Long-Time Behavior of a Stochastic SIQR Model with Markov Switching
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Long-Time Behavior of a Stochastic SIQR Model with Markov Switching Nguyen Huu Du1 · Nguyen Thanh Dieu2 · Tran Quan Ky3 · Vu Hai Sam4 Received: 8 December 2018 / Revised: 16 June 2019 / Accepted: 28 June 2019 / © Institute of Mathematics, Vietnam Academy of Science and Technology (VAST) and Springer Nature Singapore Pte Ltd. 2020
Abstract In this paper, we consider a stochastic SIQR model with a perturbed disease transmission coefficient. We determine the threshold λ that is used to classify the extinction and permanence of the model. Concretely, λ < 0 implies that the disease-free equilibrium (K, 0, 0, 0) is globally asymptotically stable; that is, the disease will disappear and the entire population will become susceptible individuals. If λ > 0, the model is strongly stochastically permanent. Our findings are considered significant improvements over the results in Liu et al. (Appl. Math. Comput. 316, 310–325 2018). Keywords SIQR model · Epidemic · Extinction · Permanence · Strongly stochastically permanent Mathematics Subject Classification (2010) 34C12 · 60H10 · 92D25
Nguyen Thanh Dieu
[email protected] Nguyen Huu Du [email protected] Tran Quan Ky [email protected] Vu Hai Sam [email protected] 1
Faculty of Mathematics, Mechanics, and Informatics, University of Science-VNU, 334 Nguyen Trai, Thanh Xuan, Hanoi, Vietnam
2
Department of Mathematics, Vinh University, 182 Le Duan, Vinh, Nghe An, Vietnam
3
Department of Mathematics, College of Education, Hue University, Hue, Vietnam
4
HUS High School for Gifted Student, VNU University of Science-VNU, 182 Luong The Vinh, Thanh Xuan, Hanoi, Vietnam
N.H. Du et al.
1 Introduction Since epidemic models were first introduced by Kermack and McKendrick in [9, 10], the study on such mathematical models and their extensions has flourished. Much attention has been drawn to analyzing, predicting the spread, and designing controls of infectious diseases in host populations (see [2–4, 6, 8–12] and the references therein). One important method to control the spread of infectious diseases is to isolate some infectives in order to reduce transmissions of the infection to susceptibles. In order to examine the effects of quarantine on endemic infectious diseases, the authors in [6] extended the standard SIR epidemic model to a SIQR epidemic model. In that model, a homogeneous host population is subdivided into four epidemiologically distinct types of individuals: • (S): The susceptible class, the class of those individuals who are capable of contracting the disease and becoming infective, • (I ): The infective class, the class of those individuals who are capable of transmitting the disease to others, • (Q): The quarantined class, the class of infected individuals who are removed and isolated either voluntarily or coercively from the infectious class, • (R): The removed class, the class of infected individuals who are dead, or have recovered, and are permanently immune, or are isolated. In that model, the susceptibles become infected and then some infected individuals
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