Epidemic SIS model in air-polluted environment
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Epidemic SIS model in air-polluted environment Tran Dinh Tuong1 Received: 26 January 2020 © Korean Society for Informatics and Computational Applied Mathematics 2020
Abstract In this paper, the epidemic model SIS under polluted environment is investigated. Study of epidemic models often focuses on determining under what conditions the disease will disappear or it will persist. With new techniques developed for periodic stochastic differential equations, we are able to classify the longtime behavior of the epidemic model by introducing a threshold value λ. To be more specific, we show that if λ < 0 then the disease-free is globally asymptotic stable, i.e., the disease will eventually disappear while the epidemic is strongly stochastically permanent provided that λ > 0. We also give some of numerical examples to illustrate our results. Moreover, the approach in this paper can be used to more general models. Keywords SIS epidemic models · Extinction · Permanence · Asymptotic stable Mathematics Subject Classification 37A25 · 60J60 · 60J75
1 Introduction The commonly used epidemic models nowadays, in which the density functions are spatially homogeneous, were first introduced in 1927 by Kermack and McKendrick in [16], known as compartment models. Compartmental models are a technique used to simplify the mathematical modeling of infectious disease. The population is divided into compartments, with the assumption that every individual in the same compartment has the same characteristics. Then the dynamics of these classes are formulated by use of a system of deterministic differential equations. One of classic epidemic models is the SIR model, which subdivides a homogeneous host population into three epidemiologically distinct types of individuals, the susceptible, the infective, and the removed, with their population sizes denoted by S, I and R, respectively. Precisely, the model assumes that the susceptible class can transform
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Tran Dinh Tuong [email protected] Faculty of Basic Sciences, Ho Chi Minh University of Transport, 2 Vo Oanh, Ho Chi Minh, Vietnam
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T. D. Tuong
into the infective class through the contact with infected persons, and the infectives can be recovered through treatment so that have permanent immunity. It is suitable for some infectious diseases of permanent or long immunity, such as chickenpox, smallpox, measles, etc. However, some research showed that some diseases, such as transmission dynamics of Ebola virus disease [36], viral diarrhea [9] and hand, foot and mouth disease [4], do not confer any long lasting immunity. The infections do not give immunization upon recovery from infection, and individuals become susceptible again. Individuals have repeat or reoccurring infections, and infected individuals return to the susceptible state. This model is known as the SIS model, see [17]. A differential equation model for this is described by following system of equations d S(t) = ω − β S(t)I (t) − μS(t) + γ I (t) dt d I (t) = β S(t)I (t) − (μ + γ − γ0 )I (t) dt where S(t) and
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