
MBW:Population Biology of Infectious DiseasesFrom MathBioContentsBackgroundAbstractPrior to the late 1970s, mathematical models, and field and experimental studies in ecology focused primarily on how populations are regulated through predator prey interactions. However, research was suggesting that parasite infection might also contribute to densitydependent regulation of host populations by reducing the natural population growth rates of hosts. Although mathematical models of pathogen transmission dynamics were well documented in the literature, most assumed that the host population remained constant. This assumption was an artifact of the evolution of disease modeling from a human disease perspective for which the population density does not change dramatically over the time course of the disease. The objective of this review article was to explore how models used primarily for describing human disease dynamics could be modified to describe the effects of parasite infections on the growth rate of host populations. The model developed eliminated the assumption of a constant population size by including parameters for loss of immunity, birth rate, death rate, and disease induced mortality. The proposed models could then be used to explore questions about the time course of a parasitic infection, whether it could be endemic or epidemic, and if it could be stably maintained in a natural host population. This classic paper serves as a framework for characterizing the transmission processes of parasites in terms of population variables and equations rather than simple natural history descriptions of parasite lifecycles and pathology. Predictions based on the proposed models have been met by current research on natural hostparasite systems. For a different look at parasite modeling look at APPM4390:Modelling the dynamics of a complex lifecycle parasite. HistoryIn ecology, field and laboratory studies, and simple mathematical models focused primarily on population regulation through predatorprey interactions (May 1972, 1973), resource limitation, and competition (Hairston et al. 1960). However, research on parasitehost interactions demonstrated the possibility that parasites, like predators could regulate host population growth and mediate species interactions (Park 1948). (for more information on recent predatorprey interactions models, please see MBW:Three Species Food Chain Modeling). In medicine, William Kermack and Anderson McKendrick proposed an early form of the SIR (Susceptible, Infected, Recovered) model in 1927 in the Proceedings of the Royal Society of London. The SIR model is the standard form used for modeling disease dynamics within a closed population. The goal of these early modeling attempts was to determine whether epidemics subside when the supply of susceptible individuals has been completely exhausted, or whether an epidemic can cease simply when the number of susceptible individuals drops below a threshold level (Kermack and McKendrick 1927). SIR ModelVariable and parameter definitions r = infection rate of susceptible individuals a = recovery rate of infected individuals S = Susceptible individuals I = Infected individuals R = Recovered individuals A conceptual diagram of the model in its simplest form is: With dynamics described by: This model involves several key assumptions: a) fixed population size (no births or deaths) b) homogenous population (no age or stage structuring) c) unlimited mixing of the population See the following projects that utilize or extend the SIR model
IntroductionAnderson and May (1979) specifically models the dynamics of directly transmitted, microparasite diseases by modifying the basic SIR model. The first model describes the dynamics of two experimental laboratory studies, one of mice and a bacterial disease and the second of mice and a viral disease. In both cases, susceptible mice were added to the population at a known rate, demonstrating the importance of changes within the host population. Subsequently, a second model is derived for natural populations with additions of associated birth rates, death rates, loss of immunity, and disease induced mortality. Based on this model, the article addresses three main questions 1) what biological characteristics of infection determine the impact on host population growth, 2) what are the population consequences of acquired immune responses and 3) what conditions lead to endemic or to epidemic infections? In part two of the two part series, the basic model was adapted to describe the dynamics of macroparasitic diseases (May and Anderson 1979). Types of ParasitesMicroparasites typically include viruses, bacteria, and protozoans, and are characterized by their small size. These parasites reach reproductive maturity quickly and produce vast numbers of offspring. Hosts that recover typically become immune to these parasites. Macroparasites, including parasitic helminths and arthropods, on the other hand, have longer generation times, and little reproduction within their hosts. These parasites are associated with persistent infection, and any immune response is shortlived and dependent on parasite abundance.
