Threshold dynamics of a time-delayed epidemic model for continuous imperfect-vaccine with a generalized nonmonotone inci
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ORIGINAL PAPER
Threshold dynamics of a time-delayed epidemic model for continuous imperfect-vaccine with a generalized nonmonotone incidence rate Isam Al-Darabsah
Received: 29 March 2020 / Accepted: 16 July 2020 © Springer Nature B.V. 2020
Abstract In this paper, we study the dynamics of an infectious disease in the presence of a continuousimperfect vaccine and latent period. We consider a general incidence rate function with a non-monotonicity property to interpret the psychological effect in the susceptible population. After we propose the model, we provide the well-posedness property and calculate the effective reproduction number R E . Then, we obtain the threshold dynamics of the system with respect to R E by studying the global stability of the disease-free equilibrium when R E < 1 and the system persistence when R E > 1. For the endemic equilibrium, we use the semi-discretization method to analyze its linear stability. Then, we discuss the critical vaccination coverage rate that is required to eliminate the disease. Numerical simulations are provided to implement a case study regarding data of influenza patients, study the local and global sensitivity of R E < 1, construct approximate stability charts for the endemic equilibrium over different parameter spaces and explore the sensitivity of the proposed model solutions. Keywords Epidemic model · Delay differential equations · Latent period · Vaccination · Persistence · Global stability Mathematics Subject Classification 92D30 · 34D20 I. Al-Darabsah (B) Department of Applied Mathematics, University of Waterloo, Waterloo, ON N2L 3G1, Canada e-mail: [email protected]
1 Introduction In 1979, Cooke introduced a “time delay” to represent the disease incubation period in studying the spread of an infectious disease transmitted by a vector in [1]. Since then, many authors have incorporated time delays in epidemic models in different scenarios, such as vaccination period [2], asymptomatic carriage period [3], immune period [4] and incubation period or latent period [3–7]. More precisely, in [3], a disease transmission model with two delays in incubation and asymptomatic carriage periods is investigated. In [4], the authors study an SEIRS epidemic model with constant latent and immune periods. In [5], a latent period and relapse are considered in a general mathematical model for disease transmission. In [7], the authors studied a time-delayed SIR model with nonlinear incidence rate and Holling functional type II treatment rate for epidemic transmission. Also, many authors studied time-delayed epidemic models with vaccination [8– 11]. For example, the authors in [9] study a vaccination model with a time delay to represent the time that an unaware susceptible individual takes to become aware of the infection. Due to the inherent complexity of epidemiological transmission, other works studied epidemic complex network models. For example, in [12], the authors studied a semi-random epidemic network and discussed the relationship between its topological structure
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