Global stability for cholera epidemic models

A comprehensive analysis of COVID-19 nonlinear mathematical model by incorporating the environment and social distancing

This research conducts a detailed analysis of a nonlinear mathematical model representing COVID-19, incorporating both environmental factors and social distancing measures. It thoroughly analyzes the model's equilibrium points, computes the basic reproductive rate, and evaluates the stability of the model at disease-free and endemic equilibrium states, both locally and globally. Additionally, sensitivity analysis is carried out. The study develops a sophisticated stability theory, primarily focusing on the characteristics of the Volterra-Lyapunov (V-L) matrices method. To understand the dynamic behavior of COVID-19, numerical simulations are essential. For this purpose, the study employs a robust numerical technique known as the non-standard finite difference (NSFD) method, introduced by Mickens. Various results are visually presented through graphical representations across different parameter values to illustrate the impact of environmental factors and social distancing measures.

On a two-strain epidemic mathematical model with vaccination

In this paper, we study mathematically a two strains epidemic model taking into account non-monotonic incidence rates and vaccination strategy. The model contains seven ordinary differential equations that illustrate the interaction between the susceptible, the vaccinated, the exposed, the infected and the removed individuals. The model has four equilibrium points, namely, disease free equilibrium, endemic equilibrium with respect to the first strain, endemic equilibrium with respect to the second strain and the endemic equilibrium with respect to both strains. The global stability of the equilibria has been demonstrated using some suitable Lyapunov functions. The basic reproduction number is found depending on the first strain reproduction number R 0 1 and the second reproduction number R 0 2 . We have shown that the disease dies out when the basic reproduction number is less than unity. It was remarked that the global stability of the endemic equilibria depends, on the strain basic reproduction number and on the strain inhibitory effect reproduction number. We have also observed that the strain with high basic reproduction number will dominate the other strain. Finally, the numerical simulations are presented in the last part of this work to support our theoretical results. We notice that our suggested model has some limitations and does not predicting the long-term dynamics for some reproduction numbers cases.

Mathematical modeling of cholera dynamics with intrinsic growth considering constant interventions

A mathematical model that describes the dynamics of bacterium vibrio cholera within a fixed population considering intrinsic bacteria growth, therapeutic treatment, sanitation and vaccination rates is developed. The developed mathematical model is validated against real cholera data. A sensitivity analysis of some of the model parameters is also conducted. The intervention rates are found to be very important parameters in reducing the values of the basic reproduction number. The existence and stability of equilibrium solutions to the mathematical model are also carried out using analytical methods. The effect of some model parameters on the stability of equilibrium solutions, number of infected individuals, number of susceptible individuals and bacteria density is rigorously analyzed. One very important finding of this research work is that keeping the vaccination rate fixed and varying the treatment and sanitation rates provide a rapid decline of infection. The fourth order Runge-Kutta numerical scheme is implemented in MATLAB to generate the numerical solutions.

An immuno-epidemiological model linking between-host and within-host dynamics of cholera

Cholera, a severe gastrointestinal infection caused by the bacterium Vibrio cholerae, remains a major threat to public health, with a yearly estimated global burden of 2.9 million cases. Although most existing models for the disease focus on its population dynamics, the disease evolves from within-host processes to the population, making it imperative to link the multiple scales of the disease to gain better perspectives on its spread and control. In this study, we propose an immuno-epidemiological model that links the between-host and within-host dynamics of cholera. The immunological (within-host) model depicts the interaction of the cholera pathogen with the adaptive immune response. We distinguish pathogen dynamics from immune response dynamics by assigning different time scales. Through a time-scale analysis, we characterise a single infected person by their immune response. Contrary to other within-host models, this modelling approach allows for recovery through pathogen clearance after a finite time. Then, we scale up the dynamics of the infected person to construct an epidemic model, where the infected population is structured by individual immunological dynamics. We derive the basic reproduction number ($ \mathcal{R}_0 $) and analyse the stability of the equilibrium points. At the disease-free equilibrium, the disease will either be eradicated if $ \mathcal{R}_0 < 1 $ or otherwise persists. A unique endemic equilibrium exists when $ \mathcal{R}_0 > 1 $ and is locally asymptotically stable without a loss of immunity.

