Introduction

Given the prominent role of medicine in today's world, it is amazing to realize that the theory that microbes (germs) were the cause of many diseases was not really established until the 1870s. Attempts to create models of epidemiology did not begin until the twentieth century. Hamer (1906) formulated a discrete time model in an attempt to understand epidemics of measles. By the late 1920s Kermack and McKendrick (1927) had published models showing that the density of individuals susceptible to a disease must exceed a critical number before an epidemic was possible. A rich literature of mathematical epidemiology developed during the middle of the twentieth century. But most ecologists were only dimly aware of this literature until the late 1970s, when Anderson and May published a series of articles with titles such as "Population biology of infectious diseases" (Anderson 1982, Anderson and May 1979, 1981, 1982). A recent review of Hethcote (2000) is recommended for students interested in a review of the history of infectious-disease models.

In this chapter we will examine models for interactions between hosts and microparasites such as viruses and bacteria. Parasitoid-host interactions (the Nicholson-Bailey models) will be covered in Chapter 10, on predator-prey relationships. Here we are primarily concerned with models for microparasitic diseases such as measles, in which the parasite reproduces quickly and reaches tremendous populations within the host. The duration of the acute infection is limited by either host defenses or host mortality, and is short relative to the host life span. Recovered hosts may have lifelong immunity. The dynamics of this relationship are driven by transmission between hosts and the essentials can be modeled by classifying host individuals as susceptible, infected, or recovered (immune). This is known as the SIR model, although there are many modifications and elaborations on the basic SIR model (Hethcote 2000). For example, in the MSEIR model, individuals are born with passive immunity (M); over time they lose this immunity and transfer to the susceptible class (S); individuals exposed (E) to the disease then become part of the infected (I) class; those who survive become part of the recovered (R) class. The actual abundance of the parasite within the host is ignored. Recall that many metapopulation models also ignore population size and dynamics within habitat patches.

By contrast, macroparasites (such as intestinal worms) typically cause chronic, persistent infections. (This is also a feature of certain microparasites such as herpesviruses and the malaria parasite.) Disease severity depends on the number of parasites present. Not only do infected vertebrate and invertebrate hosts accumulate parasites throughout their adult lives, but both the number and diversity of parasitic species also increase with host age (Dobson et al. 1992). For example, in brown pelicans (Pelecanus occidentalis), the number of helminth parasite species increases by age class, reaching 13 for birds aged over three years (Humphrey et al. 1978). The same study showed that the number of individual parasites peaked at 8000 for one-year-old pelicans, and then declined to about 4000 in birds older than three years. A survey by Dobson et al. (1992) of North American mammals showed that the average individual carried 369 macroparasites of three different species. The survey examined four orders: carnivores, lagomorphs, rodents, and artiodactyls. Carnivores carried the most diverse parasite fauna, lagomorphs the least.

Models of macroparasite dynamics must account for parasite abundance within hosts as well as host-to-host variation in parasite abundance (May and Anderson 1979). Another complication is that many macroparasites such as Schistosoma mansoni, which causes the disease known as schistosomiasis, have an asexually reproducing stage in an intermediate host. Schistosomiasis is one of the most important human diseases in tropical regions. It is estimated that 100 million people carry at least one worm. The adult schistosomes live in pockets of the intestinal blood vessels or veins of the bladder in the vertebrate host, where males and females carry out sexual reproduction. The eggs are passed out of the host through the feces and urine into a body of water, where the eggs hatch into free-swimming larvae known as miracidia. A successful miracidium penetrates a snail. Once inside the snail it undergoes asexual reproduction. After 4-7 weeks new free-swimming stages, known as cercariae, are shed from the snail into the water. The cercariae must find the vertebrate host within 48 hours and each cercaria is capable of penetrating the skin of a vertebrate host. Once inside the host they travel through the circulatory system in a journey that can take 6-12 weeks. Once they locate the proper tissue they mature into adult worms, completing the life cycle. An important aspect of this kind of life history is the asexual-reproduction stage in the snail. The large numbers of cercariae shed from the snail make it much more likely that the vertebrate host will be located and the life cycle completed. Finally, perhaps because of the complexity of the life cycle of the parasite, vertebrate hosts generally have ineffective immune responses. In fact, the pathology of schistosomiasis is a consequence of the immune response of the host, in which shed eggs are attacked and calcareous deposits laid down around them. The accumulation of these deposits blocks the spleen and excretory organs of the infected hosts. For models of macroparasites with complex life cycles, such as the parasitic helminths, see Dobson et al. (1992).

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