Analysis of cholera epidemics with bacterial growth and spatial movement

Modelling techniques in cholera epidemiology: A systematic and critical review

Diverse modelling techniques in cholera epidemiology have been developed and used to (1) study its transmission dynamics, (2) predict and manage cholera outbreaks, and (3) assess the impact of various control and mitigation measures. In this study, we carry out a critical and systematic review of various approaches used for modelling the dynamics of cholera. Also, we discuss the strengths and weaknesses of each modelling approach. A systematic search of articles was conducted in Google Scholar, PubMed, Science Direct, and Taylor & Francis. Eligible studies were those concerned with the dynamics of cholera excluding studies focused on models for cholera transmission in animals, socio-economic factors, and genetic & molecular related studies. A total of 476 peer-reviewed articles met the inclusion criteria, with about 40% (32%) of the studies carried out in Asia (Africa). About 52%, 21%, and 9%, of the studies, were based on compartmental (e.g., SIRB), statistical (time series and regression), and spatial (spatiotemporal clustering) models, respectively, while the rest of the analysed studies used other modelling approaches such as network, machine learning and artificial intelligence, Bayesian, and agent-based approaches. Cholera modelling studies that incorporate vector/housefly transmission of the pathogen are scarce and a small portion of researchers (3.99%) considers the estimation of key epidemiological parameters. Vaccination only platform was utilized as a control measure in more than half (58%) of the studies. Research productivity in cholera epidemiological modelling studies have increased in recent years, but authors used diverse range of models. Future models should consider incorporating vector/housefly transmission of the pathogen and on the estimation of key epidemiological parameters for the transmission of cholera dynamics.

Mathematical Models for Cholera Dynamics-A Review

Cholera remains a significant public health burden in many countries and regions of the world, highlighting the need for a deeper understanding of the mechanisms associated with its transmission, spread, and control. Mathematical modeling offers a valuable research tool to investigate cholera dynamics and explore effective intervention strategies. In this article, we provide a review of the current state in the modeling studies of cholera. Starting from an introduction of basic cholera transmission models and their applications, we survey model extensions in several directions that include spatial and temporal heterogeneities, effects of disease control, impacts of human behavior, and multi-scale infection dynamics. We discuss some challenges and opportunities for future modeling efforts on cholera dynamics, and emphasize the importance of collaborations between different modeling groups and different disciplines in advancing this research area.

Dynamics and strategies evaluations of a novel reaction-diffusion COVID-19 model with direct and aerosol transmission

The COVID-19 epidemic has infected millions of people and cast a shadow over the global economic recovery. To explore the epidemic's transmission law and provide theoretical guidance for epidemic prevention and control. In this paper, we investigate a novel SEIR-A reaction-diffusion COVID-19 system with direct and aerosol transmission. First, the solution's positivity and boundedness for the system are discussed. Then, the system's the basic reproduction number is defined. Further, the uniform persistence of disease when R 0 > 1 is explored. In addition, the system equilibrium's global stability based on R 0 is demonstrated. Next, the system's NSFD scheme is investigated and the discrete system's positivity, boundedness, and global properties are studied. Meantime, global sensitivity analysis on threshold R 0 is investigated. Interestingly, the effects of three strategies, including vaccination, receiving treatment, and wearing a mask, are evaluated numerically. The results suggest that the above three strategies can effectively control the peak and final scale of infection and shorten the duration of the epidemic. Finally, theoretical simulations and instance predictions are used to give several key indicators of the epidemic, including threshold R 0 , peak, time to peak, time to clear cases, and final size. The instance prediction results are as follows: (1) The basic reproduction numbers of Yangzhou and Putian in China are R 0 = 2.5107 and R 0 = 1.8846 , respectively. (2) This epidemic round in Yangzhou will peak at 56 new daily confirmed cases on the 9th day (August 5), and Putian will peat at 37 new daily confirmed cases on the 6th day (September 15). (3) The final scale of infections in Yangzhou and Putian reached 570 and 205 cases, respectively. (4) The Yangzhou epidemic is expected to be completely cleared on the 25th day (August 21). In addition, the Putian epidemic will continue for 15 days and be cleared on September 24. The analysis results mean that we should improve our immunity by actively vaccinating, reducing the possibility of aerosol transmission by wearing masks. In particular, people should maintain proper social distance, and the government should strengthen medical investment and COVID-19 project research.