Bacteria-bacteriophage cycles facilitate Cholera outbreak cycles: an indirect Susceptible-Infected-Recovered-Bacteria- Phage (iSIRBP) model-based mathematical study

Cholera is an acute enteric infectious disease caused by the Gram-negative bacterium Vibrio cholerae. Despite a huge body of research, the precise nature of its transmission dynamics has yet to be fully elucidated. Mathematical models can be useful to better understand how an infectious agent can spread and be properly controlled. We develop a compartmental model describing a human population, a bacterial population as well as a phage population. We show that there might be eight equilibrium points, one of which is a disease free equilibrium point. We carry out numerical simulations and sensitivity analyses and we show that the presence of phage can reduce the number of infectious individuals. Moreover, we discuss the main implications in terms of public health management and control strategies.

Preventive control strategy on second wave of Covid-19 pandemic model incorporating lock-down effect

This study presents an optimal control strategy through a mathematical model of the Covid-19 outbreak without lock-down. The pandemic model analyses the lock-down effect without control strategy based on the current scenario of second wave data to control the rapid spread of the virus. The pandemic model has been discussed with respect to the basic reproduction number and stability analysis of disease-free and endemic equilibrium. A new optimal control problem with treatment is framed to minimize the vulnerable situation of the second wave. This system is applied to study the effects of vaccines and treatment controls. Numerical solutions and the graphical presentation of the results predict the fate of India’s second wave situation on account of the control strategy. Lastly, a comparative study with control and without control has been analysed for the exposed phase, infective phase, and recovery phase to understand the effectiveness of the controls. This model is used to estimate the total number of infected and active cases, deaths, and recoveries in order to control the disease using this system and studying the effects of vaccines and treatment controls.

Optimal control design incorporating vaccination and treatment on six compartment pandemic dynamical system

In this paper, a mathematical model of the COVID-19 pandemic with lockdown that provides a more accurate representation of the infection rate has been analyzed. In this model, the total population is divided into six compartments: the susceptible class, lockdown class, exposed class, asymptomatic infected class, symptomatic infected class, and recovered class. The basic reproduction number (R0) is calculated using the next-generation matrix method and presented graphically based on different progression rates and effective contact rates of infective individuals. The COVID-19 epidemic model exhibits the disease-free equilibrium and endemic equilibrium. The local and global stability analysis has been done at the disease-free and endemic equilibrium based on R0. The stability analysis of the model shows that the disease-free equilibrium is both locally and globally stable when R0<1, and the endemic equilibrium is locally and globally stable when R0>1 under some conditions. A control strategy including vaccination and treatment has been studied on this pandemic model with an objective functional to minimize. Finally, numerical simulation of the COVID-19 outbreak in India is carried out using MATLAB, highlighting the usefulness of the COVID-19 pandemic model and its mathematical analysis.

Borderline microscopic organism and lockdown impacted across the borders-global shakers

Viruses are the potential cause of several diseases including novel corona virus-19, flu, small pox, chicken pox, acquired immunodeficiency syndrome, severe acute respiratory syndrome etc. The objectives of this review article are to summarize the reasons behind the epidemics caused by several emerging viruses and bacteria, how to control the infection and preventive strategies. We have explained the causes of epidemics along with their preventive measures, the impact of lockdown on the health of people and the economy of a country. Several reports have revealed the transmission of infection during epidemic from the contact of an infected person to the public that can be prevented by implementing the lockdown by the government of a country. Though lockdown has been considered as one of the significant parameters to control the diseases, however, it has some negative consequences on the health of people as they can be more prone to other ailments like obesity, diabetes, cardiac problems etc. and drastic decline in the economy of a country. Therefore, the transmission of diseases can be prevented by warning the people about the severity of diseases, avoiding their public transportation, keeping themselves isolated, strictly following the guidelines of lockdown and encouraging regular exercise.

