Journal of Mathematics, Physics and Mechanics

INTRODUCTION

The battle against cholera is far from victory yet and hence investigations both theoretically and numerically to assess cholera transmission become more and more significant. Research that could be used to enhance the control, prediction or prevention of severe cholera outbreaks is of significant importance especially because mortality is so high in these out-breaks. Mathematical models are useful tools for understanding how diseases are spread and for assessing the effectiveness of interventions in various epidemiological situations.

Cholera is an acute infection of the gut which causes severe diarrhea and vomiting due to the ingestion of the bacterium Vibrio cholerae spread by ingesting contaminated water or food. It has a short incubation period (1-5 days). After infection, Vibrio cholerae is able to produce an enterotoxin that hastens the dehydration process and renders the organisms incapable of absorbing fluids, ultimately producing death if treatment does not occur within a few hours after infection [1]. Crooks and Atesmachew report that access to potable drinking water and good environment sanitation is still a major problem in many affected areas of Africa and Asia and the incidence of cholera is still on the increase [2].

Cholera is one of the most ancient diseases to impact both children and adults, and, despite its long control, it is still causing epidemic and pandemic outbreaks. Cholera is a disease that has many causes that are environmental. Changes in the hydrological cycle can signif-icantly affect pathogen levels in water bodies as Vibrio cholerae persists and proliferates in water. The rainfall pattern and intensity can either increase or decrease the transmission with droughts or floods [3]. These environmental disruptions are usually sudden and er-ratic and result in unforeseen contamination of water sources, displacement of people, and disruption of sanitation and can lead to unexpected cases of cholera.

After onset of symptoms, the person suffering from cholera has watery diarrhea without pain, profuse vomiting, irregular heart rate and low blood pressure [4]. Many infected peo-ple do not experience any symptoms or have only mild symptoms, but continue to excrete the pathogen in their stool for 7-14 days after infection making water sources and susceptible people vulnerable to contamination [5]. About ten percent of those infected get seriously ill, with the potential for dehydration and death to become very rapid without treatment. There are both direct and indirect routes of transmission for cholera [6]. Direct transmission is defined as close contact with infected persons and indirect transmission is through inges-tion of food or water contaminated with Vibrio cholerae [7]. The environmental reservoir is thus an important factor in the perpetuation of transmission and in the occurrence of recurrent outbreaks.

Cholera epidemics have been a growing health challenge all over the world, and much re-search effort has been focused on elucidating the epidemiological dynamics of cholera. The interaction of Vibrio cholerae, humans, and the aquatic environment have demonstrated that cholera is a much more complex process of transmission than first thought [3]. This fact has led to the development of many mathematical models to study the epidemiology and endemism of cholera. One of the first deterministic models to investigate the 1973 outbreak of cholera in the Mediterranean was developed by Capasso and Paveri Fontana [8]. For this purpose, Codeco developed an extension of this model, explicitly including the aquatic environmental reservoir in the transmission. Hntsa and Kahsay studied a fractional order cholera model, and Opoku and Afriyie analyzed the effect of environmental factors and health care interventions towards the control of cholera. The work of Mukandavire was extended by Wang and Modnak to incorporate treatment, vaccination and sanitation interventions in a cholera model. Other contributions are from Mark, Ochoche, Fatima, Togbenon and Moyo, Posny and Wang, Nyabadza, and Lemospaiaao, along with others [3,4,7,9-22].

Deterministic models have made a large improvement in the understanding of cholera dynamics, but most of them assume that environmental conditions are continuously varying with time. In fact, there are random fluctuations in the environment many of which affect cholera transmission, and sudden catastrophic events like floods, heavy rainfall, failure of water supply systems, sewage overflow, population displacement and contamination episodes. While jump processes are useful for modelling abrupt and unpredictable events, continuous random fluctuations can be represented by Brownian motion. SDEs with jump diffusion processes integrate these two types of uncertainty, both Wiener process and jumps driven by a Poisson process, into a single equation, hence the name, giving a more realistic representation of an epidemic system with environmental shocks. In a cholera context, a jump diffusion could be a dramatic spike or drop in the density of pathogens, a sudden shift in transmission rates, or a sudden outbreak as a result of unforeseen events like a natural disaster.

Liang, have reported mathematical models for the analysis and prediction of dynamic behavior in the biological systems are successful [23]. Inspired by the necessity to capture the continuous variation in the environment as well as outbreak triggering events, this study extends the deterministic model proposed by Codeco to a stochastic jump-diffusion model of cholera transmission [3]. First the deterministic system is analyzed, with the derivation of the basic reproduction number (R0), positivity and invariant regions of the solutions and analysis of stability of the disease-free and endemic equilibria. The model is then modified to be a stochastic jump-diffusion model by adding Gaussian white noise and Poisson jump perturbations, so that the dynamics can include sudden events in the environment that influence cholera transmission. A resulting stochastic model is shown to have unique and existing positive solutions. Lastly, numerical simulations are carried out with the Euler-Maruyama scheme and with Poisson jump simulation methods, and implemented in python. The numerical results show the effects of the intensity of the jumps and jump size on the trajectories of the infectious individuals and the concentration of the environmental pathogen; this indicates that the representation of the jumps by the jump-diffusion formulation yields a more realistic representation of cholera outbreaks with sudden disturbances in the environment and irregular surges in the number of infected individuals.

LITERATURE REVIEW

Cholera transmission dynamics and factors that influence the persistence and control of the disease have been increasingly explored through mathematical modelling. Mathematical models which include both human populations and aquatic pathogen reservoirs have been a focus of interest as cholera is strongly associated with contaminated water and environmental conditions. Later deterministic models built on the basis of early models. Later deterministic models followed on the basis of the early models. In 1973, there was an epidemic of cholera in the European Mediterranean area, which was described mathematically by Capasso and Paveri-Fontana (1979). Their work proved the value of mathematical modelling in represent-ing the transmission and the behaviour of cholera epidemics. This early contribution was important, laying the groundwork for more recent models that added environmental factors to cholera transmission.

