Article Contents

2020, Volume 19, Issue 5: 2513-2531. Doi: 10.3934/cpaa.2020110

This issue Previous Article Global existence and non-existence analyses to a nonlinear Klein-Gordon system with damping terms under positive initial energy Next Article Paradoxical phenomena and chaotic dynamics in epidemic models subject to vaccination

A stochastic threshold for an epidemic model with isolation and a non linear incidence

1.
Dpto. Ecuaciones Diferenciales y Análisis Numérico, Facultad de Matemáticas, Universidad de Sevilla, c/ Tarfia s/n, 41012-Sevilla, Spain
2.
Département de Mathématiques, Faculté des Sciences, Université Ibn Tofail, BP 133, Kénitra, Morocco
3.
Université Chouaib Doukkali, EST Sidi Benour, El Jadida, Morocco
4.
Université Sultan Moulay Slimane, Faculté Polydisciplinaire, Beni Mellal, Morocco

^* Corresponding author
^* Corresponding author

Received: November 2018

Revised: October 2019

Published: March 2020

The first author is supported by Ministerio de Ciencia. Innovación y Universidades (Spain), FEDER (European Community) under grant PGC2018-096540-B-I00, and Consejer\'\i a de Innovación Ciencia y Empresa de la Junta de Andaluc\'\i a (Spain) under grant P12-FQM-1492. The second and the third authors are supported by Ibn Tofail University of Kénitra (Morocco)

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Abstract

In this paper, we study a stochastic epidemic model with isolation and nonlinear incidence. In particular, we propose a stochastic threshold for the model without any sharp sufficient assumptions on model parameters as compared to existing works. Firstly, we establish the uniqueness of the global positive solution according to Lyapunov function method. Secondly, we prove stochastic permanence of the solutions. Then, we establish sufficient condition for the extinction. Thirdly, we investigate necessary and sufficient conditions for persistence in mean of the disease. Finally, we provide some numerical simulations to illustrate our theoretical results.

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Mathematics Subject Classification: 92B05, 60J60, 60J65.

Citation:

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Figure 1. Paths of stochastic and deterministic systems as given in Example 1

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Figure 2. Paths of stochastic and deterministic systems as given in Example 2

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Figure 3. Paths of stochastic and deterministic systems as given in Example 3

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Figure 4. Paths of stochastic and deterministic systems as given in Example 4

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Figure 5. Effect of quarantine

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Table 1. Table of parameter used in the numerical simulation

$ \quad A $	$ \qquad \beta $	$ \qquad\ \mu $	$ \quad \gamma $	$ \; \varepsilon $
206.04	$ 2.865\times 10^{-7} $	$ 1.3736\times 10^{-3} $	0.02011	0.1

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