Aspects of Epidemiology
Darwin pictured evolution as a branching tree with lines of descent evolving independently from a common ancestor. Recent analyses show many examples of lateral transfers of genetic elements, at levels of complexity from gene fragments through genes and genomes. Darwin's picture of evolution among non-exchanging lines of descent was thus seriously incomplete. Evolution embodies a continuum of transfers, combinations, and amalgamations. Infections are important agents of lateral transfers of heritable elements. The presence of an infection in a susceptible host creates a new combined phenotype of host and agent that affects the survival, behavior, reproduction of that combined organism. The host-agent combination is sometimes vertically transmissible to the next generation (e.g. HIV, Trypanosoma cruzi, plant seeds infected with endophytic fungi). Infections are often acquired by ingestion (termites and domestic ruminants and many vector-borne diseases, for example). The three phenomena of infection, evolution and food webs are intimately interwoven but a general theory of their interaction remains to be developed.
A systematic approach to the modelling of a variety of biological systems is presented, which starts from individual-based models, and then goes on to derive from these the corresponding deterministic equations which are valid when the size of the system is large. The formalism may be used to study the stochastic dynamics of individual-based models common to a large number of systems, such as models of epidemics, metapopulations, metabolic reactions, biodiversity --- including Hubbell's neutral theory --- as well as more conventional predator-prey and competition models. In contrast to most previous studies, these processes are modelled using master equations, which allows use to be made of well-established methods from the theory of these equations to analyse their behaviours. The approach will be illustrated by formulating the SIR model with immigration as an individual-based model described as a master equation. The stochastic behaviour of this equation is then studied for relatively large system sizes, where it is shown that a simpler equation holds, which can then be solved exactly. In particular, the power spectrum of stochastic fluctuations can be found analytically. By using data on childhood infectious diseases in England and Wales, we observe that typical model parameters lie within a region of parameter space characterised by strong stochastic amplification, leading to large coherent fluctuations in the number of cases. These ideas are compared with those based on analyses carried out in a more deterministic framework.
During the past two decades, age-structured population models have been widely developed and applied to many areas in mathematical biology. I will present basic ideas for analyzing age-structured epidemic models where basic models are formulated as partial differential equation systems. Special attention will be paid to the basic reproduction ratio (threshold condition), characterization of endemicity, variable infectivity and susceptibility and the interaction between demography and epidemics. Concrete examples will be introduced using an SIR model for common childhood diseases, an HIV/AIDS epidemic model, evolutionary epidemic (influenza) model and vector-borne disease (Chagas disease) model.
The dynamics of many diseases show multi-year periods. The most widely know example is the two years epidemic period for measles. We study an evolutionary theory for why measles has a period of two years. We use standard susceptible-infected-recovered (SIR) model with seasonal variation in transmission rate. In general, when variation in transmission rate is small (i.e. strength of seasonality is weak), dynamics of infectious diseases show annual cycles, which is the same period as the seasonality. However, if we increase the strength, a period doubling bifurcation occurs and a multi-year period appears. We assume that the sensitivity of pathogens to seasonality is different among strains. Some strains have high sensitivity to the seasonality and their transmission rate varies greatly (seasonal specialist). Other strains are less sensitive and the variation in their transmission rate is smaller (seasonal generalist). If evolution prefers the former, longer periods would be observed because the seasonal variation in transmission rate is large. However, if evolution prefers the latter, shorter periods would be observed. As a result of invasion analysis, we find that there is an evolutionary stable sensitivity, and the sensitivity determines the period in the population dynamics. When we apply measles parameters, we can show that the period of two years is evolutionary stable. Our analysis gives new insight into the evolution of multi-year period in many diseases.
Seasonally forced SIR epidemiological models may have deterministic resonance solutions oscillating at an integer multiple of the forcing period. In models with two pathogen strains the occurrence and structure of these resonance solutions is influenced by immunological cross-reaction. This talk considers the impact of immunological cross-protection and cross-enhancement in an annually forced model, focusing on the phase relationship between strains. We will show that most resonant solutions have an in phase structure, with antiphase structures only occurring when the intensity of cross-immunity is within a limited range, and explain this in terms of the amplification or moderation of cross-immunity by the phase structure. Finally we will briefly apply a stochastic form of the model to data for Dengue serotypes circulating in Bangkok.
How should resistant cultivars be introduced to plant population in order to control outbreaks of an infectious disease? In this presentation I will discuss optimal mixing ratios and planting patterns of susceptible and resistant cultivars based on a simplified lattice SIR model that describes only the infection and recovery processes in the host population. First, to estimate the initial expansion of the disease and the final crop, I will compare the lattice model with the complete mixing (homogeneous) model. Then I will discuss some relevant analytical methods and the role of spatial correlation between susceptible and resistant plants, considering in particular the difference between checker board and same-cultivar cluster planting patterns.