The addition of the aquatic reservoir of Vibrio cholerae in the transmission process was a new approach in cholera modelling introduced by Code¸co (2001). The importance of the role of environmental bacteria in sustaining the endemic and epidemic dynamics of cholera were highlighted in the model. This environmental-reservoir perspective is of particular impor-tance since the density of toxigenic Vibrio cholerae in water may affect human infection and the presence of toxigenic Vibrio cholerae in water can also play a role in the survival of the disease. Later works in this series extended deterministic cholera models by including control measures and other epidemiological dynamics. Wang and Modnak (2011) studied cholera dynamics with control measures and Fatima et al. (2014) formulated a mathematical model for controlling cholera in Nigeria. The dynamics and control strategy of cholera epidemic phenomena were also investigated by Fakai (2014) with a deterministic mathematical approach. These studies showed that mathematical models can be helpful for assessing the interventions and finding mechanisms that have an impact on the spread of disease.

Several researchers have also further extended cholera models to include other control mechanisms, limited resources, periodic environments, and environmental effects. Hntsa and Kahsay (2020) explored mathematical modelling approaches to cholera epidemic control and Opoku and Afriyie (2020) explored roles of control measures and the environment in the transmission of cholera. The reproductive number is important in determining transmis-sion potential of diseases and in estimating the reproductive numbers for cholera outbreaks in Zimbabwe was shown by Mukandavire. Mark, analyzed the potential of using a bacterial virus (bacteriophage) to control the spread of cholera, and Posny and Wang examined cholera transmission in a non-continuous environment. Togbenon and Moyo added the vaccination phenomenon in cholera dynamics and Nyabadza, accounted for the effect of limited resources in cholera transmission. Lemos-Pai˜ao, investigated optimal control treatment for cholera, further demon-strating the importance of control-oriented modelling. Deterministic models have given us important insights into cholera transmission, but typically have assumed that model pa-rameters and environmental conditions are constant or can be assumed to vary according to pre-determined functions. In fact, environmental and demographic changes can impact transmission of cholera, and can add randomness to the dynamics of the disease. For this reason, the inclusion of uncertainty and random environmental changes into epidemiological systems can be designed using stochastic mathematical models.

Stochastic differential equation models have thus become an increasingly popular way to model the impact of continuous environmental stochasticity on the spread of infectious dis-eases. In these models, a typical example of continuous random fluctuations is Brownian motion. This can be very helpful with cholera as the water quality and concentration of bacteria may change continuously over time. But, in some cases, sudden environmental or epidemiological events may not be adequately captured by continued stochastic perturba-tions. Another way of modelling sudden changes in a dynamical system is to use jump processes. A jump-diffusion model is a model with continuous random fluctuation (Brown-ian motion) and discontinuous fluctuation (Poisson jump process). This framework can be used for cholera modelling where both gradual environmental variation and sudden events are deemed to be important. In this current paper, the infected human population is con-tinuously and stochastically perturbed, and also subjected to jump events, which results in the nonlinear stochastic jump-diffusion cholera model.

Scopes

The focus of this study is on the mathematical modeling and investigation of cholera transmission via contact between human and aquatic populations. The model has three main state variables: the susceptible human population χ(t), the infected human population I(t) and the amount of toxic Vibrio cholerae bacteria in the aquatic environment (F(t)). Infection occurs through contact with contaminated water and susceptible individuals are drawn into the community. Infected people excrete germs which contaminates the aquatic environment and recovers at a set rate.

This study extends the deterministic model for cholera to include two stochastic components to account for environmental uncertainty. The Poisson jump process models the abrupt and unpredictable environmental disruptions, while the continuous environmental changes are modeled by a Brownian motion. These disruptions can be triggered by an event such as floods, heavy rains, sewage overflow, pollution events or sudden changes in the environment. A deterministic cholera model is developed, the fundamental reproduction number R0 is derived, positive and invariant regions are analyzed and the stability of the endemic and disease-free equilibria is investigated.

The model is then extended to a stochastic jump-diffusion model to explore the impacts of sudden environmental shocks and continuous environmental fluctuations on cholera trans-mission. Existence and uniqueness of positive solutions are also discussed.

Finally, we conduct numerical simulations to investigate the influence of Brownian fluc-tuations, the leap size and the jump intensity on the trajectories of infected individuals and the abundance of ambient pathogens. The numerical analysis uses the Euler-Maruayama technique and Poisson jump simulation which enables the model to give several possible epidemic paths instead of one deterministic trajectory.

Model Formulation

The cholera model in adopts a classic SIR framework to explain the relationship between the human population and the aquatic environment by including the concentration of Vibrio cholerae bacteria in water F. In this paper, the random environmental fluctuations are simulated with the help of the Brownian motion (Wiener process) which leads to the stochastic extension of the deterministic model [20]. The model is composed of three compartments: bacterial concentration in water F, susceptible individuals χ and infected individuals I. The recruitment of susceptible people is at rate n. The contact of polluted water causes infection at rate $\frac{aF}{K+F}$.

The bacteria in the aquatic reservoir grow at rate b, infected individuals shed bacteria to the environment at rate e and recover at rate r.

Flow Diagram of the Aquatic Toxigenic Vibrio cholerae Model

Deterministic Model

The Deterministic Model Equations. In this model, the deterministic compartmental human population and a Vibrio cholera bacteria population. The total human population is divided into three classes which are the susceptible population χ, the cholera infectious population I and a Vibrio cholerae bacteria population F. Susceptible are recruited with the rate of n through birth, susceptible can BE infected at the rate of $\frac{aF}{K+F}$, the contact rate with untreated water and $\frac{aF}{K+F}$ is the probability of an individual to catch cholera.

Infected individuals also contribute to the enhancement of the Vibrio cholera bacteria population through excretion at the rate of e. The infected individual may recover at the rate of r. In the aquatic environment, the bacteria population also grows at a rate determined by environmental factors, and b is the size of bacteria in the aquatic environment. A susceptible individual dies naturally at a to extend the deterministic model in to a stochastic jump diffusion, we reformulate the basic model as follows. The above assumptions of the model lead to the system of ordinary differential equations shown below as in.

$$\begin{array}{cc}\hfill \frac{d\chi }{dt}& =n(H-\chi )-\frac{aF\chi }{K+F}\hfill \\ \hfill \frac{dI}{dt}& =\frac{aF\chi }{K+F}-rI\hfill \\ \hfill \frac{dF}{dt}& =F(n_q-m_q)+eI\hfill \end{array}\}\cdots \cdots \left(1\right)$$

Table 1: Model parameters and their definitions

Theorem 1

Positivity

In order to get a realistic model, our solution must be non-negative for all t. Therefore if the initial values of the parameters $\chi \left(0\right)\ge 0,I\left(0\right)\ge 0,F\left(0\right)\ge 0$, then the solution set $\{\chi \left(t\right),I\left(t\right),F\left(t\right)\}$ is nonnegative for all $t\ge 0$.

