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Investigating the effects of intervention strategies in a spatio-temporal anthrax model

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  • In this paper, we extend our previous work on optimal control applied in an anthrax outbreak in wild animals. We use a system of ordinary differential equation (ODE) and partial differential equations (PDEs) to track the change in susceptible, infected and vaccinated animals as well as the infected carcasses. In addition to the assumption that the infected animals and the infected carcasses are the main source of infection, we consider the animal movement by diffusion and see its effects in disease transmission. Two controls: vaccinating susceptible animals and disposing infected carcasses properly are applied in the model and these controls depend on both space and time. We formulate an optimal control problem to investigate the effect of intervention strategies in our spatio-temporal model in controlling the outbreak at minimum cost. Finally some numerical results for the optimal control problem are presented.

    Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.


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  • Figure 1.  Simulation results for model (1)-(4) without control $ u_1 = u_2 = 0 $. The initial population of susceptible and infected animals are considered to be uniformly distributed in $ 1\le x\le 34 $ and $ 27\le x\le 31 $ respectively while only one initial carcass is considered near an end of the domain, $ 29\le x\le 30 $. The figures in the first row show the plots for susceptible (left) and infected (right) animals; and the figure in the second row represents the carcasses

    Figure 2.  Simulation results for model (1)-(4) with optimal rates of vaccination and optimal carcass disposal rates $ 0\le u_1(x,t)\le 0.027,\; \; \text{and}\; \; 0\le u_2(x,t)\le 0.5. $. The initial population of susceptible and infected animals are considered to be uniformly distributed in $ 1\le x\le 34 $ and $ 27\le x\le 31 $ respectively while only one initial carcass is considered near an end of the domain, $ 29\le x\le30 $. The two plots in the first row represent the concentrations of susceptible(left) and infected (right) animals. The plots in the second row represents the concentrations of the infected carcasses(left) and the vaccinated animals(right). The last row represents the vaccination (left) and carcass disposal(right) rates

    Table 1.  The model parameters, their description, values and units

    Parm. Description Values Units
    $ r $ Intrinsic growth rate of healthy animals $ 5.052\times 10^{-4} $ day$ ^{-1} $
    $ \gamma $ Disease induced death rate of infecteds $ \frac{1}{7.5} $ day$ ^{-1} $
    $ \alpha $ Carcass feeding rate by scavengers $ 0 $ animal$ ^{-1} $ day$ ^{-1} $
    $ K $ Carrying capacity of animals 2000 animal
    $ p $ Carcass decay rate $ 0.02816 $ day$ ^{-1} $
    $ d $ Diffusion rate of healthy animals $ 0.12 $ $ km^2 $ day$ ^{-1} $
    $ d_1 $ Diffusion rate of infected animals $ 0.024 $ $ km^2 $ day$ ^{-1} $
    $ \theta_c $ Disease transmission rate from carcasses $ 1.65\times 10^{-3} $ carcass$ ^{-1} $ day$ ^{-1} $
    $ \theta_i $ Disease transmission rate from infected animals $ 2.05\times 10^{-2} $ animal$ ^{-1} $ day$ ^{-1} $
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