Threshold dynamics of a nonlocal and delayed cholera model in a spatially heterogeneous environment

A nonlocal and delayed cholera model with two transmission mechanisms in a spatially heterogeneous environment is derived. We introduce two basic reproduction numbers, one is for the bacterium in the environment and the other is for the cholera disease in the host population. If the basic reproduction number for the cholera bacterium in the environment is strictly less than one and the basic reproduction number of infection is no more than one, we prove globally asymptotically stability of the infection-free steady state. Otherwise, the infection will persist and there exists at least one endemic steady state. For the special homogeneous case, the endemic steady state is actually unique and globally asymptotically stable. Under some conditions, the basic reproduction number of infection is strictly decreasing with respect to the diffusion coefficients of cholera bacteria and infectious hosts. When these conditions are violated, numerical simulation suggests that spatial diffusion may not only spread the infection from high-risk region to low-risk region, but also increase the infection level in high-risk region.

Dynamical analysis of novel COVID-19 epidemic model with non-monotonic incidence function

In this study, we developed and analyzed a mathematical model for explaining the transmission dynamics of COVID-19 in India. The proposed SI u I k R model is a modified version of the existing SIR model. Our model divides the infected class I of SIR model into two classes: I u (unknown infected class) and I k (known infected class). In addition, we consider R a recovered and reserved class, where susceptible people can hide them due to fear of the COVID-19 infection. Furthermore, a non-monotonic incidence function is deemed to incorporate the psychological effect of the novel coronavirus diseases on India's community. The epidemiological threshold parameter, namely the basic reproduction number, has been formulated and presented graphically. With this threshold parameter, the local and global stability analysis of the disease-free equilibrium and the endemic proportion equilibrium based on disease persistence have been analyzed. Lastly, numerical results of long-run prediction using MATLAB show that the fate of this situation is very harmful if people are not following the guidelines issued by the authority.

Analysis of COVID-19 using a modified SLIR model with nonlinear incidence

Infectious diseases kill millions of people each year, and they are the major public health problem in the world. This paper presents a modified Susceptible-Latent-Infected-Removed (SLIR) compartmental model of disease transmission with nonlinear incidence. We have obtained a threshold value of basic reproduction number ( R 0 ) and shown that only a disease-free equilibrium exists whenR 0 < 1 and endemic equilibrium whenR 0 > 1 . With the help of the Lyapunov-LaSalle Invariance Principle, we have shown that disease-free equilibrium and endemic equilibrium are both globally asymptotically stable. The study has also provided the model calibration to estimate parameters with month wise coronavirus (COVID-19) data, i.e. reported cases by worldometer from March 2020 to May 2021 and provides prediction until December 2021 in China. The Partial Rank Correlation Coefficient (PRCC) method was used to investigate how the model parameters' variation impact the model outcomes. We observed that the most important parameter is transmission rate which had the most significant impact on COVID-19 cases. We also discuss the epidemiology of COVID-19 cases and several control policies and make recommendations for controlling this disease in China.

Modeling the Effectiveness of TV and Social Media Advertisements on the Dynamics of Water-Borne Diseases

Cholera is a serious threat to the health of human-kind all over the world and its control is a problem of great concern. In this context, a nonlinear mathematical model to control the prevalence of cholera disease is proposed and analyzed by incorporating TV and social media advertisements as a dynamic variable. It is considered that TV and social media ads propagate the knowledge among the people regarding the severe effects of cholera disease on human health along with its precautionary measures. It is also assumed that the mode of transmission of cholera disease among susceptible individuals is due to consumption of contaminated drinking water containing Vibrio cholerae. Moreover, the propagation of knowledge through TV and social media ads makes the people aware to adopt precautionary measures and also the aware people make some effectual efforts to washout the bacteria from the aquatic environment. Model analysis reveals that increase in the washout rate of bacteria due to aware individuals causes the stability switch. It is found that TV and social media ads have the potential to reduce the number of infectives in the region and thus control the cholera epidemic. Numerical simulation is performed for a particular set of parameter values to support the analytical findings.

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.

DYNAMICS OF CHOLERA EPIDEMIC MODELS IN FLUCTUATING ENVIRONMENTS

Based on our deterministic models for cholera epidemics, we propose a stochastic model for cholera epidemics to incorporate environmental fluctuations which is a nonlinear system of Itô stochastic differential equations. We conduct an asymptotical analysis of dynamical behaviors for the model. The basic stochastic reproduction valueR s is defined in terms of the basic reproduction number R 0 for the corresponding deterministic model and noise intensities. The basic stochastic reproduction value determines the dynamical patterns of the stochastic model. WhenR s < 1 , the cholera infection will extinct within finite periods of time almost surely. WhenR s > 1 , the cholera infection will persist most of time, and there exists a unique stationary ergodic distribution to which all solutions of the stochastic model will approach almost surely as noise intensities are bounded. When the basic reproduction number R 0 for the corresponding deterministic model is greater than 1, and the noise intensities are large enough such thatR s < 1 , the cholera infection is suppressed by environmental noises. We carry out numerical simulations to illustrate our analysis, and to compare with the corresponding deterministic model. Biological implications are pointed out.