Proof

Let $t_{*}=sup\{t>0:\chi \left(t\right)>0,I\left(t\right)>0,F\left(t\right)>0\}$. Since χ(t), I(t)andF(t) are continuous, then we $t_{*}>0$. If $t_{*}=+\infty$, the positivity holds however if $0<t_{*}<+\infty$, $\chi \left(t\right)=0\mathrm{\text{or}}I\left(t\right)=0\mathrm{\text{or}}F\left(t\right)=0$

Consider the model equation

$$\frac{d\chi \left(t\right)}{dt}=nH-\frac{aF\left(t\right)\chi \left(t\right)}{K+F\left(t\right)}-n\chi \left(t\right)$$

We can wrte the above model equation as

$$\frac{d\chi \left(t\right)}{dt}+(n+\lambda \left(t\right))\chi \left(t\right)=nH$$

Where $\lambda \left(t\right)=\frac{aF\left(t\right)\chi \left(t\right)}{K+F\left(t\right)}$.

$$\frac{d\chi \left(t\right)}{dt}+(n+\lambda \left(t\right))\chi \left(t\right)=nH.$$

The integrating factor is

$$\begin{array}{cc}\hfill IF& =e^{\int (n+\lambda \left(t\right))dt}\hfill \\ \hfill & \Rightarrow e^{nt+\int \lambda \left(t\right)dt}\hfill \end{array}$$

Multiplying the differential equation by the integrating factor,

$$e^{nt+\int \lambda \left(t\right)dt}\frac{d\chi }{dt}+(n+\lambda \left(t\right))\chi \left(t\right)e^{nt+\int \lambda \left(t\right)dt}=nH{e}^{nt+\int \lambda \left(t\right)dt}$$

Hence,

$$\frac{d\chi \left(t\right)}{dt}\left(\chi \left(t\right)e^{nt+\int \lambda \left(t\right)dt}\right)=nH{e}^{nt+\int \lambda \left(t\right)dt}$$

Integrating both sides from 0 to t*,

$${\int }_0^{t_{*}}\left(\frac{d}{dt}\left[\chi \left(t\right)e^{nt+\int \lambda \left(t\right)dt}\right]\right)dt={\int }_0^{t_{*}}\left(nH{e}^{nt+\int \lambda \left(t\right)dt}\right)dt$$

Therefore,

$${\chi \left(t\right)e^{nt+\int \lambda \left(t\right)dt}|}_0^{t_{*}}={\int }_0^{t_{*}}\left(nH{e}^{nt+\int \lambda \left(t\right)dt}\right)dt$$

That is,

$$\begin{array}{cc}\hfill \chi \left(t_{*}\right)e^{nt+\int \lambda \left(t\right)dt}-\chi \left(0\right)& ={\int }_0^{t_{*}}\left(nH{e}^{nt+\int \lambda \left(t\right)dt}\right)dt\hfill \\ \hfill \chi \left(t_{*}\right)e^{nt+\int \lambda \left(t\right)dt}& =\chi \left(0\right)+{\int }_0^{t_{*}}\left(nH{e}^{nt+\int \lambda \left(t\right)dt}\right)dt\hfill \\ \hfill \chi \left(t_{*}\right)& =\chi \left(0\right)e^{-(nt+\int \lambda \left(t\right)dt)}+e^{-(nt+\int \lambda \left(t\right)dt)}{\int }_0^{t_{*}}\left(nH{e}^{nt+\int \lambda \left(t\right)dt}\right)dt\hfill \\ \hfill \chi \left(t_{*}\right)& =T_1\chi \left(0\right)+T_1{\int }_0^{t_{*}}\left(nH{e}^{nt+\int \lambda \left(t\right)dt}\right)dt\hfill \end{array}$$

where $T_1=e^{-(nt+{\int }_0^t\lambda \left(t\right)dt)}>0$.

$\chi \left(0\right)>0$. Hence, the solution $\chi \left(t_{*}\right)>0$.

Implies that $\chi \left(t_{*}\right)\ne 0$.

Also,

$$\begin{array}{cc}\hfill \frac{dI}{dt}& =\frac{aF\left(t\right)\chi \left(t\right)}{K+F\left(t\right)}-rI\left(t\right)\hfill \\ \hfill \Rightarrow \frac{dI}{dt}& =\lambda \chi \left(t\right)-rI\left(t\right)\hfill \end{array}$$

where $\lambda =\frac{aF\left(t\right)}{K+F\left(t\right)}$.

$$\begin{array}{cc}\hfill \frac{dI}{dt}+rI\left(t\right)& =\lambda \chi \left(t\right)\hfill \\ \hfill IF& =e^{\int rdt}\hfill \\ \hfill \Rightarrow IF& =e^{rt}\hfill \end{array}$$

Multiplying both sides of the above equation by the integrating factor,

$$\begin{array}{cc}\hfill e^{rt}\frac{dI}{dt}+rI\left(t\right)e^{rt}& =\lambda \chi \left(t\right)e^{rt}\hfill \\ \hfill \frac{d}{dt}\left(e^{rt}I\left(t\right)\right)& =\lambda \chi \left(t\right)e^{rt}\hfill \\ \hfill {\int }_0^{t_{*}}\left(e^{rt}\frac{dI}{dt}\right)& ={\int }_0^{t_{*}}\left(\lambda \chi \left(t\right)e^{rt}\right)dt\hfill \\ \hfill {I{e}^{rt}|}_0^{t_{*}}& ={\int }_0^{t_{*}}\left(\lambda \chi \left(t\right)e^{rt}\right)dt\hfill \\ \hfill I\left(t_{*}\right)e^{rt}-I\left(0\right)& ={\int }_0^{t_{*}}\left(\lambda \chi \left(t\right)e^{rt}\right)dt\hfill \\ \hfill I\left(t_{*}\right)e^{rt}& =I\left(0\right)+{\int }_0^{t_{*}}\left(\lambda \chi \left(t\right)e^{rt}\right)dt\hfill \\ \hfill I\left(t_{*}\right)& =I\left(0\right)e^{-rt}+e^{-rt}{\int }_0^{t_{*}}\left(\lambda \chi \left(t\right)e^{rt}\right)dt\hfill \\ \hfill I\left(t_{*}\right)& =I\left(0\right)T_1+T_1{\int }_0^{t_{*}}\left(\lambda \chi \left(t\right)e^{rt}\right)dt\hfill \end{array}$$

Where $T_1=e^{-rt}>0$.

