A data-validated temporary immunity model of COVID-19 spread in Michigan

An immuno-epidemiological model with waning immunity after infection or vaccination

In epidemics, waning immunity is common after infection or vaccination of individuals. Immunity levels are highly heterogeneous and dynamic. This work presents an immuno-epidemiological model that captures the fundamental dynamic features of immunity acquisition and wane after infection or vaccination and analyzes mathematically its dynamical properties. The model consists of a system of first order partial differential equations, involving nonlinear integral terms and different transfer velocities. Structurally, the equation may be interpreted as a Fokker-Planck equation for a piecewise deterministic process. However, unlike the usual models, our equation involves nonlocal effects, representing the infectivity of the whole environment. This, together with the presence of different transfer velocities, makes the proved existence of a solution novel and nontrivial. In addition, the asymptotic behavior of the model is analyzed based on the obtained qualitative properties of the solution. An optimal control problem with objective function including the total number of deaths and costs of vaccination is explored. Numerical results describe the dynamic relationship between contact rates and optimal solutions. The approach can contribute to the understanding of the dynamics of immune responses at population level and may guide public health policies.

The emergence of a virus variant: dynamics of a competition model with cross-immunity time-delay validated by wastewater surveillance data for COVID-19

We consider the dynamics of a virus spreading through a population that produces a mutant strain with the ability to infect individuals that were infected with the established strain. Temporary cross-immunity is included using a time delay, but is found to be a harmless delay. We provide some sufficient conditions that guarantee local and global asymptotic stability of the disease-free equilibrium and the two boundary equilibria when the two strains outcompete one another. It is shown that, due to the immune evasion of the emerging strain, the reproduction number of the emerging strain must be significantly lower than that of the established strain for the local stability of the established-strain-only boundary equilibrium. To analyze the unique coexistence equilibrium we apply a quasi steady-state argument to reduce the full model to a two-dimensional one that exhibits a global asymptotically stable established-strain-only equilibrium or global asymptotically stable coexistence equilibrium. Our results indicate that the basic reproduction numbers of both strains govern the overall dynamics, but in nontrivial ways due to the inclusion of cross-immunity. The model is applied to study the emergence of the SARS-CoV-2 Delta variant in the presence of the Alpha variant using wastewater surveillance data from the Deer Island Treatment Plant in Massachusetts, USA.

A generalized distributed delay model of COVID-19: An endemic model with immunity waning

The severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) has been spreading worldwide for over two years, with millions of reported cases and deaths. The deployment of mathematical modeling in the fight against COVID-19 has recorded tremendous success. However, most of these models target the epidemic phase of the disease. The development of safe and effective vaccines against SARS-CoV-2 brought hope of safe reopening of schools and businesses and return to pre-COVID normalcy, until mutant strains like the Delta and Omicron variants, which are more infectious, emerged. A few months into the pandemic, reports of the possibility of both vaccine- and infection-induced immunity waning emerged, thereby indicating that COVID-19 may be with us for longer than earlier thought. As a result, to better understand the dynamics of COVID-19, it is essential to study the disease with an endemic model. In this regard, we developed and analyzed an endemic model of COVID-19 that incorporates the waning of both vaccine- and infection-induced immunities using distributed delay equations. Our modeling framework assumes that the waning of both immunities occurs gradually over time at the population level. We derived a nonlinear ODE system from the distributed delay model and showed that the model could exhibit either a forward or backward bifurcation depending on the immunity waning rates. Having a backward bifurcation implies that $ R_c < 1 $ is not sufficient to guarantee disease eradication, and that the immunity waning rates are critical factors in eradicating COVID-19. Our numerical simulations show that vaccinating a high percentage of the population with a safe and moderately effective vaccine could help in eradicating COVID-19.

A Scoping Review of Three Dimensions for Long-Term COVID-19 Vaccination Models: Hybrid Immunity, Individual Drivers of Vaccinal Choice, and Human Errors

The virus that causes COVID-19 changes over time, occasionally leading to Variants of Interest (VOIs) and Variants of Concern (VOCs) that can behave differently with respect to detection kits, treatments, or vaccines. For instance, two vaccination doses were 61% effective against the BA.1 predominant variant, but only 24% effective when BA.2 became predominant. While doses still confer protection against severe disease outcomes, the BA.5 variant demonstrates the possibility that individuals who have received a few doses built for previous variants can still be infected with newer variants. As previous vaccines become less effective, new ones will be released to target specific variants and the whole process of vaccinating the population will restart. While previous models have detailed logistical aspects and disease progression, there are three additional key elements to model COVID-19 vaccination coverage in the long term. First, the willingness of the population to participate in regular vaccination campaigns is essential for long-term effective COVID-19 vaccination coverage. Previous research has shown that several categories of variables drive vaccination status: sociodemographic, health-related, psychological, and information-related constructs. However, the inclusion of these categories in future models raises questions about the identification of specific factors (e.g., which sociodemographic aspects?) and their operationalization (e.g., how to initialize agents with a plausible combination of factors?). While previous models separately accounted for natural- and vaccine-induced immunity, the reality is that a significant fraction of individuals will be both vaccinated and infected over the coming years. Modeling the decay in immunity with respect to new VOCs will thus need to account for hybrid immunity. Finally, models rarely assume that individuals make mistakes, even though this over-reliance on perfectly rational individuals can miss essential dynamics. Using the U.S. as a guiding example, our scoping review summarizes these aspects (vaccinal choice, immunity, and errors) through ten recommendations to support the modeling community in developing long-term COVID-19 vaccination models.