A stochastic SIRI epidemic model with Lévy noise

Badr-eddine Berrhazi; Mohamed El Fatini; Tomás Caraballo; Roger Pettersson

doi:10.3934/dcdsb.2018057

Article Contents

2018, Volume 23, Issue 6: 2415-2431. Doi: 10.3934/dcdsb.2018057

This issue Previous Article Qualitative analysis of kinetic-based models for tumor-immune system interaction Next Article Necessary and sufficient conditions for ergodicity of CIR model driven by stable processes with Markov switching

A stochastic SIRI epidemic model with Lévy noise

1.
Département de Mathématiques, Faculté des Sciences, Université Ibn Tofail, BP 133, Kénitra, Morocco
2.
Dpto. Ecuaciones Diferenciales y Análisis Numérico, Facultad de Matemáticas, Universidad de Sevilla, Tarfia s/n, 41012-Sevilla, Spain
3.
Department of Mathematics, Linnaeus University, 351 95 Växjö, Sweden

Received: April 2017

Revised: July 2017

Early access: February 2018

Published: June 2018

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Abstract

Some diseases such as herpes, bovine and human tuberculosis exhibit relapse in which the recovered individuals do not acquit permanent immunity but return to infectious class. Such diseases are modeled by SIRI models. In this paper, we establish the existence of a unique global positive solution for a stochastic epidemic model with relapse and jumps. We also investigate the dynamic properties of the solution around both disease-free and endemic equilibria points of the deterministic model. Furthermore, we present some numerical results to support the theoretical work.

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Mathematics Subject Classification: 92B05, 60G51, 60H30, 60G57.

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Figure 1. rajectories of the solutions to the systems (1.1) and (1.3) for Moroccan zoonotic tuberculosis with $\mathcal{R}_0\leq 1$.

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Figure 2. Trajectories of the solutions to the systems (1.1) and (1.3) for bovine tuberculosis [5] with $\mathcal{R}_0>1$, $N = \mu = 0.1,\;\beta = 0.6\;and\;\gamma = \delta = 0.5$.

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Figure 3. Trajectories of the solutions to the systems (1.1) and (1.3) with various relapse rate $\delta$: $0.14, 0.2, 0.4$.

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Figure 4. Trajectories of the solutions to the systems (1.1) and (1.3) with a various recovery rate $\gamma$: $0.09, 0.18, 0.22$

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