# American Institute of Mathematical Sciences

## Journals

MBE
Mathematical Biosciences & Engineering 2014, 11(3): 449-469 doi: 10.3934/mbe.2014.11.449
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.
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DCDS-B
Discrete & Continuous Dynamical Systems - B 2016, 21(8): 2851-2866 doi: 10.3934/dcdsb.2016076
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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