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Optimal control of non-autonomous SEIRS models with vaccination and treatment

  • * Corresponding author: delfim@ua.pt

    * Corresponding author: delfim@ua.pt

Mateus was partially supported by FCT through CMA-UBI (project UID/MAT/00212/2013), Rebelo by FCT through CMA-UBI (project UID/MAT/00212/2013), Rosa by FCT through IT (project UID/EEA/50008/2013), Silva by FCT through CMA-UBI (project UID/MAT/00212/2013), and Torres by FCT through CIDMA (project UID/MAT/04106/2013) and TOCCATA (project PTDC/EEI-AUT/2933/2014 funded by FEDER and COMPETE 2020)

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  • We study an optimal control problem for a non-autonomous SEIRS model with incidence given by a general function of the infective, the susceptible and the total population, and with vaccination and treatment as control variables. We prove existence and uniqueness results for our problem and, for the case of mass-action incidence, we present some simulation results designed to compare an autonomous and corresponding periodic model, as well as the controlled versus uncontrolled models.

    Mathematics Subject Classification: Primary: 34H05, 92D30; Secondary: 37B55, 49M05.

    Citation:

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  • Figure 1.  SEIRS autonomous model (${\rm{per}} = 0$ in (28) and (29)): uncontrolled (dashed lines) versus optimally controlled (continuous lines).

    Figure 2.  SEIRS periodic model (${\rm{per}} = 0.8$ in (28) and (29)): uncontrolled (dashed lines) versus optimally controlled (continuous lines).

    Figure 3.  SEIRS model subject to optimal control: autonomous (${\rm{per}} = 0$) versus periodic (${\rm{per}} = 0.8$) cases.

    Figure 4.  SEIRS model without control measures: autonomous (${\rm{per}} = 0$) versus periodic (${\rm{per}} = 0.8$) cases.

    Figure 5.  The optimal controls ${\mathbb{T}}^*$ (5) (treatment) and ${\mathbb{V}}^*$ (6) (vaccination): autonomous (${\rm{per}} = 0$) versus periodic (${\rm{per}} = 0.8$) cases.

    Figure 6.  Variation of infected individuals $I^*(t)$ and optimal measures of treatment ${\mathbb{T}}^*(t)$ and vaccination ${\mathbb{V}}^*(t)$, in both autonomous (${\rm{per}} = 0$) and periodic (${\rm{per}} = 0.8$) cases, with the natural death $\mu \in [0, 0.1]$.

    Figure 7.  Variation of infected individuals $I^*(t)$ and optimal measures of treatment ${\mathbb{T}}^*(t)$ and vaccination ${\mathbb{V}}^*(t)$, in both autonomous (${\rm{per}} = 0$) and periodic (${\rm{per}} = 0.8$) cases, with the rate of recovery $\gamma \in [0, 0.1]$.

    Figure 8.  Variation of infected individuals $I^*(t)$ and optimal measures of treatment ${\mathbb{T}}^*(t)$ and vaccination ${\mathbb{V}}^*(t)$, in both autonomous (${\rm{per}} = 0$) and periodic (${\rm{per}} = 0.8$) cases, with the infectivity rate $\varepsilon \in [0, 0.1]$.

    Figure 9.  Variation of infected individuals $I^*(t)$ and optimal measures of treatment ${\mathbb{T}}^*(t)$ and vaccination ${\mathbb{V}}^*(t)$, in both autonomous (${\rm{per}} = 0$) and periodic (${\rm{per}} = 0.8$) cases, with the loss of immunity rate $\eta \in [0, 0.1]$.

    Table 1.  Values of the parameters for problem (P) used in Section 6.

    NameDescriptionValue
    $S_0$Initial susceptible population0.98
    $E_0$Initial exposed population0
    $I_0$Initial infective population0.01
    $R_0$Initial recovered population0.01
    $\mu$natural deaths0.05
    $\varepsilon$infectivity rate0.03
    $\gamma$rate of recovery0.05
    $\eta$loss of immunity rate0.041
    $k_1$weight for the number of infected1
    $k_2$weight for treatment0.01
    $k_3$weight for vaccination0.01
    $\tau_{\max}$maximum rate of treatment0.1
    $v_{\max}$maximum rate of vaccination0.4
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