Introduction

Vaccination is one of the most notable successes of modern medicine. Smallpox has been eradicated, and many serious infectious diseases of childhood have been brought under control, with a vast reduction in the associated morbidity and mortality. To achieve this required placing a huge selection pressure upon the associated pathogens. Despite this pressure, there has been little evolution of the pathogen strains that escape from vaccine-induced immunity.

In this chapter, I first present a modeling approach that allows consideration of competition between strains of pathogens and their responses to changes in the balance of competition that are imposed by a vaccination campaign. This framework allows the calculation of conditions that would allow the emergence of a vaccineresistant strain. The numerical simulation of the evolution of vaccine resistance gives interesting insights into the time scale over which it might occur. Finally, I discuss four case studies from infectious diseases of humans.

Theoretical Framework

This section describes the basic theoretical framework on which the discussion in this chapter is built.

Basic reproduction ratio

The community-level impact of vaccines is best considered within the context of the basic reproduction ratio R0, which is defined as the number of secondary cases caused by one infectious individual introduced into a community in which everyone is susceptible. R0 can be generalized to Rp, the number of secondary cases caused by one infectious individual introduced into a community where a fraction p have been vaccinated and everyone else is susceptible.

The lifespan of T lymphocytes is of particular interest because of their central role in immunological memory. Is the recall of a vaccination or early infection, which may be demonstrated clinically up to 50 years after antigen exposure, retained by a long-lived cell, or its progeny? Using the observation that T lymphocyte expression of isoforms of CD45 corresponds with their ability to respond to recall antigens, we have investigated the lifespan of both CD45RO (the subset containing responders, or ‘memory’ cells) and CD45RA (the unresponsive, or ‘naive’ subset) lymphocytes in a group of patients after radiotherapy (Michie et al. 1992). We have found a rapid loss of unstable chromosomes (which result in cell death in mitosis) from the CD45RO but not the CD45RA pool. Immunological memory therefore apparently resides in a population with a more rapid rate of division. The survival curves for the two populations are best described by a model in which there is also reversion in vivo from the CD45RO to the CD45RA phenotype. Expression of CD45RO in T cells may therefore be reversible. Further data showing survival curves of T lymphocytes with stable radiation damage (passed to one daughter cell during mitosis) is also considered. These curves show very little loss of such cells. The difference between the two populations (stable and unstable damage) allows an estimate of their proliferation rates and death rates. These parameter estimates may be of interest to people modelling the dynamics of the immune response as they give some rough indicators of the timescales on which T lymphocytes turn over (McLean and Michie 1993).

What's so special about within-host dynamics?

When an epidemiology conference hosts a session on ‘within-host dynamics’, three questions immediately come to a discussant's mind: ‘Why are we doing this?’, ‘What are we doing here?’ and ‘What difference does being within a host make?’ The first of these three questions is answered by the quality of the papers presented in this session. There are many fascinating questions about the pathogenesis of infectious diseases, and about the dynamics of host responses to infectious organisms. These questions often involve highly nonlinear interactions between host and pathogen within the host organism. The rigour and clarity of thought required by mathematical description of such interactions is a great aid in developing an intuitive understanding of which processes are important, and of what patterns those processes might generate.

The subject matter of the four talks: two on HIV, one on malaria and one on schistosomiasis is probably a fair representation of the field. The enigma of HIV's pathogenesis has prompted many theoretical (and empirical) investigations. Nowak's theory is one elegant example of the numerous theories proposed to explain the long period between infection with HIV and illness with AIDS (reviewed in McLean 1993). In contrast to the care and rigour with which Nowak's theory has been expounded, some of the ‘verbal theories’ of HIV's pathogenesis are classic examples of why biologists ought to make mathematical models; so that they can see when the predictions made by their verbal models simply cannot be matched up with the patterns they aim to explain. A cogent argument for the use of mathematical models in an exploratory fashion by biologists is given by Hillis (1993).