# American Institute of Mathematical Sciences

December  2018, 11(6): 1179-1199. doi: 10.3934/dcdss.2018067

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

 1 Research Unit for Inland Development (UDI), Polytechnic Institute of Guarda, 6300-559 Guarda, Portugal 2 Centro de Matemática e Aplicações da Universidade da Beira Interior (CMA-UBI), Departamento de Matemática, Universidade da Beira Interior, 6201-001 Covilhã, Portugal 3 Departamento de Matemática and Instituto de Telecomunicações (IT), Universidade da Beira Interior, 6201-001 Covilhã, Portugal 4 Center for Research and Development in Mathematics and Applications (CIDMA), Department of Mathematics, University of Aveiro, 3810-193 Aveiro, Portugal

* Corresponding author: delfim@ua.pt

Received  April 2017 Revised  June 2017 Published  June 2018

Fund Project: 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).

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.

Citation: Joaquim P. Mateus, Paulo Rebelo, Silvério Rosa, César M. Silva, Delfim F. M. Torres. Optimal control of non-autonomous SEIRS models with vaccination and treatment. Discrete & Continuous Dynamical Systems - S, 2018, 11 (6) : 1179-1199. doi: 10.3934/dcdss.2018067
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##### References:
SEIRS autonomous model (${\rm{per}} = 0$ in (28) and (29)): uncontrolled (dashed lines) versus optimally controlled (continuous lines).
SEIRS periodic model (${\rm{per}} = 0.8$ in (28) and (29)): uncontrolled (dashed lines) versus optimally controlled (continuous lines).
SEIRS model subject to optimal control: autonomous (${\rm{per}} = 0$) versus periodic (${\rm{per}} = 0.8$) cases.
SEIRS model without control measures: autonomous (${\rm{per}} = 0$) versus periodic (${\rm{per}} = 0.8$) cases.
The optimal controls ${\mathbb{T}}^*$ (5) (treatment) and ${\mathbb{V}}^*$ (6) (vaccination): autonomous (${\rm{per}} = 0$) versus periodic (${\rm{per}} = 0.8$) cases.
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 $\mu \in [0, 0.1]$.
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 $\gamma \in [0, 0.1]$.
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 $\varepsilon \in [0, 0.1]$.
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 $\eta \in [0, 0.1]$.
Values of the parameters for problem (P) used in Section 6.
 Name Description Value $S_0$ Initial susceptible population 0.98 $E_0$ Initial exposed population 0 $I_0$ Initial infective population 0.01 $R_0$ Initial recovered population 0.01 $\mu$ natural deaths 0.05 $\varepsilon$ infectivity rate 0.03 $\gamma$ rate of recovery 0.05 $\eta$ loss of immunity rate 0.041 $k_1$ weight for the number of infected 1 $k_2$ weight for treatment 0.01 $k_3$ weight for vaccination 0.01 $\tau_{\max}$ maximum rate of treatment 0.1 $v_{\max}$ maximum rate of vaccination 0.4
 Name Description Value $S_0$ Initial susceptible population 0.98 $E_0$ Initial exposed population 0 $I_0$ Initial infective population 0.01 $R_0$ Initial recovered population 0.01 $\mu$ natural deaths 0.05 $\varepsilon$ infectivity rate 0.03 $\gamma$ rate of recovery 0.05 $\eta$ loss of immunity rate 0.041 $k_1$ weight for the number of infected 1 $k_2$ weight for treatment 0.01 $k_3$ weight for vaccination 0.01 $\tau_{\max}$ maximum rate of treatment 0.1 $v_{\max}$ maximum rate of vaccination 0.4
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