A dynamic optimal control model for COVID-19 and cholera co-infection in Yemen

In this work, we propose a new dynamic mathematical model framework governed by a system of differential equations that integrates both COVID-19 and cholera outbreaks. The estimations of the model parameters are based on the outbreaks of COVID-19 and cholera in Yemen from January 1, 2020 to May 30, 2020. Moreover, we present an optimal control model for minimizing both the number of infected people and the cost associated with each control. Four preventive measures are to be taken to control the outbreaks: social distancing, lockdown, the number of tests, and the number of chlorine water tablets (CWTs). Under the current conditions and resources available in Yemen, various policies are simulated to evaluate the optimal policy. The results obtained confirm that the policy of providing resources for the distribution of CWTs, providing sufficient resources for testing with an average social distancing, and quarantining of infected individuals has significant effects on flattening the epidemic curves.

Mathematical modelling for scarlet fever with direct and indirect infections

Scarlet fever is an acute respiratory infectious disease and the incidence rate is increasing from 2011 throughout the world. In this paper, the mathematical models are proposed, which incorporate both direct transmissions and indirect transmissions of scarlet fever. The threshold conditions for disease invasion are obtained in terms of the basic reproduction number. The peak value, final size and epidemic time in a seasonal prevalence are investigated numerically. Furthermore, the effects of seasonal fluctuations on disease outbreak are also studied on the basis of real data in China.

Asymptomatic transmission shifts epidemic dynamics

Asymptomatic transmission of infectious diseases has been recognized recently in several epidemics or pandemics. There is a great need to incorporate asymptomatic transmissions into traditional modeling of infectious diseases and to study how asymptomatic transmissions shift epidemic dynamics. In this work, we propose a compartmental model with asymptomatic transmissions for waterborne infectious diseases. We conduct a detailed analysis and numerical study with shigellosis data. Two parameters, the proportion $p$ of asymptomatic infected individuals and the proportion $k$ of asymptomatic infectious individuals who can asymptomatically transmit diseases, play major rules in the epidemic dynamics. The basic reproduction number $\mathscr{R}_{0}$ is a decreasing function of parameter $p$ when parameter $k$ is smaller than a critical value while $\mathscr{R}_{0}$ is an increasing function of $p$ when $k$ is greater than the critical value. $\mathscr{R}_{0}$ is an increasing function of $k$ for any value of $p$. When $\mathscr{R}_{0}$ passes through 1 as $p$ or $k$ varies, the dynamics of epidemics is shifted. If asymptomatic transmissions are not counted, $\mathscr{R}_{0}$ will be underestimated while the final size may be overestimated or underestimated. Our study provides a theoretical example for investigating other asymptomatic transmissions and useful information for public health measurements in waterborne infectious diseases.

An Approach for the Global Stability of Mathematical Model of an Infectious Disease

The global stability analysis for the mathematical model of an infectious disease is discussed here. The endemic equilibrium is shown to be globally stable by using a modification of the Volterra–Lyapunov matrix method. The basis of the method is the combination of Lyapunov functions and the Volterra–Lyapunov matrices. By reducing the dimensions of the matrices and under some conditions, we can easily show the global stability of the endemic equilibrium. To prove the stability based on Volterra–Lyapunov matrices, we use matrices with the symmetry properties (symmetric positive definite). The results developed in this paper can be applied in more complex systems with nonlinear incidence rates. Numerical simulations are presented to illustrate the analytical results.

