To date, our knowledge of Rift Valley fever disease spread and maintenance is still limited, as flooding, humid weather and presence of biting insects such as mosquitoes, have not completely explained RVF outbreaks. We propose a model that includes livestock, mosquitoes and ticks compartments structured according to their questing and feeding behaviour in order to study the possible role of ticks on the dynamics of RVF. To quantify disease transmission at the initial stage of the epidemic, we derive an explicit (...) formula of the basic reproductive number, $$R_0$$ R 0. Using the concept of Metzler matrix, we state necessary conditions for global asymptotic stability of the disease-free equilibrium. Results suggest that although host-ticks interactions may serve as disease reservoirs or disease amplifiers, the Aedes reproductive number should be kept under unity if disease post-epizootics activities are to be controlled. Results of both local and global sensitivity analysis of selected model parameters indicate that $$R_0$$ R 0 is more sensitive to the ticks attachment and detachment rates, probability of transmission from ticks to host and from host to ticks, length of infection in livestock and ticks death rate. Furthermore, when comparing the mean value of $$R_0$$ R 0 with that from previous studies which did not include ticks we found that our $$R_0$$ R 0 is very much larger resulting in an increase in the exponential phase of an outbreak. These findings suggest that if ticks are capable of transmitting the virus, they may be contributing to disease outbreaks and endemicity. (shrink)

Thresholds for disease extinction provide essential information for the prevention and control of diseases. In this paper, a stochastic epidemic model, a continuous-time Markov chain, for the transmission dynamics of West Nile virus in birds is developed based on the assumptions of its analogous deterministic model. The branching process is applied to derive the extinction threshold for the stochastic model and conditions for disease extinction or persistence. The probability of disease extinction computed from the branching process is shown to be (...) in good agreement with the probability approximated from numerical simulations. The disease dynamics of both models are compared to ascertain the effect of demographic stochasticity on West Nile virus dynamics. Analytical and numerical results show differences in model predictions and asymptotic dynamics between stochastic and deterministic models that are crucial for the prevention of disease outbreaks. It is found that there is a high probability of disease extinction if the disease emerges from exposed mosquitoes unlike if it emerges from infectious mosquitoes and birds. Finite-time to disease extinction is estimated using sample paths and it is shown that the epidemic duration is shortest if the disease is introduced by exposed mosquitoes. (shrink)