A reaction-advection-diffusion model of cholera epidemics with seasonality and human behavior change

Cholera is a water- and food-borne infectious disease caused by V. cholerae. To investigate multiple effects of human behavior change, seasonality and spatial heterogeneity on cholera spread, we propose a reaction-advection-diffusion model that incorporates human hosts and aquatic reservoir of V. cholerae. We derive the basic reproduction number [Formula: see text] for this system and then establish a threshold type result on its global dynamics in terms of [Formula: see text]. Further, we show that the bacterial loss at the downstream end of the river due to water flux can reduce the disease risk, and describe the asymptotic behavior of [Formula: see text] for small and large diffusion in a special case (where the diffusion rates of infected human and the pathogen are constant). We also study the transmission dynamics at the early stage of cholera outbreak numerically, and find that human behavior change may lower the infection level and delay the disease peak. Moreover, the relative rate of bacterial loss, together with convection rate, plays an important role in identifying the severely infected areas. Meanwhile spatial heterogeneity may dilute or amplify cholera infection, which in turn would increase the complexity of disease spread.

Stochastic models of infectious diseases in a periodic environment with application to cholera epidemics

Seasonal variation affects the dynamics of many infectious diseases including influenza, cholera and malaria. The time when infectious individuals are first introduced into a population is crucial in predicting whether a major disease outbreak occurs. In this investigation, we apply a time-nonhomogeneous stochastic process for a cholera epidemic with seasonal periodicity and a multitype branching process approximation to obtain an analytical estimate for the probability of an outbreak. In particular, an analytic estimate of the probability of disease extinction is shown to satisfy a system of ordinary differential equations which follows from the backward Kolmogorov differential equation. An explicit expression for the mean (resp. variance) of the first extinction time given an extinction occurs is derived based on the analytic estimate for the extinction probability. Our results indicate that the probability of a disease outbreak, and mean and standard derivation of the first time to disease extinction are periodic in time and depend on the time when the infectious individuals or free-living pathogens are introduced. Numerical simulations are then carried out to validate the analytical predictions using two examples of the general cholera model. At the end, the developed theoretical results are extended to more general models of infectious diseases.

MODELING CHOLERA TRANSMISSION UNDER DISEASE CONTROL MEASURES

We present a new mathematical model to investigate the transmission dynamics of cholera under disease control measures that include education programs and water sanitation. The model incorporates the impact of education programs into the disease transmission rates and that of water sanitation into the environmental pathogen dynamics. We conduct a detailed analysis to the autonomous system of the model and establish the local and global stabilities of its equilibria that characterize the threshold dynamics of cholera. We then perform an optimal control study on the general model with time-dependent controls and explore effective approaches to implement the education programs and water sanitation while balancing their costs. Our analysis and simulation highlight the complex interaction among the direct and indirect transmission pathways of the disease, the intrinsic growth of the environmental pathogen and the impact of multiple control measures, and their roles in collectively shaping the transmission dynamics of cholera.

Modelling the aqueous transport of an infectious pathogen in regional communities: application to the cholera outbreak in Haiti

A mathematical model is developed to describe the dynamics of the spread of a waterborne disease among communities located along a flowing waterway. The model is formulated as a system of reaction-diffusion-advection partial differential equations in this spatial setting. The compartments of the model consist of susceptible, infected, and recovered individuals in the communities along the waterway, together with a term representing the pathogen load in each community and a term representing the spatial concentration of pathogens flowing along the waterway. The model is applied to the cholera outbreak in Haiti in 2010.

Parameter and State Estimation in a Cholera Model with Threshold Immunology: A Case Study of Senegal

It is often impossible to measure all states affecting spread of a disease. In cholera, asymptomatic and cholera pathogen densities are not practically measurable despite playing a big role in its transmission. They are referred to as inaccessible states of the model and can only be manipulated using the measurable states of the given model. Our interest lies in estimating such states and the parameters catalyzing the spread. A mathematical model for cholera dynamics consisting of five compartments (susceptible, symptomatic, asymptomatic, recovered and bacteria population) with a minimum infection dose (MID) is considered. A method based on observer (from modern control theory) is proposed to estimate the state variables not accessible to measurement and the time-dependent parameters from real data of Senegal. We suppose that the total population of Senegal, the monthly reported cholera-induced deaths and the monthly recovered individuals are known inputs obtainable from real data, and the monthly new cholera cases the system output. An auxiliary system is used, an observer whose solutions converge exponentially to those of an original system and solely utilize known inputs and output of the model. Thus, the estimation of the unmeasured state variables like the pathogen density and the yearly asymptomatic population within the human community playing an important role in the spread of cholera is possible. We derive the expressions for time-dependent infection rate, induced cholera death rate and symptomatic recovery rate and their estimations done using real data. Numerical simulations are then performed for the validation of estimation results. The system together with the observer designed is detectable but is not observable. The observer delivered estimates reflect a close trend already ascertained by other researchers. They indicate the existence of bacteria and asymptomatic individuals in the population of Senegal at any single time for the duration of collection of the data. The ever existence of cholera pathogen explains the endemicity of Cholera in Senegal and other sub-Saharan-African countries owing to role played by the asymptomatic individual in the bacteria density. As such, the heath authorities in Senegal need to educate the general public on hygiene irrespective of observable symptoms to lower the possible number of new infections. We have analytically showed and numerically confirmed the exponential convergence to zero of the estimation errors resulting from the observer model hence the high quality of the estimates.

