In this survey, we propose an overview on Lyapunov functions for a variety of compartmental models in epidemiology. We exhibit the most widely employed functions, and provide a commentary on their use. Our aim is to provide a comprehensive starting point to readers who are attempting to prove global stability of systems of ODEs. The focus is on mathematical epidemiology, however some of the functions and strategies presented in this paper can be adapted to a wider variety of models, such as prey–predator or rumor spreading.

We propose a compartmental model for a disease with temporary immunity and secondary infections. From our assumptions on the parameters involved in the model, the system naturally evolves in three time scales. We characterize the equilibria of the system and analyze their stability. We find conditions for the existence of two endemic equilibria for some cases in which [Formula: see text]. Then, we unravel the interplay of the three time scales, providing conditions to foresee whether the system evolves in all three scales, or only in the fast and the intermediate ones. We conclude with numerical simulations and bifurcation analysis to complement our analytical results.

Standard-dose quadrivalent influenza vaccines (QIV) are designed to provide protection against all four influenza strains. Adjuvanted QIV (aQIV), indicated for individuals aged 65+ years, combines MF59^® adjuvant (an oil-in-water emulsion of squalene oil) with a standard dose of antigen, and is designed to produce stronger and longer immune response, especially in the elderly where immunosenescence reduces vaccine effectiveness. This study evaluated the cost-effectiveness of aQIV vs. egg-based standard-dose QIV (QIVe) in the elderly population, from the payer and societal perspective in Spain. A dynamic transmission model, which accounts for herd protection, was used to predict the number of medically attended infections in Spain. A decision tree structure was used to forecast influenza-related costs and benefits. Influenza-related probabilities of outpatient visit, hospitalization, work absenteeism, mortality, and associated utilities and costs were extracted from Spanish and European published literature. Relative vaccine effectiveness (rVE) was sourced from two different meta-analyses: the first meta-analysis was informed by laboratory-confirmed influenza studies only, resulting in a rVE = 34.6% (CI95% 2–66%) in favor of aQIV; the second meta-analysis included real world evidence influenza-related medical encounters outcomes, resulting in a rVE = 13.9% (CI95% 4.2–23.5%) in benefit of aQIV. All costs were expressed in 2021 euros. Results indicate that replacing QIVe with aQIV in the Spanish elderly population would prevent on average 43,664 influenza complicated cases, 1111 hospitalizations, and 569 deaths (with a rVE = 34.6%) or 19,104 influenza complicated cases, 486 hospitalizations, and 252 deaths (with a rVE = 13.9%). When the rVE of aQIV vs. QIVe is 34.6%, the incremental cost per quality adjusted life years (QALY) gained was €2240 from the payer; from the societal perspective, aQIV was cost saving compared with QIVe. If the rVE was 13.9%, the incremental cost per QALY was €6694 and €3936 from the payer and societal perspective, respectively. Sensitivity analyses validated the robustness of these findings. Results indicate that replacing QIVe with aQIV in the Spanish elderly population is a cost-effective strategy for the Spanish healthcare system.

With the premise that social interactions are described by power-law distributions, we study the stochastic dynamics of SIR (Susceptible-Infected-Removed) compartmental models on static scale-free random networks generated via the configuration model. We compare simulations of our model to analytical results, providing a closed formula and a lower bound for the probability of having a minor epidemic of the disease. We explore the variability in disease spread by stochastic simulations. In particular, we demonstrate how important epidemic indices change as a function of the contagiousness of the disease and the connectivity of the network. Our results quantify the role of the starting node’s degree in determining these indices, commonly used to describe epidemic spread. Our results and implementation set a baseline for studying epidemic spread on networks, showing how analytical methods can help in the interpretation of stochastic simulations.