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Optimal control of normalized SIMR models with vaccination and treatment

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  • We study a model based on the so called SIR model to control the spreading of a disease in a varying population via vaccination and treatment. Since we assume that medical treatment is not immediate we add a new compartment, $M$, to the SIR model. We work with the normalized version of the proposed model. For such model we consider the problem of steering the system to a specified target. We consider both a fixed time optimal control problem with $L^1$ cost and the minimum time problem to drive the system to the target. In contrast to the literature, we apply different techniques of optimal control to our problems of interest.Using the direct method, we first solve the fixed time problem and then proceed to validate the computed solutions using both necessary conditions and second order sufficient conditions. Noteworthy, we perform a sensitivity analysis of the solutions with respect to some parameters in the model. We also use the Hamiltonian Jacobi approach to study how the minimum time function varies with respect to perturbations of the initial conditions. Additionally, we consider a multi-objective approach to study the trade off between the minimum time and the social costs of the control of diseases. Finally, we propose the application of Model Predictive Control to deal with uncertainties of the model.

    Mathematics Subject Classification: Primary: 49K15, 49N90; Secondary: 49M37.


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  • Figure 1.  SIMR compartmental model with treatment and vaccination.

    Figure 2.  Case 1: state variables for $(P_1)$ with parameters as in Table 1.

    Figure 4.  Case 1 with parameters as in Table 1. (left) comparison between the computed values $u_*(t)$ and the values of the singular control $u_{sing}(x_*(t),p_*(t))$ (19), (right) control $v_*(t)$ and switching function $\phi_2$ showing that the control law (14) for $v_*$ is precisely satisfied.

    Figure 5.  Optimal controls for $(P_1)$. Left: Control $u_*(t)$ for Case 1 in red and, for Case 2, control $u_*(t)$ in blue and its switching function $\phi_1(t)$ in dashed blue. Right: Control $v_*(t)$ for Case 1 in red and, for Case 2, control $v_*(t)$ in blue and its switching function $\phi_2(t)$ in dashed blue,

    Figure 3.  Case 1: controls for $(P_1)$ with parameters as in Table 1.

    Figure 6.  Value of the minimum time function for different values of $(s_0,i_0)$. The vertical axis corresponds to the values of the minimum time.

    Figure 7.  State variables for optimal trajectories corresponding to different values of the initial position $(s_0,i_0,m_0)$.

    Figure 8.  Case 1 :$A=10, B=1, C=3$. Minimize the scalarized objective $J^{(w)}$ (23). Top row: (left) Pareto front $(J_1,T)$ for $w \in [0,1]$, (right) objective $J_1$ as function of $w \in [0,1]$. Bottom row: (left) Distance of Pareto curve to the origin, (right) objective $J_2=T$ as function of $w \in [0,1]$.

    Figure 9.  Case 2 : $A=10, B=3, C=1$. Minimize the scalarized objective $J^{(w)}$ (23) for $w \in [0,1]$. Top row: (left) Pareto front $(J_1,J_2)$ for $w \in [0,1]$, (right) objective $J_1(u,v)$ as function of $w \in [0,1]$. Bottom row: (left) Distance of Pareto curve to the origin, (right) objective $J_2=T$ as function of $w \in [0,1]$.

    Figure 10.  Perturbed incidence rate $c_p$.

    Figure 11.  Six computed values of the state variables for the Fixed Horizon MPC described above assuming that the "real" system differs from the model of $(P_1)$ solely with respect to the incidence rate $c$. From left to right on the first row, we show the $s$ and $i$ state variables; on the second row we show the $m$ state variable. All the three graphs have different scales.

    Table 1.  Parameters for the normalized SIMR model (7)-(10)

    Parameter Description Value
    $ b$ Natural birth rate 0.01
    $ c$ Incidence coefficient 1.1
    $ a_1$ Infection induced death rate 0.08
    $ a_2$ Treatment induced death rate 0.005
    $ g_1$ Recovery rate of those infected 0.02
    $ g_2$ Recovery rate of those under treatment 0.5
    $ \eta$ Efficiency of vaccine 0.8
    $ u_{max}$ Maximum rate of vaccination 0.8
    $ v_{max}$ Maximum rate of treatment 0.8
    $ s_0$ Initial percentage of susceptible population 0.95
    $ i_0$ Initial percentage of infected population 0.05
    $ m_0$ Initial percentage of infected population under treatment 0
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    Numerical results for $(P_1)$ with $ A=10, B=1, C=3 : {\text{Cost}} \ J_1= 22.02571 ,$
    $ s(T)= 1.201593 e-01 ,$ $i( T)= 5e-04 ,$ $m(T)= 8.932 084 e-06$
    $p_s(0)= -6.3486 ,$ $p_i(0) = -7.6544e+01,$ $p_m(0) = -6.2821e-02,$
    $p_s(T)= -3.2237e-03,$ $p_i(T) = -5.8613e+03,$ $p_m(T) = -1.4648e-05 .$
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    Numerical results fors $(P_1)$ with $ A=10, B=3, C=1 : {\text{Cost}} J_1= 15.56816 ,$
    $s(T)= 3.337 73e-01 ,$ $i( T)= 5e-04 ,$ $m(T)= 1.370 10 e-05 $
    $p_s(0)= -5.614 47 ,$ $p_i(0) = -3.432 49e+01,$ $p_m(0) = -5.360 07 e-02,$
    $p_s(T)= -5.4088e-04 ,$ $p_i(T) = -1.9668 e+03 ,$ $p_m(T) = -2.4584e-06.$
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    parameter $dt_1/dq$ $dt_2/dq$ $dt_3/dq$ $du_c/dq$
    $ q = B$ -0.5892, 0.7584, 0.7584, -0.1868,
    $ q = C$ 0.1520, -0.6556, -0.6556, 0.07068,
    $q = a_1 $ -1.960, -23.52, -23.52, 0.4620.
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    parameter $dt_1/dq$ $dt_2/dq$
    $ q = B$ -0.49896, 1.45249
    $ q = C$ 1.4525, -5.4816
    $q = a_1 $ 0.27758, -24.112
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