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# Stability and Hopf bifurcation in an age-structured SIR epidemic model with relapse

• *Corresponding author: Zhidong Teng

This work was supported by the Natural Science Foundation of China (Grant Nos. 11861065 and 11771373), the Natural Science Foundation of Xinjiang (Grant No. 2021D01A98), the China Postdoctoral Science Foundation (Grant No. 2020M683714XB), the Natural Science Basic Research Plan in Shaanxi Province of China (Grant No. 2021JM-445), the Natural Science Basic Research Plan in Shaanxi Province of China (Grant No. 2022JM-023)

• An SIR epidemic model with age structure in the infected class is investigated. The model is transformed into a non-densely defined abstract Cauchy problem. The positivity and boundedness of solutions, basic reproduction number $\mathcal R_0$ and the existence of equilibria are established. The linearized system and characteristic equation at an equilibrium from the corresponding abstract Cauchy problem are obtained. When $\mathcal R_0\leq 1$, local and global stability of the disease-free equilibrium is proved, and hence the disease will be deracinated. For the model with the latent period described by infection age, when $\mathcal R_0>1$ and Assumptions 5.1 and 5.2 are satisfied, local stability of the endemic equilibrium and the existence of Hopf bifurcation are established. This shows that the disease with the latent period of infection age has complex dynamical behavior at the endemic equilibrium. Finally, numerical examples are presented to verify the theoretical results.

Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.

 Citation:

• Figure 1.  Flow diagram of model (1)

Figure 2.  $(S(t), i(t, a), R(t))$ in model (1) converges to the disease-free equilibrium $P^0$ as $t\to\infty$

Figure 3.  $(i(t, a), s(t, a), r(t, a))$ in system (3) converges to the disease-free equilibrium $w^0(a)$ as $t\to\infty$

Figure 4.  $(S(t), i(t, a), R(t))$ converges to the endemic equilibrium $P^*$ as $t\to\infty$

Figure 5.  $(a)$ $(S(t), i(t, a), R(t))$ converges to the equilibrium $P^*$ for $\tau = 0.5<\tau_{1}$; $(b)$ $(S(t), i(t, a), R(t))$ converges to a periodic solution for $\tau = 1.5\in(\tau_{1}, \tau_{2})$; $(c)$ $(S(t), i(t, a), R(t))$ converges to the equilibrium $P^*$ for $\tau = 4\in(\tau_{2}, \tau_{3})$; $(d)$ $(S(t), i(t, a), R(t))$ converges to a periodic solution for $\tau = 7\in(\tau_{3}, \tau_{4})$; $(e)$ $(S(t), i(t, a), R(t))$ converges to the equilibrium $P^*$ for $\tau = 9.7>\tau_{4}$

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