Modes of TransmissionParasite transmission can occur either directly or indirectly. Direct transmission is the spread of disease from one host to the next through physical contact between hosts, or by transmission stages which are inhaled, ingested, or penetrate the host. Conversely, indirect transmission occurs when a disease passes from one host to the next through an intermediate host. Intermediate hosts are organisms through which a parasite passes, but in which the parasite does not reproduce sexually. These can include biting vectors such as flies, mosquitoes, and ticks. Other organisms can also serve as intermediate hosts when they are eaten by the final host, or release parasites into the environment to infect the final hosts. Intermediate hosts are necessary hosts in the parasite lifecycle where asexual production occurs. A special case of direct transmission is vertical transmission, which occurs when a parent transmits a disease to their offspring. All other forms of transmission are referred to as horizontal. (For more information about the life cycle and transmission of parasites, please see APPM4390:Modelling the dynamics of a complex lifecycle parasite). Types of ImmunityOrganisms can respond to parasite infection through the innate immune system based on the genetic or physiological characteristics of an individual and not from previous exposure. However, most microparasite infections also stimulate the acquired immune system, which confers longlasting immunity from previous infection. When a portion of the population possesses an immune response to a particular pathogen, other individuals may be protected from infection by the parasite because the number of susceptible individuals is below the threshold needed to support the transmission of the parasite. This is referred to as herd immunity.
Disease DynamicsThe dynamics of the interactions between parasites and hosts can show two particular patterns. Endemic diseases are persistent within populations and infect a relatively constant percentage of individuals at any given time. Alternatively, epidemic diseases show a rapid increase in the percentage of individuals infected with a disease, followed by disease subsidence for a period of time. (For more information on modeling epidemics, please see APPM4390:Stochastic Epidemic Modeling). Anderson and May Model: Microparasitic infections as regulators of natural populationsAnderson and May (1979) builds on the simple SIR model in two key ways: 1) They remove the assumption that population size is fixed. To do this they add terms for the natural birth rate (a), natural mortality rate (b), and diseaseinduced mortality (α) 2) They remove the assumption that recovered individuals develop lifelong immunity by adding a term for loss of immunity (γ). Variable and parameter definitions X = Uninfected or Susceptible Y = Infected Z = Immune N = Total population A= Rate at which new individuals were introduced to the population a = Per capita birth rate b = Natural mortality rate t = Time ß = Transmission coefficient = Mortality rate caused by disease = Rate of loss due to immunity = Recovery rate A conceptual diagram of the model in its simplest form With dynamics described by: Therefore, the total population of hosts, N = X + Y + Z, obeys Analysis and interpretationFit of model to experimental dataAnderson and May (1979) first analyze the modified SIR model by allowing for the number of susceptible individuals to change. The objective was to use laboratory data to test how well populations of mice infected with microparasites conform to the predictions of the model. The laboratory study involved experimental addition of new susceptible mice rather than natural births or deaths, so instead of the parameter, a, above, they use A to represent the rate at which new susceptible mice are added to the experiment. Steadystate analysis of the equations yields the following: The disease is only maintained in the population if: If this condition is met then the total population size is reduced to: Through a combination of life history tables and estimates from graphing experimental data for N* vs. A, the authors estimate each of the parameters in the model. The graphs of model predictions matched experimental results quite well, suggesting that the model has good predictive power. These results have implications for host population size at different levels of disease severity and immigration rates. A graph of stable host population size (N*) vs. disease mortality rate (α) shows that diseases N* is smallest at intermediate levels of pathogenicity (disease severity). This is because when pathogenicity is small it has little effect on population size, while high pathogenicity reduces the susceptible population size below the threshold needed for disease persistence, and the disease subsides. Furthermore, the higher the immigration rate, A, the more the stable population size (N*) becomes depressed, because there are more susceptible individuals, allowing the disease to persist for longer. Microparasites and natural population regulationThe authors also consider the scenario of natural birth rather than artificial, experimental addition of new susceptible organisms. This analysis provides insight into parasitic regulation of natural populations. In particular it allows assessment of how infection characteristics might influence population growth, and what conditions lead to diseases existing as epidemics or endemics. A steadystate analysis on the SIR model that includes a parameter for natural birth rates yields equation similar to those above. The disease is only maintained in the population if: If this condition is met then the total population size is reduced to: If the condition is not met, the disease cannot persist, and the population has no other densitydependent regulatory factors, and grows exponentially according to: These equations