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.
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.
We consider a model of population dynamics with age dependence and spatial structure but unknown birth rate. Using the notion of Low-regret [1], we prove that we can bring the state of the system to a desired state by acting on the system via a localized distributed control. We provide the optimality systems that characterize the Low-regret control. Moreover, using an appropriate Hilbert space, we prove that the family of Low-regret controls tends to a so-called Noregret control, which we, in turn, characterize.
We consider a bilinear optimal control for an evolution equation involving the fractional Laplace operator of order 0 < s < 1. We first give some existence and uniqueness results for the considered evolution equation. Next, we establish some weak maximum principle results allowing us to obtain more regularity of our state equation. Then, we consider an optimal control problem which consists to bring the state of the system at final time to a desired state. We show that this optimal control problem has a solution and we derive the first and second order optimality conditions. Finally, under additional assumptions on the initial datum and the given target, we prove that local uniqueness of optimal solutions can be achieved.
We consider two evolution equations involving the space fractional Laplace operator of order
0
<
s
<
1
{0<s<1}
. We first establish some existence and uniqueness results for the considered evolution equations. Next, we give some comparison theorems and prove that, if the data of each equation are data bounded, then the solutions are also bounded.
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