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

• * Corresponding author

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

• 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.

 Citation: • • 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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