can then be modified based on various disease characteristics (Table 1). Analysis of these modified equations shows disease characteristics that lend themselves to population regulation. Population growth is most affected by diseases that have short incubation times, a high mortality rate, and low recovery rates. Further analysis can provide insight into conditions that lead to epidemic and endemic diseases. The most important characteristic that determines disease dynamics is the host birth rate, because this introduces new susceptible individuals into the population. When host growth rate (r) is much less than the mortality due to disease (α), then infection causes an epidemic. This is because the high mortality causes the population size to drop below the threshold required for disease persistence, and the disease subsides. This suggests that shortlived diseases that cause longterm immunity in a population are most likely to occur as epidemics. Analysis of the model allows for predictions of disease dynamics in different systems. One of the predictions is that diseases which affect the reproductive capacities of infected hosts are more liable to suppress population growth. Control of population cycles by parasite infection reducing host reproduction has been shown in Red grouse population dynamics and nematode parasites (Hudson et al. 1998). Predictions about the threshold density of susceptibles along with population consequences of herd immunity have been observed in populations of wild African ungulates and the disease virus rinderpest (McCallum and Dobson 1995). Populations of wild animals increased after domestic animals were vaccinated for the disease preventing the disease from successfully transmitting to wild species (McCallum and Dobson 1995). However, there are still examples in few natural systems, especially in wildlife. Future research can investigate the interplay of stochastic forces that can result in population extinction and determine whether different host species support different types of disease dynamics because of susceptible host turn over through different population dynamics. Further complications arise when pathogens can infect multiple host species. For more on this topic, visit MBW:Population Dynamics of Pathogens with Multiple Host Species. ReferencesAnderson, R.M. and R.M. May. 1979. Population biology of infectious diseases: Part I. Nature 280:361367. Hairston, N.G., F.E. Smith, and L.B. Slobodkin. 1960. Community structure, population control and competition. The American Naturalist 94:421425. Kermack, W.O., and A. G. McKendrick. 1927. A contribution to the mathematical theory of epidemics. Proceedings of the Royal Society of London 115:700721. May, R.M. 1972. Limit Cycles in predatoryprey communities. Science 177:900902. May, R.M. 1973. Complexity and stability of model ecosystems. Princeton University Press, Princeton, NJ. May, R.M., and R.M. Anderson. 1979. Population biology of infectious diseases: Part II. Nature 280:455461. Park,T. 1948. Interspecies competition in populations of Trilobium confusum Duval and Trilobium castaneum Herbst. Ecological Monographs 18:265307. Hudson, P.J., A.P. Dobson, and D. Newborn. 1998. Prevention of population cycles by parasite removal. Science 282:22562258. McCallum, H. and A. Dobson. 1995. Detecting disease and parasite threats to endangered species and ecosystems. Trends in Ecology and Evolution 10:190194. Paper SummaryMathematics Used: ordinary differential equations, nonlinear system, separation of variables, transcendental questions, vital dynamics. Type of Model: SIR model with vital dynamics, parasites (microparasites / macroparasites). Biological System Studied: Parasites (both microparasites and macroparasites) to include transmission, immunity, birth, death, life cycles and their impact on population regulation.
Modeling Pathogen TransmissionThis is a summary of the article How should pathogen transmission be modeled? in which a copy can be found by clicking here. How should pathogen transmission be modeled? cites the article by Anderson and May which is the topic of the project above. In 1995 a researcher by the name of de Jong introduced a term called “pseudo mass action.” He felt as though the SIR model, specifically the BSI (or transmission) did not accurately reflect the density dependent nature necessary for true mass action. Instead, he felt that BSI/N more accurately reflected true mass action. Since his article was published, both transmission terms above are used in research, but how does this pertain to the Anderson and May article? Anderson and May found that there was a host density threshold below which a pathogen could no longer infect a population. This threshold is the corresponding population size that would reduce the basic reproductive ratio to one which transitions the outbreak away from an epidemic. The concept of host density threshold cannot exist if the model is depicted as BSI/N because the basic reproductive ratio will be independent of N where N is the total population size. In order to account for the density dependent requirement of true mass action, if Anderson and May changed the population size in an experiment they would also make a corresponding change in the size of the arena so that the density would remain constant. McCallum, Barlow, and Hone, authors of How should pathogen transmission be modeled? close their paper with recommendations. They suggest that the term "pseudo mass action" no longer be used because of its confusing nature. Additionally, they recommend that when constructing models the researcher explicitly state the form of transmission used, mainly that S, I, and N are numbers or densities. External linksAuthors Websites: Professor Sir Roy M. Anderson http://www1.imperial.ac.uk/medicine/people/roy.anderson/ Robert, Professor Lord May of Oxford http://www.zoo.ox.ac.uk/staff/academics/may_r.htm 