Global analysis of an environmental disease transmission model linking within-host and between-host dynamics

In this paper, a multi-scale mathematical model for environmentally transmitted diseases is proposed which couples the pathogen-immune interaction inside the human body with the disease transmission at the population level. The model is based on the nested approach that incorporates the infection-age-structured immunological dynamics into an epidemiological system structured by the chronological time, the infection age and the vaccination age. We conduct detailed analysis for both the within-host and between-host disease dynamics. Particularly, we derive the basic reproduction number R ₀ for the between-host model and prove the uniform persistence of the system. Furthermore, using carefully constructed Lyapunov functions, we establish threshold-type results regarding the global dynamics of the between-host system: the disease-free equilibrium is globally asymptotically stable when R ₀ < 1, and the endemic equilibrium is globally asymptotically stable when R ₀ > 1. We explore the connection between the within-host and between-host dynamics through both mathematical analysis and numerical simulation. We show that the pathogen load and immune strength at the individual level contribute to the disease transmission and spread at the population level. We also find that, although the between-host transmission risk correlates positively with the within-host pathogen load, there is no simple monotonic relationship between the disease prevalence and the individual pathogen load.

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.

Analysis of a mathematical model for the transmission dynamics of human melioidosis

A deterministic model for the transmission dynamics of melioidosis disease in human population is designed and analyzed. The model is shown to exhibit the phenomenon of backward bifurcation, where a stable disease-free equilibrium co-exists with a stable endemic equilibrium when the basic reproduction number [Formula: see text] is less than one. It is further shown that the backward bifurcation dynamics is caused by the reinfection of individuals who recovered from the disease and relapse. The existence of backward bifurcation implies that bringing down [Formula: see text] to less than unity is not enough for disease eradication. In the absence of backward bifurcation, the global asymptotic stability of the disease-free equilibrium is shown whenever [Formula: see text]. For [Formula: see text], the existence of at least one locally asymptotically stable endemic equilibrium is shown. Sensitivity analysis of the model, using the parameters relevant to the transmission dynamics of the melioidosis disease, is discussed. Numerical experiments are presented to support the theoretical analysis of the model. In the numerical experimentations, it has been observed that screening and treating individuals in the exposed class has a significant impact on the disease dynamics.

Modeling and analyzing cholera transmission dynamics with vaccination age

A new mathematical model is formulated to investigate the transmission dynamics of cholera under vaccination, with a focus on the impact of vaccination age. The basic reproduction number is derived and proved to be a sharp control threshold determining whether or not the infection is persistent. We conduct a rigorous analysis on the local and global stability properties of the equilibria in system. Meanwhile, we compare the results to those of the simplified model based on ordinary differential equations where the effects of vaccination age are not incorporated. Numerical simulation results verify our analytical findings.

Global dynamics of an age-structured cholera model with multiple transmissions, saturation incidence and imperfect vaccination

In this paper, an age-structured cholera model with multiple transmissions, saturation incidence and imperfect vaccination is proposed. In the model, we consider both the infection age of infected individuals and the biological age of Vibrio cholerae in the aquatic environment. Asymptotic smoothness is verified as a necessary argument. By analysing the characteristic equations, the local stability of disease-free and endemic steady states is established. By using Lyapunov functionals and LaSalle's invariance principle, it is proved that the global dynamics of the model can be completely determined by basic reproduction number. The study of optimal control helps us seek cost-effective solutions of time-dependent vaccination strategy against cholera outbreaks. Numerical simulations are carried out to illustrate the corresponding theoretical results.

Modelling the dynamics of direct and pathogens-induced dysentery diarrhoea epidemic with controls

In this paper, the dysentery dynamics model with controls is theoretically investigated using the stability theory of differential equations. The system is considered as SIRSB deterministic compartmental model with treatment and sanitation. A threshold number R0 is obtained such that R0≤ 1 indicates the possibility of dysentery eradication in the community while R0>1 represents uniform persistence of the disease. The Lyapunov-LaSalle method is used to prove the global stability of the disease-free equilibrium. Moreover, the geometric approach method is used to obtain the sufficient condition for the global stability of the unique endemic equilibrium for R0>1 . Numerical simulation is performed to justify the analytical results. Graphical results are presented and discussed quantitatively. It is found out that the aggravation of the disease can be decreased by using the constant controls treatment and sanitation.

