonlinear models. Dynamic models consisted of compartmental differential equation models including direct (fast) and indirect (delayed) disease transmission, where indirect transmission was forced by empirical rainfall data. Data sources include cholera case and hospitalization time series from the Haitian Ministry of Public Health, the United Nations Water, Sanitation and Health Cluster, International Organization for Migration, and Hôpital Albert Schweitzer. Rainfall data was obtained from rain gauges from the U.S. Geological Survey and Haiti Regeneration Initiative, and remote sensing rainfall data from the National Aeronautics and Space Administration Tropical Rainfall Measuring Mission. A strong relationship between rainfall and cholera was found for all spatial scales and locations examined. Increased rainfall was significantly correlated with increased cholera incidence 4-7 days later. Forcing the dynamic models with rainfall data resulted in good fits to the cholera case data, and rainfall-based predictions from the dynamic models closely matched observed cholera cases. These models provide a tool for planning and managing the epidemic as it continues.

Dynamics of an age-of-infection cholera model

A new model for the dynamics of cholera is formulated that incorporates both the infection age of infectious individuals and biological age of pathogen in the environment. The basic reproduction number is defined and proved to be a sharp threshold determining whether or not cholera dies out. Final size relations for cholera outbreaks are derived for simplified models when input and death are neglected.

GLOBAL DYNAMICS OF A CHOLERA MODEL WITH TIME DELAY

The global dynamics of a cholera model with delay is considered. We determine a basic reproduction number R0 which is chosen based on the relative ODE model, and establish that the global dynamics are determined by the threshold value R0. If R0 < 1, then the infection-free equilibrium is global asymptotically stable, that is, the cholera dies out; If R0 > 1, then the unique endemic equilibrium is global asymptotically stable, which means that the infection persists. The results obtained show that the delay does not lead to periodic oscillations. Finally, some numerical simulations support our theoretical results.

Identifiability and estimation of multiple transmission pathways in cholera and waterborne disease

Cholera and many waterborne diseases exhibit multiple characteristic timescales or pathways of infection, which can be modeled as direct and indirect transmission. A major public health issue for waterborne diseases involves understanding the modes of transmission in order to improve control and prevention strategies. An important epidemiological question is: given data for an outbreak, can we determine the role and relative importance of direct vs. environmental/waterborne routes of transmission? We examine whether parameters for a differential equation model of waterborne disease transmission dynamics can be identified, both in the ideal setting of noise-free data (structural identifiability) and in the more realistic setting in the presence of noise (practical identifiability). We used a differential algebra approach together with several numerical approaches, with a particular emphasis on identifiability of the transmission rates. To examine these issues in a practical public health context, we apply the model to a recent cholera outbreak in Angola (2006). Our results show that the model parameters-including both water and person-to-person transmission routes-are globally structurally identifiable, although they become unidentifiable when the environmental transmission timescale is fast. Even for water dynamics within the identifiable range, when noisy data are considered, only a combination of the water transmission parameters can practically be estimated. This makes the waterborne transmission parameters difficult to estimate, leading to inaccurate estimates of important epidemiological parameters such as the basic reproduction number (R0). However, measurements of pathogen persistence time in environmental water sources or measurements of pathogen concentration in the water can improve model identifiability and allow for more accurate estimation of waterborne transmission pathway parameters as well as R0. Parameter estimates for the Angola outbreak suggest that both transmission pathways are needed to explain the observed cholera dynamics. These results highlight the importance of incorporating environmental data when examining waterborne disease.

Indirect transmission and the effect of seasonal pathogen inactivation on infectious disease periodicity

The annual occurrence of many infectious diseases remains a constant burden to public health systems. The seasonal patterns in respiratory disease incidence observed in temperate regions have been attributed to the impact of environmental conditions on pathogen survival. A model describing the transmission of an infectious disease by means of a pathogenic state capable of surviving in an environmental reservoir outside of its host organism is presented in this paper. The ratio of pathogen lifespan to the duration of the infectious disease state is found to be a critical parameter in determining disease dynamics. The introduction of a seasonally forced pathogen inactivation rate identifies a time delay between peak pathogen survival and peak disease incidence. The delay is dependent on specific disease parameters and, for influenza, decreases with increasing reproduction number. The observed seasonal oscillations are found to have a period identical to that of the seasonally forced inactivation rate and which is independent of the duration of infection acquired immunity.

