Stochastic and deterministic mathematical model of cholera disease dynamics with direct transmission

Getachew Teshome Tilahun,Aychew Wondifraw,Woldegebriel Assefa Woldegerima

doi:10.1186/s13662-020-03130-w

Getachew Teshome Tilahun, Aychew Wondifraw + Show 1 more

Open Access

https://doi.org/10.1186/s13662-020-03130-w

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Abstract

In this paper we develop a stochastic mathematical model of cholera disease dynamics by considering direct contact transmission pathway. The model considers four compartments, namely susceptible humans, infectious humans, treated humans, and recovered humans. Firstly, we develop a deterministic mathematical model of cholera. Since the deterministic model does not consider the randomness process or environmental factors, we converted it to a stochastic model. Then, for both types of models, the qualitative behaviors, such as the invariant region, the existence of a positive invariant solution, the two equilibrium points (disease-free and endemic equilibrium), and their stabilities (local as well as global stability) of the model are studied. Moreover, the basic reproduction numbers are obtained for both models and compared. From the comparison, we obtained that the basic reproduction number of the stochastic model is much smaller than that of the deterministic one, which means that the stochastic approach is more realistic. Finally, we performed sensitivity analysis and numerical simulations. The numerical simulation results show that reducing contact rate, improving treatment rate, and environmental sanitation are the most crucial activities to eradicate cholera disease from the community.

Highlights

According to WHO [15] the occurrence of an infectious disease causes death of thousands of individuals in a population
Like common cold and tuberculosis, are air-borne, whereas some are water-borne like acute water diarrhea (AWD) or cholera (Ochoche [12])
Since the above deterministic model in equation (2) does not consider stochastic environmental factors and lacks realistic conditions, we extended it to a stochastic model

Summary

Introduction

According to WHO [15] the occurrence of an infectious disease causes death of thousands of individuals in a population. Proof To prove Theorem 2, let us take the first equation of model (1): dS = π + δR – (αI + μ)S. 3.3 Disease-free equilibrium point To obtain a disease-free equilibrium point, set model equation (1) to zero and there are no infectious individuals in the population, which means I = 0, T = 0, R = 0. 3.4.1 Basic reproduction number for deterministic model In view of that, first let us take the newly infectious class dI = αIS – (μ + τ + σ )I. dt by the principle of generation matrix approach, we obtain f = αIS, v = (μ + τ + σ )I. Our disease-free equilibrium point is locally asymptotically stable if and only if RD0 < 1

Local stability of disease-free equilibrium in the case of stochastic model

Endemic equilibrium point