Global dynamics of a cholera model with age structures and multiple transmission modes

In this paper, we propose and analyze a cholera model. The model incorporates both direct transmission (person-to-person transmission) and indirect transmission (contaminated environment-to-person transmission: hyper-infectivity and lower-infectivity). Moreover, we employ general nonlinear incidences and introduce infection age of infectious individuals and biological ages of pathogens in the environment. After considering the well-posedness of the system, we study the existence and local stability of steady states, which is determined by the basic reproduction number. To establish the attractivity of the infection steady state, we also get the uniform persistence and existence of compact global attractors. The main result is a threshold dynamics obtained by applying the Fluctuation Lemma and the approach of Lyapunov functionals. When the basic reproduction number is less than one, the infection-free steady state is globally asymptotically stable while when the basic reproduction number is larger than one, the infection steady state attracts each solution with nonzero infection force at some time point. The effect of multiple transmission modes on the disease dynamics is also discussed.

A CHOLERA METAPOPULATION MODEL INTERLINKING MIGRATION WITH INTERVENTION STRATEGIES — A CASE STUDY OF ZIMBABWE (2008–2009)

Cholera is a water-borne disease and a major threat to human society affecting about 3–5 million people annually. A considerable number of research works have already been done to understand the disease transmission route and preventive measures in spatial or non-spatial scale. However, how the control strategies are to be linked up with the human migration in different locations in a country are not well studied. The present investigation is carried out in this direction by proposing and analyzing cholera meta-population models. The basic dynamical properties including the domain basic reproduction number are studied. Several important model parameters are estimated using cholera incidence data (2008–2009) and inter-provincial migration data from Census 2012 for the five provinces in Zimbabwe. By defining some migration index, and interlinking these indices with different cholera control strategies, namely, promotion of hand-hygiene and clean water supply and treatment, we carried out an optimal cost effectiveness analysis using optimal control theory. Our analysis suggests that there is no need to provide control measures for all the five provinces, and the control measures should be provided only to those provinces where in-migration flow is moderate. We also observe that such selective control measures which are also cost effective may reduce the overall cases and deaths.

A MATHEMATICAL MODEL FOR EBOLA EPIDEMIC WITH SELF-PROTECTION MEASURES

A mathematical model presented in Berge T, Lubuma JM-S, Moremedi GM, Morris N Shava RK, A simple mathematical model for Ebola in Africa, J Biol Dyn 11(1): 42–74 (2016) for the transmission dynamics of Ebola virus is extended to incorporate vaccination and change of behavior for self-protection of susceptible individuals. In the new setting, it is shown that the disease-free equilibrium is globally asymptotically stable when the basic reproduction number [Formula: see text] is less than or equal to unity and unstable when [Formula: see text]. In the latter case, the model system admits at least one endemic equilibrium point, which is locally asymptotically stable. Using the parameters relevant to the transmission dynamics of the Ebola virus disease, we give sensitivity analysis of the model. We show that the number of infectious individuals is much smaller than that obtained in the absence of any intervention. In the case of the mass action formulation with vaccination and education, we establish that the number of infectious individuals decreases as the intervention efforts increase. In the new formulation, apart from supporting the theory, numerical simulations of a nonstandard finite difference scheme that we have constructed suggests that the results on the decrease of the number of infectious individuals is valid.

Dynamics of cholera epidemics with impulsive vaccination and disinfection

Waterborne diseases have a tremendous influence on human life. The contaminated drinking water causes water-borne disease like cholera. Pulse vaccination is an important and effective strategy for the elimination of infectious diseases. A waterborne disease like cholera can also be controlled by using impulse technique. In this paper, we have proposed a delayed SEIRB epidemic model with impulsive vaccination and disinfection. We have studied the pulse vaccination strategy and sanitation to control the cholera disease. The existence and stability of the disease-free and endemic periodic solution are investigated both analytically and numerically. It is shown that there exists an infection-free periodic solution, using the impulsive dynamical system defined by the stroboscopic map. It is observed that the infection-free periodic solution is globally attractive when the impulse period is less than some critical value. From the analysis of the model, we have obtained a sufficient condition for the permanence of the epidemic with pulse vaccination. The main highlight of this paper is to introduce impulse technique along with latent period into the SEIRB epidemic model to investigate the role of pulse vaccination and disinfection on the dynamics of the cholera epidemics.

