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MBE

Infection age is an important factor affecting the transmission of
infectious diseases. In this paper, we consider an SIRS model
with infection age, which is described by a mixed system of
ordinary differential equations and partial differential
equations. The expression of the basic reproduction number
$\mathscr {R}_0$ is obtained. If $\mathscr{R}_0\le 1$ then the
model only has the disease-free equilibrium, while if
$\mathscr{R}_0>1$ then besides the disease-free equilibrium the
model also has an endemic equilibrium. Moreover, if
$\mathscr{R}_0<1$ then the disease-free equilibrium is globally
asymptotically stable otherwise it is unstable; if
$\mathscr{R}_0>1$ then the endemic
equilibrium is globally asymptotically stable under additional conditions. The local stability
is established through linearization. The global stability of the
disease-free equilibrium is shown by applying the fluctuation
lemma
and that of the endemic equilibrium is proved by employing Lyapunov functionals.
The theoretical results are illustrated with numerical simulations.

DCDS-B

In this paper, a two-strain epidemic model on a complex network is proposed. The two strains are the drug-sensitive strain and the drug-resistant strain. The related basic reproduction numbers $R_s$ and $R_r$ are obtained. If $R_0=\max\{R_s,R_r\}<1$, then the disease-free equilibrium is globally asymptotically stable. If $R_r>1$, then there is a unique drug-resistant strain dominated equilibrium $E_r$, which is locally asymptotically stable if the invasion reproduction number $R_r^s<1$. If $R_s>1$ and $R_s>R_r$, then there is a unique coexistence equilibrium $E^*$. The persistence of the model is also proved. The theoretical results are supported with numerical simulations.

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