Periodic solutions of an age-structured epidemic model with periodic infection rate

Hao Kang; Qimin Huang; Shigui Ruan

doi:10.3934/cpaa.2020220

Article Contents

2020, Volume 19, Issue 10: 4955-4972. Doi: 10.3934/cpaa.2020220

This issue Previous Article Semilinear elliptic problems involving exponential critical growth in the half-space Next Article Mass concentration phenomenon to the two-dimensional Cauchy problem of the compressible Magnetohydrodynamic equations

Periodic solutions of an age-structured epidemic model with periodic infection rate

1.
Department of Mathematics, University of Miami, Coral Gables, FL 33146, USA
2.
Department of Mathematics, Applied Mathematics and Statistics, Case Western Reserve University, Cleveland, OH 44106, USA

^* Corresponding author
^* Corresponding author

Received: December 31, 2019

Revised: April 30, 2020

Published: July 2020

Research was partially supported by National Science Foundation (DMS-1853622)

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Abstract

In this paper we consider an age-structured epidemic model of the susceptible-exposed-infectious-recovered (SEIR) type. To characterize the seasonality of some infectious diseases such as measles, it is assumed that the infection rate is time periodic. After establishing the well-posedness of the initial-boundary value problem, we study existence of time periodic solutions of the model by using a fixed point theorem. Some numerical simulations are presented to illustrate the obtained results.

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Mathematics Subject Classification: Primary:35Q92, 35C15;Secondary:92B05.

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Figure 1. Behavior of the model when $ \mathscr{R}_0<1 $: (a) Total exposed population $ \int_{0}^{80} e(t , a)da $ versus time $ t $; (b) Total infected population $ \int_{0}^{80} i(t , a)da $ versus time $ t. $

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Figure 2. Behavior of solutions when $ \mathscr{R}_0>1 $ and the vaccination rate is zero: (a) Total susceptible population $ \int_{0}^{80} s(t , a)da $ versus time t; (b) Total exposed population $ \int_{0}^{80} e(t , a)da $ versus time t; (c) Total infected population $ \int_{0}^{80} i(t , a)da $ versus time $ t. $

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Figure 3. Effect of vaccination on the behavior of the solutions (the total exposed population $ \int_{0}^{80} e(t , a)da $ and the total infected population $ \int_{0}^{80} i(t , a)da $ versus time $ t $) and different vaccination rate $ \rho $: (a) $ \rho = 0 $; (b) $ \rho = 0.5 $; (c) $ \rho = 0.7 $; (d) $ \rho = 0.9 $. All other parameters are fixed

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Figure 4. Plots of the infected population $ i(t, a) $ and exposed population $ e(t, a) $ versus age $ a $ and time $ t $ (in three periods)

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Figure 5. Age distribution of the infected population at the peak of a periodic solution ($ t = 60.3 $)

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