$I\left(0\right)>0$, implies that the solution $I\left(t_{*}\right)>0$, hence $I\left(t_{*}\right)\ne 0$ The model equation,

$$\begin{array}{cc}\hfill \frac{dF}{dt}& =F\left(t\right)(m_q-n_q)+eI\left(t\right)\hfill \\ \hfill \Rightarrow \frac{dF}{dt}+F\left(t\right)(m_q-n_q)& =eI\left(t\right)\hfill \end{array}$$

To find the integrating factor of the above differential equation,

$$IF=e^{\int (m_q-n_q)dt}$$

Multiplying through by the integrating factor,

$$\begin{array}{cc}\hfill e^{\int (m_q-n_q)dt}\frac{dF}{dt}+F\left(t\right)(m_q-n_q)e^{\int (m_q-n_q)dt}& =eI\left(t\right)e^{\int (m_q-n_q)dt}\hfill \\ \hfill \Rightarrow \frac{d}{dt}\left[F\left(t\right)e^{\int (m_q-n_q)dt}\right]& =eI\left(t\right)e^{\int (m_q-n_q)dt}\hfill \end{array}$$

Integrating both sides from 0 to t,

$$F\left(t\right)e^{{\int }_0^t(m_q-n_q)d\chi }-F\left(0\right)={\int }_0^{t_{*}}eI\left(\chi \right)e^{{\int }_0^t(m_q-n_q)dt}d\chi +{\int }_0^{t_{*}}{e}^{{\int }_0^s(m_q-n_q)dt}d\chi$$

Hence,

$$\begin{array}{cc}\hfill F\left(t\right)e^{{\int }_0^t(m_q-n_q)d\chi }& =F\left(0\right)+{\int }_0^{t_{*}}\left(eI\left(\chi \right)e^{{\int }_0^t(m_q-n_q)dt}\right)d\chi \hfill \\ \hfill & +{\int }_0^{t_{*}}\left(\chi e^{{\int }_0^t(m_q-n_q)dt}\right)d\chi \hfill \end{array}$$

Multiplying by $e^{-{\int }_0^t(m_q-n_q)d\chi }$.

The equation becomes

$$F\left(t_{*}\right)=F\left(0\right)e^{-{\int }_0^t(m_q-n_q)d\chi }+{\int }_0^{t_{*}}eI\left(t\right)$$

Let $k_1=e^{-{\int }_0^t(m_q-n_q)d\chi }$.

$$\begin{array}{cc}\hfill F\left(t\right)& =F\left(0\right)e^{-\int (m_q-n_q)dt}+e^{-{\int }_0^t(m_q-n_q)dt}{\int }_0^{t_{*}}{e}^{\int (m_q-n_q)dt}\left(eI\left(t\right)\right)dt\hfill \\ \hfill e^{-\int (m_q-n_q)dt}& =k_1\hfill \\ \hfill F\left(t\right)& =F\left(0\right)k_1+k_1{\int }_0^{t_{*}}{e}^{\int (m_q-n_q)dt}eI\left(t\right)dt>0\hfill \\ \hfill k_1& =\frac{1}{e^{{\int }_0^t(m_q-n_q)dt}}>0\hfill \\ \hfill & \Rightarrow m_q-n_q>0\mathrm{\text{and}}F\left(0\right)>0\hfill \end{array}$$

Therefore

$$\begin{array}{cc}\hfill k_{1}F\left(0\right)+k_1{\int }_0^{t_{*}}{e}^{\int (m_q-n_q)dt}eI\left(t\right)dt& >0\hfill \\ \hfill k_{1}F\left(0\right)+k_1{\int }_0^{t_{*}}\left(eI\left(t\right)\right)e^{\int (m_q-n_q)dt}dt& >0\hfill \\ \hfill F\left(t\right)=k_{1}F\left(0\right)+k_1{\int }_0^{t_{*}}eI\left(t\right)e^{\int (m_q-n_q)dt}dt& >0\hfill \\ \hfill F\left(0\right)>0,I\left(t\right)>0& \Rightarrow F\left(t\right)>0\hfill \\ \hfill F\left(t\right)& \ne 0\hfill \end{array}$$

By definition t* is not finite, which implies that $t_{*}=+\infty$. Therefore the model solution of the system (1) is always positive

Theorem 2

Invariant Region for the Deterministic Model

In this section, we obtain the invariant region of the model Equation (1). The model solution are in the region

$$\Omega =\{(\chi ,I,F):\chi \ge 0,I\ge 0,0\le \chi +I\le \frac{\Delta }{n},0\le F\le \frac{e\Delta }{nq}\}\forall t_{*}\ge 0$$

Consider the total human population

$$\begin{array}{cc}\hfill H\left(t\right)& =\chi \left(t\right)+I\left(t\right)\hfill \\ \hfill \Rightarrow \frac{dH\left(t\right)}{dt}& =\frac{d\chi \left(t\right)}{dt}+\frac{dI\left(t\right)}{dt}\hfill \\ \hfill \frac{dH\left(t\right)}{dt}& =nH\left(t\right)-\frac{qF\left(t\right)\chi \left(t\right)}{K+F\left(t\right)}-nS\left(t\right)+\frac{qF\left(t\right)\chi \left(t\right)}{K+F\left(t\right)}-rI\left(t\right)\hfill \end{array}$$

Let $nH=\Delta$, $\chi =H-I$.

