I MMUNOLOGICAL MODELS OF E PIDEMICS Oscar Angulo Torga ♭ , Fabio Augusto Milner † and Laurentiu Mircea Sega ♯ ♭ Departamento de Matem ´ atica Aplicada, Universidad de Valladolid, Espa˜ na, oscar@mat.uva.es † School of Mathematical and Statistical Sciences, Arizona State University, USA, milner@asu.edu ♯ Department of Mathematics, Georgia Regents University, USA, lsega@gru.edu Abstract: Mathematical models for analyzing the spread of a disease are usually epidemiological or immunological. The former are mostly ODE-based models that use classes like susceptibles, recovered, infectives, latently infected, etc to describe the evolution of an epidemic in a population. Some of them also use structure variables, such as size or age. The latter describe the evolution of the immune system/pathogen in the infected host—usually resulting in death, recovery or chronic infection. There is valuable insight to be gained from combining these two types of models, as that may lead to a better understanding of the severity of an epidemic. In this article we propose a new type of model that combines the two by using variables of immunological nature as structure variables for epidemiological models. We then describe a practical application of the model to HIV infection. Keywords: immunology, epidemiology, dynamical systems, PDEs, mathematical biology 2000 AMS Subject Classification: 92D25 - 92D30 - 35Q80 1 I NTRODUCTION In almost all epidemiological models, the death rates and the recovery rates are either constant or de- pendent on chronological age or age of infection. However, what ultimately decides whether an individual recovers from an infection (with or without immunity), endures it as a chronic disease or dies because of it, is how the immune system of that individual protects her/him from the disease [1]. With HIV, for example, the depletion of CD4+ T cells, not age, is the best indicator of an individual’s state of health and prognosis [7]. Therefore, a model that takes into account the immune status of individuals in the population seems reasonable and desirable. A similar philosophy of nesting one model inside another was used in [5]. The authors presented a framework for modeling host-parasite coevolution using nested modeling, and illustrated it by analysing a simple host-parasite system. [2] and [4] considered models similar to the one presented in this paper, although their approach was geared more towards understanding the evolution of virulence and natural selection. [6] presented a model of viral spread in a population that embeds a immune response model in an epidemic network model. 2 AGENERAL I MMUNOLOGICAL SIR- TYPE MODEL We present the general form of our model resembling so-called size-structured models, and then present a model that is specific to HIV. We will allow any number of structure variables that will be represented by a vector, and assume the total population is undergoing logistic growth thorugh linearly increasing mortality but no reduction in fertility. The goal of this model is to provide detailed immunological information about individuals in a population suffering from an epidemic of a viral disease for the purpose of informed public health planning. Consider a vector Y that has as components the immunological variables that will be used as structure variables for the epidemiological model, which evolve according to a system of ODEs, for example Y = (T,T * ,V ) and dT dt = s + pT  1 − T +T * Tmax  − d T T − kV T, dT * dt = kV T − δT * , dV dt = bδT * − cV, (1) where T , T * and V are, respectively, the densities of CD4+ T cells, infected CD4+ T cells and virus in the plasma of an individual. The model parameters in this model have the following meaning and approximate values [3]: