Abstract
This paper is concern with modeling cholera epidemic. Despite the advances made in understanding this disease and its treatment, cholera continues to be a major public health problem in many countries. Deterministic and stochastic models emerged in modeling of cholera epidemic, in order to understand the mechanism by which cholera disease spread, conditions for cholera disease to have minor and large outbreaks. We formulate a continuous time Markov chain model for cholera epidemic transmission from the deterministic model. The basic reproduction number (R0) and the extinction thresholds of corresponding cholera continuous time Markov chain model are derived under certain assumptions. We find that, the probability of extinction (no outbreak) is 1 if R0 < 1, but less than 1 if R0 > 1. We also carry out numerical simulations using Gillespie algorithm and Runge–Kutta method to generate the sample path of cholera continuous time Markov chain model and the solution of ordinary differential equation respectively. The results show that the sample path of continuous time Markov chain model fluctuates within the solution of the ordinary differential equation.
Highlights
Cholera is a virulent disease that affects both adults and children, and can kill within few hours if remain untreated
The results show that the sample path of continuous time Markov chain model fluctuates within the solution of the ordinary differential equation
Cholera bacterium is found in water or food sources that have been contaminated by feaces from a person who is infected with cholera
Summary
Cholera is a virulent disease that affects both adults and children, and can kill within few hours if remain untreated. Continuous time Markov chain models help to determine the probability of an outbreak or disease extinction through the use of branching process theory. Extend the deterministic model developed in (Marwa et al, 2018) by formulating an equivalent cholera continuous time Markov chain model. The formulated cholera continuous time Markov chain (CTMC) model will be analyzed using branching process approximation and probability of extinction. The advantage of using continuous time Markov chain (CTMC) model over deterministic model in epidemic modelling is that the CTMC model yields an estimate of the probability of extinction and outbreaks of cholera disease (Zevika & Soewono, 2018). This section, introduce a deterministic S IsIaR − B cholera model formulated by (Marwa et al, 2018) In their model, S (t) denotes number susceptible individuals, Is(t) denotes the number of symptomatic infected individuals, Ia(t) denotes.
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