TAILIEUCHUNG - Dynamics of a stochastic epidemic model with markov switching and general incidence rate

In this paper, the stochastic SIR epidemic model with Markov switching and general incidence rate is investigated. We classify the model by introducing a threshold value λ. To be more specific, we show that if λ 0 | Trường Dại học Vinh Tạp chí khoa học Tập 47 Số 3A 2018 tr. 17-27 DYNAMICS OF A STOCHASTIC EPIDEMIC MODEL WITH MARKOV SWITCHING AND GENERAL INCIDENCE RATE Nguyen Thanh Dieu 1 Nguyen Duc Toan 2 Vuong Thi Hai Ha 3 1 School of Natural Sciences Education Vinh University 2High School for Gifted Students Vinh University 3 Fundametal Sciences Faculty Vinh Medical University Received on 30 10 2018 accepted for publication on 28 11 2018 Abstract In this paper the stochastic SIR epidemic model with Markov switching and general incidence rate is investigated. We classify the model by introducing a threshold value A. To be more specific we show that if A 0 then the disease-free is globally asymptotic stable . the disease will eventually disappear while the epidemic is strongly stochastically permanent provided that A 0. We also give some of numerical examples to illustrate our results. 1 Introduction The idea of using mathematical models to investigate disease transmissions and behavior of epidemics was first introduced by Kermack and McKendrick in 11 12 . Since then much attention has been devoted to analyzing predicting the spread and designing controls of infectious diseases in host populations see 2 3 4 13 14 16 and the references therein . 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. It is suitable for some infectious diseases of permanent or long immunity such as chickenpox smallpox measles etc. As we all know the incidence rate of a disease is the number of new cases per unit time and it plays an important role in the investigation of mathematical epidemiology. Therefore during the last few decades a number of realistic transmission functions have become the focus of considerable attention. Concreterly in 10 authors studied a deterministic SIR model with

• Toán học
• Stochastic epidemic model
• Markov switching
• General incidence rate
• Disease free
• Threshold value λ