Therefore we have

$$\begin{array}{cc}\hfill \frac{dH\left(t\right)}{dt}& =\Delta -n(H-I)-rI\left(t\right)\hfill \\ \hfill \frac{dH\left(t\right)}{dt}& =\Delta -nH\left(t\right)+nI\left(t\right)-rI\left(t\right)\hfill \\ \hfill \frac{dH\left(t\right)}{dt}& \le \Delta -nH\left(t\right)\hfill \\ \hfill \Rightarrow H\left(t\right)& \le \frac{\Delta }{n}+K{e}^{-nt}\hfill \\ \hfill \mathrm{\text{At}}t=0,H\left(0\right)-\frac{\Delta }{n}& \le K\hfill \\ \hfill H\left(t\right)& \le \frac{\Delta }{n}+(H\left(0\right)-\frac{\Delta }{n})e^{-nt}\hfill \\ \hfill \Rightarrow H\left(t\right)& \le \frac{\Delta }{n}\mathrm{\text{and}}0\le H\left(t\right)\le \frac{\Delta }{n}\hfill \\ \hfill \mathrm{\text{Similarly}}\frac{dF\left(t\right)}{dt}& =eI\left(t\right)-F(m_q-n_q)\hfill \end{array}$$

$$\begin{array}{cc}\hfill \frac{dF\left(t\right)}{dt}& =eI\left(t\right)-qF(m_q-n_q),q=(m_q-n_q)\hfill \\ \hfill \frac{dF\left(t\right)}{dt}+qF& =eI\left(t\right)\hfill \end{array}$$

To find the integrating factor of the above first order DE

$$\begin{array}{cc}\hfill IF& =e^{\int qdt}\hfill \\ \hfill \int \frac{d}{dt}\left(F{e}^{\int qdt}\right)& =\int eI\left(t\right)e^{\int qdt}dt\hfill \\ \hfill F{e}^{\int qdt}& =\frac{eI\left(t\right)}{q}{e}^{\int qdt}+K\hfill \\ \hfill F\left(t\right)& =\frac{eI}{q}+K{e}^{-qt}\hfill \\ \hfill \mathrm{\text{At}}t=0,F\left(0\right)-\frac{e\Delta }{nq}& \le K\hfill \end{array}$$

Hence the feasible solution set of the system of equation (1) is in the region $\Omega =\{(\chi ,I,F):\chi \ge 0,I\ge 0,0\le \chi +I\le \frac{\Delta }{n},0\le F\le \frac{e\Delta }{nq}\}$. Therefore the model is well posed and have biological sense.

Existence and Stability of disease-Free equilibrium

The model's disease free equilibrium is obtained by equating the susceptible χ, infected I and the disease compartment F.

$$\frac{d\chi }{dt}=\frac{dI}{dt}=\frac{dF}{dt}=0$$

This implies that,

$$\begin{array}{cc}\hfill \frac{d\chi }{dt}& =n(H-\chi )-\frac{aF\chi }{K+F}=0\hfill \\ \hfill \frac{dI}{dt}& =\frac{aF\chi }{K+F}-rI=0\hfill \\ \hfill \frac{dF}{dt}& =eI+F(n_q-m_q)=0\hfill \end{array}$$

Hence, the equilibrium equations are

$$n(H-\chi )-\frac{aF\chi }{K+F}=0\cdots \left(2\right)$$

$$\frac{aF\chi }{K+F}-rI=0\cdots \cdots \cdots \left(3\right)$$

$$eI+F(n_q-m_q)=0\cdots \left(4\right)$$

This implies that there exists no infection at the disease-free equilibrium; Thus, $I=0$

By substituting $I=0$ into equation (4), we have, $F(n_q-m_q)=0$ it follows that $F=0$.

Provided that $(n_q-m_q)\ne 0$.

Also, substituting $F=0$ into equation (2), we obtain $n(H-\chi )=0$. Since $n\ne 0$, $\chi =H$.

Therefore, the disease-free equilibrium is $E_0=(H,0,0)$.

$$\begin{array}{cc}\hfill F\left(t\right)& \le \frac{e\Delta }{nq}+(F\left(0\right)-\frac{e\Delta }{nq})e^{-qt}\hfill \\ \hfill \underset{t\to \infty }{lim}F\left(t\right)& \le \frac{e\Delta }{nq}\hfill \\ \hfill 0\le F\left(t\right)& \le \frac{e\Delta }{nq}\hfill \\ \hfill \Omega & =\{(\chi ,I,F):\chi \ge 0,I\ge 0,0\le \chi +I\le \frac{\Delta }{n},0\le F\le \frac{e\Delta }{nq}\}\hfill \end{array}$$

Hence, the model is well-posed and biologically meaningful, as the jump size alters F.

Disease-Free Equilibrium Stability

To analyze the disease-free equilibrium stability, let us consider taking the Jacobian of the model equations (2) and (4) to prove that they are locally asymptotically stable around the equilibrium point as in [3]. When it comes to disease spread, local asymptotic stability means that if a small change or perturbation is introduced into the system, the system will still return to the disease-free equilibrium.

Basic Reproduction Number for the Deterministic Model

The basic reproduction number R0 is calculated from the disease compartment as follows:

• F_i: rate of appearance of new infections,
• V_i^-: transfer rate out of the disease compartment,
• V_i⁺: transfer rate into the disease compartment by other means.

$$\begin{array}{cc}\hfill V_i^{-}& =\left(\begin{array}{c}\hfill rI\hfill \\ \hfill 0\hfill \end{array}\right),V_i^{+}=\left(\begin{array}{c}\hfill 0\hfill \\ \hfill (m_q-n_q)+eI\hfill \end{array}\right)\hfill \\ \hfill V_i=V_i^{-}-V_i^{+}& =\left(\begin{array}{c}\hfill rI-0\hfill \\ \hfill -(m_q-n_q)-eI\hfill \end{array}\right)\hfill \\ \hfill V_i& =\left(\begin{array}{c}\hfill rI\hfill \\ \hfill (m_q-n_q)-eI\hfill \end{array}\right)\hfill \end{array}$$

$\frac{\partial F\left(E_0\right)}{\partial {\chi }_i}$, the Jacobian of F with respect to the disease compartment (χi) evaluated at the disease-free equilibrium.

