This article reviews researches on the spread of infectious diseases with mathematical models. First, the basic mathematical model of infectious diseases is introduced and its fundamental characteristcs are explained. After reviewing the available data of the incidence of infectious diseases, we describe how these data have been explained by mathematical models including the effect of the vaccination programs. We then move more practical models with increased reality, especially the study of sexually transmitted diseases such as HIV/AVIDS. One of the most important aspects of infectious disease spread is the behavioral heterogeneity of hosts. A population consisting of a small number of very active hosts among majority of rather inactive hosts may show spread pattern different from a uniform population consisting of moderately active hosts even though the two populations have the same average activity. Models of increased reality include the effect of different mixing patterns of sexual partners.
The models of malaria transmission have been developed for a century. Based on the Ross-Macdonald model that treats patient increase rate and infectious mosquito increase rate as two differential equation, enormous extended models have been branched. Among them, the DMT model is important because it has established SEIR framework and because involves latent period and human immunity against malaria parasites. After the DMT, major extensions included (1) intervention, (2) age-structure of human population, (3) heterogeneity of environmental settings, (4) asymptomatic infection in endemic area, (5) effect of human behavior, (6) interaction between drug-tolerant malaria parasite and mass drug administration, and (7) circulation of multi-strain parasites in a host population. Most of those could catch the actual malaria transmission better than ever, but the difficulty of valid estimation of model parameters made them less general. Heterogeneity and variability of host populations are so important that the further extension of models should be done for human behavior and agent-based framework.
The purpose of this article is to clarify the relation between the reaction -diffusion models and the integral equation models in the study of the spatial spread of epidemics and to discuss the speeds of spatial spread of an infectious disease through the reaction-diffusion models. The speeds of the epidemic waves are often derived heuristically from the linearization of the model equations, which is called 'the linear conjecture'. For the diffusive Kermack-McKendrick model, we show the validity of the linear conjecture by the use of the existence results of traveling wave solutions. Then, we give an example of the reaction-diffusion epidemic model which has a traveling wave solution with the speed greater than the predicted values of the speed by the linear conjecture, and examine it in the more general setting of reaction-diffusion modeling.
Following infection to a host, some pathogens including human immunodeficiency virus(HIV)and Trypanosoma brucei repeatedly alter their antigen expression, thereby escaping host immune defence (antigenic drift/switching). I here theoretically study intra-host dynamics of pathogen antigenic variants and immune response. In the first model, I assume that antigenicity of a pathogen strain can be indexed in a one-dimensional lattice (the charge-state model) , and analyzes the traveling waves of pathogen antigen variants and specific immune response in antigen-type space. In the second model, I assume that an epitope sequence of a finite length mounts a specific immune response (finite-site model), and focus on the effect of demographic stochasticity and phylogenetic pattern of pathogen variants generated by this escape-and-chase dynamics in sequence space. I found that, 'phylogenetic R_0', the mean number of new escaping mutants generated by a antigen variant before it is wiped out by mounted immune response, gives an accurate predictor for the phylogenetic patterns of antigenic drift.