MBE

We propose and study a model for sexually transmitted infections on uncorrelated networks, where both differential susceptibility and infectivity are considered. We first establish the spreading threshold, which exists even in the infinite networks. Moreover, it is possible to have backward bifurcation. Then for bounded hard-cutoff networks, the stability of the disease-free equilibrium and the permanence of infection are analyzed. Finally, the effects of two immunization strategies are compared. It turns out that, generally, the targeted immunization is better than the proportional immunization.

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MBE

We concern with a vector-borne disease model with horizontal transmission and infection age in the host population. With the approach of Lyapunov functionals, we establish a threshold dynamics, which is completely determined by the basic reproduction number. Roughly speaking, if the basic reproduction number is less than one then the infection-free equilibrium is globally asymptotically stable while if the basic reproduction number is larger than one then the infected equilibrium attracts all solutions with initial infection. These theoretical results are illustrated with numerical simulations.

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, we are concerned with an age-structured HIV infection model incorporating latency and cell-to-cell transmission. The model is a hybrid system consisting of coupled ordinary differential equations and partial differential equations. First, we address the relative compactness and persistence of the solution semi-flow, and the existence of a global attractor. Then, applying the approach of Lyapunov functionals, we establish the global stability of the infection-free equilibrium and the infection equilibrium, which is completely determined by the basic reproduction number.

DCDS-B

This paper is concerned with the problem of optimal contraception control
for a nonlinear population model with size structure.
First, the existence of separable solutions is established, which is crucial in obtaining
the optimal control strategy. Moreover, it is shown that the population density
depends continuously on control parameters.
Then, the existence of an optimal control strategy is proved via
compactness and extremal sequence. Finally, the conditions of the
optimal strategy are derived by means of normal cones and adjoint systems.

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.

MBE

In this paper, we propose and analyze a delayed HIV-1 model with CTL immune response and virus waning. The two discrete delays stand for the time for infected cells to produce viruses after viral entry and for the time for CD$8^+$ T cell immune response to emerge to control viral replication. We obtain the positiveness and boundedness of solutions and find the basic reproduction number $R_0$. If $R_0<1$, then the infection-free steady state is globally asymptotically stable and the
infection is cleared from the T-cell population; whereas if $R_0>1$, then the system is uniformly persistent and the viral concentration maintains at some constant level. The global dynamics when $R_0>1$ is complicated. We establish the local stability of the infected steady state and show that Hopf bifurcation can occur. Both analytical and numerical results indicate that if, in the initial infection stage,
the effect of delays on HIV-1 infection is ignored, then the risk of HIV-1 infection (if persists) will be underestimated. Moreover, the viral load differs from that without virus waning. These results highlight the important role of delays and virus waning on HIV-1 infection.