The model disease-free equilibrium point is obtained by setting

$$\begin{array}{cc}\hfill \frac{d\chi \left(t\right)}{dt}=\frac{dI\left(t\right)}{dt}=\frac{dF\left(t\right)}{dt}& =0\hfill \\ \hfill nH\left(t\right)-\frac{aF\left(t\right)\chi \left(t\right)}{K+F\left(t\right)}-n\chi \left(t\right)& =0\hfill \\ \hfill \frac{aF\left(t\right)\chi \left(t\right)}{K+F\left(t\right)}-rI\left(t\right)& =0\hfill \\ \hfill (n_q-m_q)F\left(t\right)+eI\left(t\right)& =0\hfill \\ \hfill nH\left(t\right)-(\frac{aF\left(t\right)}{K+F\left(t\right)}+n)\chi \left(t\right)& =0\hfill \end{array}$$

Therefore

$$\begin{array}{cc}\hfill {\chi }^{*}& =\frac{nH}{\frac{a{F}^{*}}{K+F^{*}}+n}\hfill \\ \hfill I^{*}& =\frac{a{F}^{*}{S}^{*}}{r(K+F^{*})}\hfill \\ \hfill F^{*}& =\frac{e{I}^{*}}{(n_q-m_q)}\hfill \end{array}$$

From the equation above, $I^{*}=0$ and $(n_q-m_q)=0$ provided that $(n_q-m_q)\ne 0$. By putting $F^{*}=0$ in to ${\chi }^{*}=\frac{nH}{\frac{a{F}^{*}}{K+F^{*}}+n}$, we have ${\chi }^{*}=H$.

Therefore,

$$E_0=(\chi \left(t\right),I\left(t\right),F\left(t\right))=(H,0,0)$$

The Jacobian matrix of is

$$F=\left(\begin{array}{cc}\hfill \frac{\partial m_1}{\partial I}\hfill & \hfill \frac{\partial m_1}{\partial F}\hfill \\ \hfill \frac{\partial m_2}{\partial I}\hfill & \hfill \frac{\partial m_2}{\partial F}\hfill \end{array}\right).$$

where $m_1=\frac{aF\left(t\right)\chi \left(t\right)}{K+F\left(t\right)}-rI\left(t\right)$ and $m_2=(n_q-m_q)+eI\left(t\right)$. Thus,

$$F=\left(\begin{array}{cc}\hfill 0\hfill & \hfill \frac{a\chi F}{{(K+F)}^2}\hfill \\ \hfill 0\hfill & \hfill 0\hfill \end{array}\right).$$

At the disease-free equilibrium $\chi =H,F=0$. Hence

$$F=\left(\begin{array}{cc}\hfill 0\hfill & \hfill \frac{aH}{K}\hfill \\ \hfill 0\hfill & \hfill 0\hfill \end{array}\right).$$

Also, $V=\frac{\partial V_i\left(E_0\right)}{\partial {\chi }_i}$, which is the Jacobian matrix of Vi with respect to the disease compartments χi, evaluated at the disease-free equilibrium (D.F.E).

Therefore,

$$V=\left(\begin{array}{cc}\hfill r\hfill & \hfill 0\hfill \\ \hfill -e\hfill & \hfill (m_q-n_q)\hfill \end{array}\right)$$

$$V^{-1}=\frac{1}{r(m_q-n_q)}\left(\begin{array}{cc}\hfill (m_q-n_q)\hfill & \hfill 0\hfill \\ \hfill e\hfill & \hfill r\hfill \end{array}\right)$$

$$V^{-1}=\left(\begin{array}{cc}\hfill \frac{1}{r}\hfill & \hfill 0\hfill \\ \hfill \frac{e}{r(m_q-n_q)}\hfill & \hfill \frac{1}{(m_q-n_q)}\hfill \end{array}\right)$$

$$\begin{array}{cc}\hfill F{V}^{-1}& =\left(\begin{array}{cc}\hfill 0\hfill & \hfill \frac{qH}{K}\hfill \\ \hfill 0\hfill & \hfill 0\hfill \end{array}\right)\left(\begin{array}{cc}\hfill \frac{1}{r}\hfill & \hfill 0\hfill \\ \hfill \frac{e}{r(m_q-n_q)}\hfill & \hfill \frac{1}{(m_q-n_q)}\hfill \end{array}\right)\hfill \\ \hfill & =\left(\begin{array}{cc}\hfill 0\hfill & \hfill \frac{qH}{K}\hfill \\ \hfill 0\hfill & \hfill 0\hfill \end{array}\right)\left(\begin{array}{cc}\hfill \frac{1}{r}\hfill & \hfill 0\hfill \\ \hfill \frac{e}{r(m_q-n_q)}\hfill & \hfill \frac{1}{(m_q-n_q)}\hfill \end{array}\right)\hfill \\ \hfill & =\left(\begin{array}{cc}\hfill \frac{aeH}{Kr(m_q-n_q)}\hfill & \hfill \frac{qH}{K(m_q-n_q)}\hfill \\ \hfill 0\hfill & \hfill 0\hfill \end{array}\right)\hfill \end{array}$$

The eigenvalues satisfy

$$\begin{array}{cc}\hfill \mid \begin{array}{cc}\hfill \frac{qHe}{Kr(m_q-n_q)}-\lambda \hfill & \hfill \frac{qH}{K(m_q-n_q)}\hfill \\ \hfill 0\hfill & \hfill -\lambda \hfill \end{array}\mid & =0\hfill \\ \hfill (\frac{qHe}{Kr(m_q-n_q)}-\lambda )\left(-\lambda \right)& =0\hfill \\ \hfill \frac{qHe}{Kr(m_q-n_q)}-\lambda & =0\mathrm{\text{or}}\lambda =0\hfill \\ \hfill {\lambda }_1& =\frac{qHe}{Kr(m_q-n_q)},{\lambda }_2=0\hfill \end{array}$$

Clearly λ1 is the dominant eigenvalue and therefore becomes the basic reproduction number R0 of the model:

$$R_0=\frac{aeH}{Kr(m_q-n_q)}$$

Theorem 3

The disease-free equilibrium point of the system is asymptotically stable if and only if $R_0<1$.

proof

$$J=\left(\begin{array}{ccc}\hfill -n-\frac{aF}{K+F}\hfill & \hfill 0\hfill & \hfill -\frac{a\chi K}{{(K+F)}^2}\hfill \\ \hfill \frac{aF}{K+F}\hfill & \hfill -r\hfill & \hfill \frac{a\chi K}{{(K+F)}^2}\hfill \\ \hfill 0\hfill & \hfill e\hfill & \hfill (n_q-m_q)\hfill \end{array}\right)$$

Also, at the disease-free equilibrium point, $E_0=(H,0,0)=({\chi }_0,I_0,F_0)$. We have

$$J\left(E_0\right)=\left(\begin{array}{ccc}\hfill -n\hfill & \hfill 0\hfill & \hfill -\frac{aH}{K}\hfill \\ \hfill 0\hfill & \hfill -r\hfill & \hfill \frac{aH}{K}\hfill \\ \hfill 0\hfill & \hfill e\hfill & \hfill (n_q-m_q)\hfill \end{array}\right)$$

