A reaction-diffusion system modeling the spread of resistance to an antimalarial drug
Nicolas Bacaër Cheikh Sokhna
Mathematical Biosciences & Engineering 2005, 2(2): 227-238 doi: 10.3934/mbe.2005.2.227
A mathematical model representing the diffusion of resistance to an antimalarial drug is developed. Resistance can spread only when the basic reproduction number of the resistant parasites is bigger than the basic reproduction number of the sensitive parasites (which depends on the fraction of infected people treated with the antimalarial drug). Based on a linearization study and on numerical simulations, an expression for the speed at which resistance spreads is conjectured. It depends on the ratio of the two basic reproduction numbers, on a coefficient representing the diffusion of mosquitoes, on the death rate of mosquitoes infected by resistant parasites, and on the recovery rate of nonimmune humans infected by resistant parasites.
keywords: propagation speed. malaria reaction-diffusion resistance
On the basic reproduction number $R_0$ in sexual activity models for HIV/AIDS epidemics: Example from Yunnan, China
Nicolas Bacaër Xamxinur Abdurahman Jianli Ye Pierre Auger
Mathematical Biosciences & Engineering 2007, 4(4): 595-607 doi: 10.3934/mbe.2007.4.595
Heterogeneity in sexual behavior is known to play an important role in the spread of HIV. In 1986, a mathematical model based on ordinary differential equations was introduced to take into account the distribution of sexual activity. Assuming proportionate mixing, it was shown that the basic reproduction number $R_0$ determining the epidemic threshold was proportional to $M+V/M$, where $M$ is the mean and $V$ the variance of the distribution. In the present paper, we notice that this theoretical distribution is different from the one obtained in behavioral surveys for the number of sexual partnerships over a period of length $\tau$. The latter is a ''mixed Poisson distribution'' whose mean $m$ and variance $v$ are such that $M=m/\tau$ and $V=(v-m)/\tau^2$. So $M+V/M=(m+v/m-1)/\tau$. This way, we improve the link between theory and data for sexual activity models of HIV/AIDS epidemics. As an example, we consider data concerning sex workers and their clients in Yunnan, China, and find an upper bound for the geometric mean of the transmission probabilities per partnership in this context.
keywords: basic reproduction number HIV mixed Poisson distribution.
Persistence in some periodic epidemic models with infection age or constant periods of infection
Carlota Rebelo Alessandro Margheri Nicolas Bacaër
Discrete & Continuous Dynamical Systems - B 2014, 19(4): 1155-1170 doi: 10.3934/dcdsb.2014.19.1155
Much recent work has focused on persistence for epidemic models with periodic coefficients. But the case where the infected compartments satisfy a delay differential equation or a partial differential equation does not seem to have been considered so far. The purpose of this paper is to provide a framework for proving persistence in such a case. Some examples are presented, such as a periodic SIR model structured by time since infection and a periodic SIS delay model.
keywords: seasonality epidemiological models delay differential equation. Persistence

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