Modeling ebola virus disease transmissions with reservoir in a complex virus life ecology

We propose a new deterministic mathematical model for the transmission dynamics of Ebola Virus Disease (EVD) in a complex Ebola virus life ecology. Our model captures as much as possible the features and patterns of the disease evolution as a three cycle transmission process in the two ways below. Firstly it involves the synergy between the epizootic phase (during which the disease circulates periodically amongst non-human primates populations and decimates them), the enzootic phase (during which the disease always remains in fruit bats population) and the epidemic phase (during which the EVD threatens and decimates human populations). Secondly it takes into account the well-known, the probable/suspected and the hypothetical transmission mechanisms (including direct and indirect routes of contamination) between and within the three different types of populations consisting of humans, animals and fruit bats. The reproduction number R0 for the full model with the environmental contamination is derived and the global asymptotic stability of the disease free equilibrium is established when R0andlt;1. It is conjectured that there exists a unique globally asymptotically stable endemic equilibrium for the full model when R0andgt;1. The role of a contaminated environment is assessed by comparing the human infected component for the sub-model without the environment with that of the full model. Similarly, the sub-model without animals on the one hand and the sub-model without bats on the other hand are studied. It is shown that bats influence more the dynamics of EVD than the animals. Global sensitivity analysis shows that the effective contact rate between humans and fruit bats and the mortality rate for bats are the most influential parameters on the latent and infected human individuals. Numerical simulations, apart from supporting the theoretical results and the existence of a unique globally asymptotically stable endemic equilibrium for the full model, suggest further that: (1) fruit bats are more important in the transmission processes and the endemicity level of EVD than animals. This is in line with biological findings which identified bats as reservoir of Ebola viruses; (2) the indirect environmental contamination is detrimental to human beings, while it is almost insignificant for the transmission in bats.

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.

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.

Modeling HIV-1 viral capsid nucleation by dynamical systems

There are two stages generally recognized in the viral capsid assembly: nucleation and elongation. This paper focuses on the nucleation stage and develops mathematical models for HIV-1 viral capsid nucleation based on six-species dynamical systems. The Particle Swarm Optimization (PSO) algorithm is used for parameter fitting to estimate the association and dissociation rates from biological experiment data. Numerical simulations of capsid protein (CA) multimer concentrations demonstrate a good agreement with experimental data. Sensitivity and elasticity analysis of CA multimer concentrations with respect to the association and dissociation rates further reveals the importance of CA trimer-of- dimers in the nucleation stage of viral capsid self- assembly.

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).

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.

Global stability of an SEIQV epidemic model with general incidence rate

In this paper, a class of SEIQV epidemic model with general nonlinear incidence rate is investigated. By constructing Lyapunov function, it is shown that the disease-free equilibrium is globally asymptotically stable if the basic reproduction number ℛ0≤ 1. If ℛ0> 1, we show that the endemic equilibrium is globally asymptotically stable by applying Li and Muldowney geometric approach.

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.

A co-infection model of malaria and cholera diseases with optimal control

In this paper we formulate a mathematical model for malaria-cholera co-infection in order to investigate their synergistic relationship in the presence of treatments. We first analyze the single infection steady states, calculate the basic reproduction number and then investigate the existence and stability of equilibria. We then analyze the co-infection model, which is found to exhibit backward bifurcation. The impact of malaria and its treatment on the dynamics of cholera is further investigated. Secondly, we incorporate time dependent controls, using Pontryagin's Maximum Principle to derive necessary conditions for the optimal control of the disease. We found that malaria infection may be associated with an increased risk of cholera but however, cholera infection is not associated with an increased risk for malaria. Therefore, to effectively control malaria, the malaria intervention strategies by policy makers must at the same time also include cholera control.

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.

Simulating optimal vaccination times during cholera outbreaks

The use of cholera vaccines has been increasingly recognized as an effective control measure in cholera endemic countries. However, guidelines for using vaccination during cholera outbreaks are still to be established, and it remains an open question as to how and when the vaccines should be deployed to best control ongoing cholera outbreaks. Here we formulate a new optimal control model to assess the value of cholera vaccines in epidemic settings and cost-effective optimal times to deploy a vaccine. Our results suggest that as long as the vaccine prices are sufficiently low, vaccination should always start from or immediately after the onset of a cholera outbreak.