a b s t r a c tAn HIV/AIDS epidemic model with treatment is investigated. The model allows for some infected individuals to move from the symptomatic phase to the asymptomatic phase by all sorts of treatment methods. We first establish the ODE treatment model with two infective stages. Mathematical analyses establish that the global dynamics of the spread of the HIV infectious disease are completely determined by the basic reproduction number R 0 . If R 0 ≤ 1, the disease-free equilibrium is globally stable, whereas the unique infected equilibrium is globally asymptotically stable if R 0 > 1. Then, we introduce a discrete time delay to the model to describe the time from the start of treatment in the symptomatic stage until treatment effects become visible. The effect of the time delay on the stability of the endemically infected equilibrium is investigated. Moreover, the delay model exhibits Hopf bifurcations by using the delay as a bifurcation parameter. Finally, numerical simulations are presented to illustrate the results.

The risk to humans of contracting tick-borne zoonotic diseases depends on the risk of a bite from an infected tick, which can be broken down into its component parts as the number of host-seeking ticks in the environment, in particular nymphs, and the prevalence of tick-borne pathogens they are carrying. In turn, the prevalence of tick-borne pathogens is dependent upon tick biting intensity on hosts that support transmission between ticks; namely rodents. These ticks once fed moult into the next life stage and search for the next blood meal, thus posing a zoonotic risk. Here, we analyse tick biting intensity on rodents in a known tick-borne encephalitis (TBE) focus in Trentino (northern Italy). We examine patterns of tick demography and the influence of host densities and climate on ticks' generation time, development rates, tick density and intensity. During the period 2000-2004, a population of the yellow-necked mouse, Apodemus flavicollis, the most important TBE transmission host, was intensively monitored. Ticks feeding on individual rodents were counted, distinguishing between the larval and nymph life-stages. Local temperature and relative humidity was calculated using both data-loggers in the field site and regional weather stations. We investigated which factors had a predictive value both on feeding tick intensity and on the overall density of larvae or nymphs feeding on rodents in a year. We observed a negative effect of rodent density on tick intensity, while temperature influenced positively both larvae and nymph intensity. Overall larval density was higher in the years and trapping grids where rodent density was higher, while for nymphs no such effect was observed. The best explanatory variable for nymph density was the larval density in the previous year, confirming the discrete nature of tick demography. This provides important information in terms of monitoring the risk to humans of acquiring pathogen-infected ticks.

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In this paper, a simple semi-discrete (ticks' feeding is assumed to occur only during summers of each year) model for tick population dynamics is presented. Conditions for existence, uniqueness, and stability of a positive equilibrium are found; the system is then studied numerically using parameter estimates calibrated for the tick Ixodes ricinus in Trentino, Italy, and the sensitivity to parameters is examined. Then, this model is extended to consider a tick-transmitted infection of one species of hosts, while other hosts are incompetent to the infection. Assuming, for simplicity, that the infection is not affecting the total number either of hosts or ticks, a threshold condition for infection persistence is obtained. The dependence of the equilibrium infection prevalence on parameters is studied numerically; in particular, we considered how infection prevalence depends on host densities. This analysis reveals that a 'dilution effect' occurs both for competent and for incompetent hosts; this means that, besides a lower threshold for host densities for infection to persist, there exists also an upper threshold: if host densities were higher than the upper threshold, the infection would go to extinction. Numerically, it is found that, for realistic parameter values, the upper threshold is not much higher than observed densities.

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