Global well-posedness of infectious disease models without life-time immunity: the cases of cholera and avian influenza

We study the systems of partial differential equations with diffusion that model the dynamics of infectious diseases without life-time immunity, in particular the cases of cholera from Wang & Wang (2015, J. Biol. Dyn., 9, 233-261) and avian influenza from Vaidya et al. (2012, Discrete Contin. Dyn. Syst. Ser. B, 17, 2829-2848). In both works, similarly to all others in the literature on various models of infectious diseases and more, it had to be assumed for a technical reason that the diffusivity coefficients of the susceptible, infected and recovered individuals, humans or birds, had to be identical in order to prove the existence of their unique solutions for all time. Considering that such uniform diffusivity strengths among the susceptible, infected and recovered hosts may not always be plausible in real world, we investigate the global well-posedness issue when such conditions are relaxed. In particular for the cholera model from Wang & Wang (2015, J. Biol. Dyn., 9, 233-261), we prove the global well-posedness with no condition on the diffusivity coefficients at all. For the avian influenza model from Vaidya et al. (2012, Discrete Contin. Dyn. Syst. Ser. B, 17, 2829-2848), we prove the global well-posedness with no condition on the diffusivity coefficients if the spatial dimension is one, and under a partial condition that the diffusivity coefficients of the susceptible and the infected hosts are same otherwise.

Threshold dynamics of reaction–diffusion partial differential equations model of Ebola virus disease

We study the reaction–diffusion Ebola PDE model that consists of equations that govern the evolution of susceptible, infected, recovered and deceased human individuals, as well as Ebola virus pathogens in the environment, with diffusive terms in all except the equation of the deceased human individuals. Under the setting of a spatial domain that is bounded, we prove the global well-posedness of the system; in contrast to the previous work on similar models such as cholera, avian influenza, malaria and dengue fever, diffusion coefficients may be different. Moreover, we derive its basic reproduction number, and under the condition that the diffusion coefficients of the susceptible and infected hosts are same, we prove the global stability of the disease-free-equilibrium, and uniform persistence in cases when the basic reproduction number lies beneath and above one, respectively. Again, we do not require that the diffusion coefficients of the recovered hosts be the same as the diffusion coefficients of the susceptible and infected hosts, in contrast to previous work on other models of infectious diseases. Another technical difficulty in our model is that the solution semiflow is not compact due to the lack of diffusion in the equation of the deceased human individuals; we overcome this difficulty using functional analysis techniques concerning Kuratowski measure of non-compactness.

A multi-scale cholera model linking between-host and within-host dynamics

We propose a multi-scale modeling framework to investigate the transmission dynamics of cholera. At the population level, we employ a SIR model for the between-host transmission of the disease. At the individual host level, we describe the evolution of the pathogen within the human body. The between-host and within-host dynamics are connected through an environmental equation that characterizes the growth of the pathogen and its interaction with the hosts outside the human body. We put a special emphasis on the within-host dynamics by making a distinction for each individual host. We conduct both mathematical analysis and numerical simulation for our model in order to explore various scenarios associated with cholera transmission and to better understand the complex, multi-scale disease dynamics.

Modeling the within-host dynamics of cholera: bacterial-viral interaction

Novel deterministic and stochastic models are proposed in this paper for the within-host dynamics of cholera, with a focus on the bacterial-viral interaction. The deterministic model is a system of differential equations describing the interaction among the two types of vibrios and the viruses. The stochastic model is a system of Markov jump processes that is derived based on the dynamics of the deterministic model. The multitype branching process approximation is applied to estimate the extinction probability of bacteria and viruses within a human host during the early stage of the bacterial-viral infection. Accordingly, a closed-form expression is derived for the disease extinction probability, and analytic estimates are validated with numerical simulations. The local and global dynamics of the bacterial-viral interaction are analysed using the deterministic model, and the result indicates that there is a sharp disease threshold characterized by the basic reproduction number [Formula: see text]: if [Formula: see text], vibrios ingested from the environment into human body will not cause cholera infection; if [Formula: see text], vibrios will grow with increased toxicity and persist within the host, leading to human cholera. In contrast, the stochastic model indicates, more realistically, that there is always a positive probability of disease extinction within the human host.

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.

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.

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).