This paper proposes a model for brucellosis transmission. The model takes into account the age of infection and waning immunity, that is, the progressive loss of immunity after recovery. Three routes of transmissions are considered: vertical transmission, and both direct and indirect routes of horizontal transmission. According to the well-posedness results, we provide explicit formulas for the equilibria. Next, we derive the basic reproduction number R0 and prove some stability results depending on the basic reproductive number. Finally, we perform numerical simulations using model parameters estimated from biological data to confirm our theoretical results. The results of these simulations suggest that for certain values of parameters, there will be periodic outbreaks of epidemics, and the disease will not be eradicated from the population. Our results also highlight the fact that the birth rate of cattle significantly influences the dynamics of the disease. The proposed model can be of a good use in studying the effects of vaccination on the cattle population.

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In this article, we study the average control of a population dynamic model with age dependence and spatial structure in a bounded domain Ω ⊂ R 3 . We assume that we can act on the system via a control in a sub-domain ω of Ω. We prove that we can bring the average of the state of our model at time t = T to a desired state. By means of Euler-Lagrange first order optimality condition, we expressed the optimal control in terms of average of an appropriate adjoint state that we characterize by an optimality system.