The equilibrium point is asymptotically stable if the following condition (Routh–Hurwitz) holds for the characteristic polynomial and its determinant

$$\begin{array}{cc}\hfill |J({\chi }_0,I_0,F_0)-\lambda I|& =\mid \begin{array}{ccc}\hfill -n-\lambda \hfill & \hfill 0\hfill & \hfill -\frac{aH}{K}\hfill \\ \hfill 0\hfill & \hfill -r-\lambda \hfill & \hfill \frac{aH}{K}\hfill \\ \hfill 0\hfill & \hfill e\hfill & \hfill (n_q-m_q)-\lambda \hfill \end{array}\mid \hfill \\ \hfill & ={\lambda }^3+(n+r){\lambda }^2+\left((n_q-m_q)\right){\lambda }^2+((n_q-m_q)(n+r)-rn+\frac{aHe}{K})\lambda \hfill \\ \hfill & +(\frac{aHe}{K}+(n_q-m_q)r)n=0\hfill \end{array}$$

It is observed that the Jacobian matrix characteristic equation has three roots. Now we write the equation of characteristic as

$$P\left(\lambda \right)={\lambda }^3+A{\lambda }^2+\lambda +C=0.$$

where

$$\begin{array}{cc}\hfill A& =n+r+(m_q-n_q)\hfill \\ \hfill F& =nr+(n+r)(m_q-n_q)-\frac{aHe}{K}\times \frac{r(m_q-n_q)}{r-(m_q-n_q)}\hfill \\ \hfill & =nr+(n+r)(m_q-n_q)-\frac{aHe}{K}\hfill \\ \hfill & =nr+(n+r)(m_q-n_q)-\frac{aHe}{K}\times \frac{r(m_q-n_q)}{Kr(m_q-n_q)}\hfill \\ \hfill F& =nr+(m_q-n_q+r)+(m_q-n_q)(1-R_0)\hfill \\ \hfill C& =nr(m_q-n_q)-\frac{aHen}{K}=nr(m_q-n_q)(1-R_0)\hfill \\ \hfill AB-C& =((m_q-n_q){\delta }_3+n+r)(n(m_q-n_q)+r)-(m_q-n_q)(1-R_0)\hfill \end{array}$$

Hence the eigenvalues that correspond to the equilibrium E0 are

$$\begin{array}{cc}\hfill {\lambda }_1& =-n\hfill \\ \hfill {\lambda }_{2,3}& =\frac{-b\pm \sqrt{b^2-4ac}}{2a}\hfill \\ \hfill {\lambda }_{2,3}& =\frac{-(m_q-n_q+r)\pm \sqrt{{\left((m_q-n_q+r)\right)}^2-4(m_q-n_q)(1-R_0)}}{2}\hfill \end{array}$$

We have $A>0,B>0,C>0,AB-C>0$ and according to the Routh-Hurwitz criterion, all the characteristic equation roots have negative real part when $R_0<1$ and $(m_q-n_q)>0$. Hence, the disease-free equilibrium is stable, given that $R_0<1$.

Existence and stability of endemic equilibrium

In the endemic equilibrium state $I\ne 0$, we use equations: 2, 3, 4 to find the endemic equilibrium state.

$$n(H-\chi )-\frac{aF\chi }{K+F}=0,\left(2\right)$$

$$\frac{qF\chi }{K+F}-rI=0,\left(3\right)$$

$$F(m_q-n_q)+eI=0,\left(4\right)$$

This implies,

$$F=\frac{eI}{(m_q-n_q)}.\left(5\right)$$

$$\chi =\frac{(K+F)rI}{aF}.\left(6\right)$$

$$nH-n\chi +\frac{aF\chi }{K+F}=0.\left(7\right)$$

Substituting the value of χ in equation (7)

$$\begin{array}{cc}\hfill nH-\frac{(K+F)rI}{aF}(n+\frac{aF}{K+F})& =0\hfill \\ \hfill nH-\left(\frac{K+F}{qF}rI\right)n-\left(\frac{K+F}{aF}rI\right)\left(\frac{aF}{K+F}\right)& =0\hfill \\ \hfill nH+\left(\frac{K+F}{aF}rI\right)n-rI& =0\hfill \\ \hfill nH-rI(n\left(\frac{K+F}{aF}\right)+1)& =0\hfill \\ \hfill nH-rI\left(\frac{n(K+F)+aF}{aF}\right)& =0\hfill \end{array}$$

Since $F=\frac{eI}{n_q-m_q}$.

We obtain

$$I=\frac{nHa-rn(n_q-m_q)K}{e(a+n)}$$

Or equivalently,

$$I=\frac{nHa-rn(n_q-m_q)K}{er(a+n)}$$

Suppose $I>0$. Then $nHa-rn(n_q-m_q)K>0$, which implies $R_0>1$ since

$$R_0=\frac{aHe}{rK(n_q-m_q)}>1$$

We observed that from the analysis, if $R_0>0$ and $(m_q-n_q)>0$, there exist a positive endemic equilibrium state.

Theorem 4

The endemic equilibrium state E* is locally asymptotically stable if $R_0>1$.

Proof

Let $E^{*}=({\chi }^{*},I^{*},F^{*})$. The Jacobian matrix evaluated at the endemic equilibrium is

$$J\left(E^{*}\right)=\left(\begin{array}{ccc}\hfill -n-u\hfill & \hfill 0\hfill & \hfill -U\hfill \\ \hfill u\hfill & \hfill -r\hfill & \hfill U\hfill \\ \hfill 0\hfill & \hfill e\hfill & \hfill -A\hfill \end{array}\right).$$

