SIRSi-vaccine dynamical model for the Covid-19 pandemic.

Covid-19, caused by severe acute respiratory syndrome coronavirus 2, broke out as a pandemic during the beginning of 2020. The rapid spread of the disease prompted an unprecedented global response involving academic institutions, regulatory agencies, and industries. Vaccination and nonpharmaceutical interventions including social distancing have proven to be the most effective strategies to combat the pandemic. In this context, it is crucial to understand the dynamic behavior of the Covid-19 spread together with possible vaccination strategies. In this study, a susceptible-infected-removed-sick model with vaccination (SIRSi-vaccine) was proposed, accounting for the unreported yet infectious. The model considered the possibility of temporary immunity following infection or vaccination. Both situations contribute toward the spread of diseases. The transcritical bifurcation diagram of alternating and mutually exclusive stabilities for both disease-free and endemic equilibria were determined in the parameter space of vaccination rate and isolation index. The existing equilibrium conditions for both points were determined in terms of the epidemiological parameters of the model. The bifurcation diagram allowed us to estimate the maximum number of confirmed cases expected for each set of parameters. The model was fitted with data from São Paulo, the state capital of SP, Brazil, which describes the number of confirmed infected cases and the isolation index for the considered data window. Furthermore, simulation results demonstrate the possibility of periodic undamped oscillatory behavior of the susceptible population and the number of confirmed cases forced by the periodic small-amplitude oscillations in the isolation index. The main contributions of the proposed model are as follows: A minimum effort was required when vaccination was combined with social isolation, while additionally ensuring the existence of equilibrium points. The model could provide valuable information for policymakers, helping define disease prevention mitigation strategies that combine vaccination and non-pharmaceutical interventions, such as social distancing and the use of masks. In addition, the SIRSi-vaccine model facilitated the qualitative assessment of information regarding the unreported infected yet infectious cases, while considering temporary immunity, vaccination, and social isolation index.

SIRSi compartmental model for COVID-19 pandemic with immunity loss.

The coronavirus disease 2019 (Covid-19) outbreak led the world to an unprecedented health and economic crisis. In an attempt to respond to this emergency, researchers worldwide are intensively studying the dynamics of the Covid-19 pandemic. In this study, a Susceptible - Infected - Removed - Sick (SIRSi) compartmental model is proposed, which is a modification of the classical Susceptible - Infected - Removed (SIR) model. The proposed model considers the possibility of unreported or asymptomatic cases, and differences in the immunity within a population, i.e., the possibility that the acquired immunity may be temporary, which occurs when adopting one of the parameters ( γ ) other than zero. Local asymptotic stability and endemic equilibrium conditions are proved for the proposed model. The model is adjusted to the data from three major cities of the state of São Paulo in Brazil, namely, São Paulo, Santos, and Campinas, providing estimations of duration and peaks related to the disease propagation. This study reveals that temporary immunity favors a second wave of infection and it depends on the time interval for a recovered person to be susceptible again. It also indicates the possibility that a greater number of patients would get infected with decreased time for reinfection.