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

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

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  • 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.

    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. $

    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. $

    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

    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)

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

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