Therefore,

$$\begin{array}{cc}\hfill |J\left(E^{*}\right)-\lambda I|& =\mid \begin{array}{ccc}\hfill -n-u-\lambda \hfill & \hfill 0\hfill & \hfill -U\hfill \\ \hfill u\hfill & \hfill -r-\lambda \hfill & \hfill U\hfill \\ \hfill 0\hfill & \hfill e\hfill & \hfill -A-\lambda \hfill \end{array}\mid \hfill \\ \hfill & =-{\lambda }^3-{\lambda }^2(n+u+r+A)-\lambda (nr+nA+ur+uA+rA-ue)\hfill \\ \hfill & -nrA-urA+nue+uAe=0\hfill \\ \hfill {\lambda }^3+{\lambda }^2(n+u+r+A)+\lambda (nr+nA+ur+uA+rA-ue)& +nrA+urA-nue-uAe=0\hfill \\ \hfill {\lambda }^3+a_1{\lambda }^2+a_2\lambda +a_3& =0\hfill \\ \hfill a_1& =n+u+r+A\hfill \\ \hfill a_2& =nr+nA+ur+uA+rA-ue\hfill \\ \hfill a_3& =nrA+urA-nue-uAe\hfill \end{array}$$

From the Routh–Hurwitz criterion, the equilibrium is locally asymptotically stable provided

$$\begin{array}{cc}\hfill a_0& >0,a_1>0,a_2>0,a_3>0,\hfill \\ \hfill A& =\left(\begin{array}{cc}\hfill a_0\hfill & \hfill a_1\hfill \\ \hfill a_2\hfill & \hfill a_3\hfill \end{array}\right)\hfill \\ \hfill a_1{a}_2-a_0{a}_3& >0\hfill \end{array}$$

Which implies that the endemic equilibrium state is locally asymptotically stable. That is all the roots of the characteristic polynomial have negative real parts.

Clearly, $a_0>0,a_1>0$ since $n,u>0$ and $R_0>1$. Similarly, $a_2>0,a_3>0$ since $n,e,r,q>0$ and $(nl-mwI)>0$.

And $R_0>1$ if and only if $nr+nA+ur+rA>ue$ and $nrA+urA>nue$. Now, we need to show that $a_1{a}_2-a_0{a}_3>0$.

$$\begin{array}{cc}\hfill & (n+u+r+A)(nr+nA+ur+uA+rA-ue)-(nrA+urA-nue)\hfill \\ \hfill & =n^{2}r+n^{2}A+nur+nuA+n{r}^2+nrA-nue+unr+unA+u^{2}r+u^{2}A+urA-u^{2}e\hfill \\ \hfill & +n{r}^2+nrA+r^{2}u+urA+A{r}^2-rue+nAr+n{A}^2+urA+u{A}^2+r{A}^2-Aue-nrA-urA+nue>0.\hfill \end{array}$$

Therefore, for $R_0>1$, the model system (1) has a unique endemic equilibrium E* and it is stable if

$$\begin{array}{c}\hfill n^{2}r+n^{2}v+nwr+nuA+wnr+wuA+w^{2}r+w^{2}A+wrA+n{r}^2+nrA+w{r}^2+wrA+A{r}^2+nAr\\ \hfill +n{A}^2+wrA+wuA+r{A}^2& >(wue+rue+Aue),\hfill \\ \hfill (mA-nA)& >0.\hfill \end{array}$$

Transition Probabilities

Consider the stochastic differential equation with jumps

$$dX\left(t\right)=f(X\left(t\right),t)dt+g(X\left(t\right),t)dW\left(t\right)+h(X\left(t\right),t)dN\left(t\right)$$

where f(X, t) is the deterministic (drift) part, g(X(t), t) is the diffusion part and h(X(t), t) is the jump-size part.

The drift coefficient is given by

$$f(X\left(t\right),t)=E\left[\frac{\Delta X}{\Delta t}\right]$$

The diffusion coefficient is

$$g(X\left(t\right),t)=\sqrt{V}=\sqrt{E\left[\frac{\Delta X{\left(\Delta X\right)}^T}{\Delta t}\right]}$$

The jump-size coefficient is $h(X\left(t\right),t)=\lambda J\left(t\right)$.

Since time is a continuous variable, the state variables χ(t) and I(t) are continuous random variables. Let $\Delta X=X(t+\Delta t)-X\left(t\right)$ and $\Delta I=I(t+\Delta t)-I\left(t\right)$. These increments are approximately normally distributed,

$$\begin{array}{cc}\hfill \Delta X\left(t\right)& \sim N(\mu \left(X\right)\Delta t,{\sigma }^2\left(X\right)\Delta t)\mathrm{\text{and}}\hfill \\ \hfill \Delta I\left(t\right)& \sim N(\mu \left(I\right)\Delta t,{\sigma }^2\left(I\right)\Delta t)\hfill \end{array}$$

For small time intervals Δt, assume $X\left(t\right)={(X_1,X_2)}^T$ where X1 and X2 correspond to the number of individuals χ(t) and I(t), respectively, with $\chi \left(t\right),I\left(t\right)\in (0,H)$ and F(t) denotes the concentration of Vibrio cholerae bacteria.

$$E\left(\Delta X\right)=\sum _{i=1}^4{P}_i\Delta X_i+\lambda J\left(t\right).$$

According to Allen [?], $g=\sqrt{V}$. Hence,

$$g=V^{\frac{1}{2}}=\frac{1}{\eta }\left(\begin{array}{cc}\hfill \gamma +\Theta \hfill & \hfill \rho \hfill \\ \hfill \rho \hfill & \hfill \psi +\Theta \hfill \end{array}\right)$$

Where $\Theta =\sqrt{\gamma \psi -{\rho }^2}$, $\eta =\sqrt{\gamma +\psi +2\Theta }$ and

$$\gamma =n(H+\chi )+\frac{aF\chi }{K+B},\rho =-\frac{aF\chi }{K+F},\psi =\frac{aF\chi }{K+F}+rI.$$

Substituting the expressions for γ, ρ, ψ

$$\Theta =\sqrt{(n(H+S)+\frac{aF\chi }{K+F})(\frac{aF\chi }{K+F}+rI)-{\left(-\frac{aF\chi }{K+F}\right)}^2}.$$

Expanding,

$$\Theta =\sqrt{n(H+\chi )\frac{aF\chi }{K+F}+\frac{aF\chi }{K+F}rI+n(H+\chi )rI}.$$

Similarly ,

$$\eta =\sqrt{n(H+\chi )+\frac{a}{K+F}+\frac{aF\chi }{K+F}+rI+2\Theta }$$

The stochastic differential equations are

$$d\chi \left(t\right)=(n(H-\chi )-\frac{aF\chi }{K+F})dt+\left(\frac{\gamma +\theta }{\eta }\right)dW\left(t\right)+\frac{\rho }{\eta }dW\left(t\right)+JdN\left(t\right)$$