An epidemic model structured by the time since last infection
Epidemiological models structured by time since infection have their origin in the seminal work of 1927 by Kermack and McKendrick. Compared to ordinary differential equations (ODE) models, they are able to capture differences in infectivity of the individuals in a more suitable manner. Their use declined in the second half of the 20th century, probably because the theory for ODE models is more robust, complete and has proved successful in providing insights and predictions for many epidemiological problems. Nevertheless, it is important to understand in what occasions the inclusion of time since infection may alter the outcomes in a significant way, specially when studying the problem of reinfections and waning immunity, where the literature is more scarce. Lower respiratory infections (including pneumonia and influenza) and tuberculosis are two examples of diseases where reinfections are common. Their study is of critical importance since, according to the World Health Organization, they represent, respectively, the number one and number fourth most common causes of death by infectious diseases in the world. The main objective of this work is to present a time since last infection (TSLI) model that can be used to investigate the dynamics of many types of infectious pathogens that are able to reinfect hosts, and to apply it directly to the cases of influenza and tuberculosis. The model has two classes (never-infected and once-infected), with the transmission rate and susceptibility to reinfection being dependent on the TSLI, and optional vital dynamics. The TSLI model serves as a motivation to develop some general theory to treat models structured by several time-like variables as age-dependent population problems. This theory is applied to the TSLI model to prove that regular solutions exist and to study the conditions for existence and stability of equilibrium solutions. The TSLI model is simple, yet it has rich dynamics, such as multiple endemic equilibria, endemic equilibria in the absence of vital dynamics, asymptotically stable endemic equilibria even when [special characters omitted] < 1, and non-global asymptotic behavior. It has been suggested that reinfections with tuberculosis are more likely than first time infections. If this is the case, solutions of the TSLI model applied to tuberculosis converge to the disease-free or endemic equilibrium depending on the initial state of the population. Moreover, under this assumption, the disease can persist in the population even when [special characters omitted] < 1. Knowing this fact can be of vital importance for eradication strategies, since efforts have to be focused on not only lowering the [special characters omitted] but also changing the existing conditions of the population. The dynamics for influenza are heavily influenced by the drifting of the virus strains in the population and the periodic appearance of pandemic strains. Without a new pandemic strain, the TSLI model applied to influenza convergences to the endemic equilibrium in an very slow oscillatory manner. This suggests that the periodicity observed for influenza between pandemics can be partially linked to the reinfection and waning immunity process. Additionally, the TSLI model shows that a pandemic can have a significantly different attack rate in a population with existing protection to a similar strain and therefore, differences in susceptibility of the population should be an important factor to consider when modeling influenza pandemics. Another important feature of influenza is that the dynamics observed vary considerably between different latitudes. This motivated the study of an additional ODE Susceptible, Infected, Recovered model. The model incorporates appearance of new strains due to antigenic drift, and partial immunity to reinfection with related circulating strains. It also includes optional seasonal forcing of the transmission rate of the virus, which allows for comparison between temperate zones and the tropics. The model is capable of reproducing observed qualitative influenza patterns such as the overall annual outbreaks in the temperate region, a reduced magnitude and an increased frequency of outbreaks in the tropics, and the herald wave phenomenon, that is, a strain that produces a small outbreak in the summer and later dominates the winter influenza season. The herald wave phenomenon plays an important role in determining possible strains for future vaccines, and thus extensions of this model could be helpful for vaccine development.
Feng, Purdue University.
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