Distinguishing between indirect and direct modes of transmission using epidemiological time series

Pathogen transmission can involve direct and/or indirect pathways. Using theoretical models, in this study we ask, "do directly and indirectly transmitted pathogens yield different population-level epidemiological dynamics?" and "can the transmission pathway be inferred from population-level epidemiological data?" Our approach involves comparing the continuous-time dynamics of a class of compartmental epidemiological models with direct versus environmentally mediated indirect transmission pathways. Combing analytical theory and numerical simulations we show that models with direct and indirect transmission can produce quantitatively similar time series when the pathogen cannot reproduce in the environment, particularly when the environmental pathogen dynamics are fast. We apply these results to a previous study on chronic wasting disease and show that identifying the transmission pathway is more difficult than previously acknowledged. Our analysis and simulations also yield conditions under which numerical differences can potentially identify the transmission route in oscillating endemic systems and systems where the environmental pathogen dynamics are not fast. This work begins to identify how differences in the transmission pathway can result in quantitatively different epidemiological dynamics and how those differences can be used to identify the transmission pathway from population level time series.

Enhanced surveillance on food-borne disease outbreaks: dynamics of cross-contamination in biocidal wash procedure

Understanding the geographic and temporal spread of food-borne diseases associated with fresh produce is crucial for informing adequate surveillance and control. As a first step towards this goal, we develop and analyze a novel three stage model at the processing/sanitization juncture in the fresh produce supply chain. The key feature of our model is its ability to describe the dynamics of cross-contamination during commercial wash procedures. In general, we quantify the degree of cross-contamination in terms of model parameters. Applying these results in the case of Escherichia coli O157:H7 contamination of fresh-cut romaine lettuce, we identify the mean wash time and free chlorine concentration as critical parameters. In addition to showing how these parameters affect contamination levels, we recommend that in order to prevent potential source misidentification, at least 2.2 mg/L of free chlorine should be used during a wash lasting at least 30s.

Heterogeneity in multiple transmission pathways: modelling the spread of cholera and other waterborne disease in networks with a common water source

Many factors influencing disease transmission vary throughout and across populations. For diseases spread through multiple transmission pathways, sources of variation may affect each transmission pathway differently. In this paper we consider a disease that can be spread via direct and indirect transmission, such as the waterborne disease cholera. Specifically, we consider a system of multiple patches with direct transmission occurring entirely within patch and indirect transmission via a single shared water source. We investigate the effect of heterogeneity in dual transmission pathways on the spread of the disease. We first present a 2-patch model for which we examine the effect of variation in each pathway separately and propose a measure of heterogeneity that incorporates both transmission mechanisms and is predictive of R(0). We also explore how heterogeneity affects the final outbreak size and the efficacy of intervention measures. We conclude by extending several results to a more general n-patch setting.

Social and spatial processes associated with childhood diarrheal disease in Matlab, Bangladesh

We develop novel methods for conceptualizing geographic space and social networks to evaluate their respective and combined contributions to childhood diarrheal incidence. After defining maternal networks according to direct familial linkages between females, and road networks using satellite imagery of the study area, we use a spatial econometrics model to evaluate the significance of correlation terms relating childhood diarrheal incidence to the incidence observed within respective networks. Disease was significantly clustered within road networks across time, but only inconsistently correlated within maternal networks. These methods could be widely applied to systems in which both social and spatial processes jointly influence health outcomes.

A comparison of a deterministic and stochastic model for hepatitis C with an isolation stage

We formulate a deterministic epidemic model for the spread of Hepatitis C containing an acute, chronic and isolation class and analyse the effects of the isolation class on the transmission dynamics of the disease. We calculate the basic reproduction number R(0) and show that for R(0)≤1, the disease-free equilibrium is globally asymptotically stable. In addition, it is shown that for a special case when R(0)>1, the endemic equilibrium is locally asymptotically stable. Furthermore, an analogous stochastic epidemic model for Hepatitis C is formulated using a continuous time Markov chain. Numerical simulations are used to estimate the mean, variance and probability distributions of the discrete random variables and these are compared to the steady-state solutions of the deterministic model. Finally, the expected time to disease extinction is estimated for the stochastic model and the impact of isolation on the time to extinction is explored.