Modelling and optimal control of multi strain epidemics, with application to COVID-19

Abstract

Reinfection and multiple viral strains are among the latest challenges in the current COVID-19 pandemic. In contrast, epidemic models often consider a single strain and perennial immunity. To bridge this gap, we present a new epidemic model that simultaneously considers multiple viral strains and reinfection due to waning immunity. The model is general, applies to any viral disease and includes an optimal control formulation to seek a trade-off between the societal and economic costs of mitigation. We validate the model, with and without mitigation, in the light of the COVID-19 epidemic in England and in the state of Amazonas, Brazil. The model can derive optimal mitigation strategies for any number of viral strains, whilst also evaluating the effect of distinct mitigation costs on the infection levels. The results show that relaxations in the mitigation measures cause a rapid increase in the number of cases, and therefore demand more restrictive measures in the future.

Optimal Control Applied to Vaccination and Testing Policies for COVID-19

“…Regarding the input controls used in the formulation of the corresponding OCPs, some previous works have considered only non-pharmaceutical interventions against COVID-19, such as isolation of the population [27]; quarantining, hospitalization interventions, and treatment of infected people [15,17]; isolation, quarantine, and public health education [16], sensitization, quarantine, diagnosis, and monitoring and psychological support [19]; public health education, treatment of infected individuals, and health care measures for asymptomatic infectious people [21]; and use of face-masks, hand sanitizer, and social distancing; treatment of patients and active screening with testing; and prevention against recurrence and reinfection of people who have recovered [26]. Fewer works have focused on pharmaceutical measures, such as allocation of the treatment [18] and vaccine administration [20,[23][24][25].…”

“…Regarding the objective functional used in the formulation of the corresponding OCPs, some previous works have considered just a single optimality criterion, such as the minimization of the number of infected people [18], the number of deaths [27], or the years of life lost due to premature mortality and the years lost due to disability [23]. Some other works have considered and compared several optimality criteria, such as minimizing the number of new infections, the number of deaths, the life years lost, and the quality-adjusted life years lost due to death [20], or minimizing the number of symptomatic infections, the number of deaths, the number of cases requiring non-ICU hospitalization, and the number of cases requiring ICU hospitalization [24].…”

“…The objective functional J (•) in ( 25) is given in Bolza form, which combines a Mayer term M(•) and a Lagrange term L(•). The set of differential Equation ( 26) describes the dynamical system, Equation (27) represents the set of algebraic constraints, and Equation (28) denotes the boundary conditions. Notice that the Lagrange term can be rewritten as a Mayer term by introducing a new state variable, x M (t), and adding a new differential equation with its associated initial condition:…”

“…A direct transcription approach is used to obtain numerical solutions of the OCP ( 25)- (28), which makes it easy to deal with inequality and equality constraints. Specifically, the Hermite-Simpson collocation method [28] is employed to transcribe the OCP ( 25)-( 28) into a NLP problem, the constraints of which consist of the Hermite-Simpson system constraints that correspond to the differential constraint (26), the algebraic constraints (27), and the boundary conditions (28). The resulting NLP problem is then solved by means of the open-source IPOPT solver [29].…”

“…In [26], the OCP of a susceptible-exposed-infectedquarentined-recovered (SEIQR) model is considered, which incorporates three types of control strategies, namely, the use of face-masks, hand sanitizer, and social distancing; the treatment of COVID-19 patients and active screening with testing; and prevention against recurrence and reinfection of humans who have recovered from COVID-19. The OCP of an epidemic model that simultaneously considers multiple viral strains and reinfection due to waning immunity is formulated in [27], which seeks a trade-off between the societal and economic costs of mitigation. This paper addresses the problem of finding and comparing the best vaccination and testing policies for an epidemic model of COVID-19 transmission developed from the compartmental model presented in [10].…”