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- A dynamically-consistent nonstandard finite difference scheme for the SICA modelPublication . Vaz, Sandra; Torres, Delfim F. M.In this work, we derive a nonstandard finite difference scheme for the SICA (Susceptible-Infected-Chronic-AIDS) model and analyze the dynamical properties of the discretized system. We prove that the discretized model is dynamically consistent with the continuous, maintaining the essential properties of the standard SICA model, namely, the positivity and boundedness of the solutions, equilibrium points, and their local and global stability.
- Parameter Estimation, Sensitivity Analysis and Optimal Control of a Periodic Epidemic Model with Application to HRSV in FloridaPublication . Rosa, Silvério; Torres, Delfim F. M.A state wide Human Respiratory Syncytial Virus (HRSV) surveillance system was implemented in Florida in 1999 to support clinical decision-making for prophylaxis of premature infants. The research presented in this paper addresses the problem of fitting real data collected by the Florida HRSV surveillance system by using a periodic SEIRS mathematical model. A sensitivity and cost-effectiveness analysis of the model is done and an optimal control problem is formulated and solved with treatment as the control variable.
- Optimal Control and Sensitivity Analysis of a Fractional Order TB ModelPublication . Rosa, Silvério; Torres, Delfim F. M.A Caputo fractional-order mathematical model for the transmission dynamics of tuberculosis (TB) was recently proposed in [Math. Model. Nat. Phenom. 13 (2018), no. 1, Art. 9]. Here, a sensitivity analysis of that model is done, showing the importance of accuracy of parameter values. A fractional optimal control (FOC) problem is then formulated and solved, with the rate of treatment as the control variable. Finally, a cost-effectiveness analysis is performed to assess the cost and the effectiveness of the control measures during the intervention, showing in which conditions FOC is useful with respect to classical (integer-order) optimal control.
- Discrete-Time System of an Intracellular Delayed HIV Model with CTL Immune ResponsePublication . Vaz, Sandra; Torres, Delfim F. M.In [Math. Comput. Sci. 12 (2018), no. 2, 111--127], a delayed model describing the dynamics of the Human Immunodeficiency Virus (HIV) with Cytotoxic T Lymphocytes (CTL) immune response is investigated by Allali, Harroudi and Torres. Here, we propose a discrete-time version of that model, which includes four nonlinear difference equations describing the evolution of uninfected, infected, free HIV viruses, and CTL immune response cells and includes intracellular delay. Using suitable Lyapunov functions, we prove the global stability of the disease free equilibrium point and of the two endemic equilibrium points. We finalize by making some simulations and showing, numerically, the consistence of the obtained theoretical results.
- Optimal control of a fractional order epidemic model with application to human respiratory syncytial virus infectionPublication . Rosa, Silvério; Torres, Delfim F. M.A human respiratory syncytial virus surveillance system was implemented in Florida in 1999, to support clinical decision-making for prophylaxis of premature newborns. Recently, a local periodic SEIRS mathematical model was proposed in [Stat. Optim. Inf. Comput. 6 (2018), no.1, 139--149] to describe real data collected by Florida's system. In contrast, here we propose a non-local fractional (non-integer) order model. A fractional optimal control problem is then formulated and solved, having treatment as the control. Finally, a cost-effectiveness analysis is carried out to evaluate the cost and the effectiveness of proposed control measures during the intervention period, showing the superiority of obtained results with respect to previous ones.
- A Discrete-Time Compartmental Epidemiological Model for COVID-19 with a Case Study for PortugalPublication . Vaz, Sandra; Torres, Delfim F. M.Recently, a continuous-time compartmental mathematical model for the spread of the Coronavirus disease 2019 (COVID-19) was presented with Portugal as case study, from 2 March to 4 May 2020, and the local stability of the Disease Free Equilibrium (DFE) was analysed. Here, we propose an analogous discrete-time model and, using a suitable Lyapunov function, we prove the global stability of the DFE point. Using COVID-19 real data, we show, through numerical simulations, the consistence of the obtained theoretical results.
- Optimal Control Of Non-Autonomous Seirs Models With Vaccination And TreatmentPublication . Mateus, Joaquim; Rebelo, Paulo; Rosa, Silvério; Silva, César M.; Torres, Delfim F. M.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.