Patterns in the effects of infectious diseases on population growth

Abstract

An infectious disease may reduce or even stop the exponential growth of a population. We consider two very simple models for microparasitic and macroparasitic diseases, respectively, and study how the effect depends on a contact parameter kappa. The results are presented as bifurcation diagrams involving several threshold values of kappa. The precise form of the bifurcation diagram depends critically on a second parameter xi, measuring the influence of the disease on the fertility of the hosts. A striking outcome of the analysis is that for certain ranges of parameter values bistable behaviour occurs: either the population grows exponentially or it oscillates periodically with large amplitude.

PIP: The dynamics of epidemic, models of mechanisms and resulting phenomena are presented: a model of the SI type for microparasitic diseases and a model for the host parasitic system. The population is assumed to be growing during which time the disease is introduced. Patterns of changes in dynamical behavior will be explored as an increasing contact rate parameter. It is expected that the dynamical behavior of diseases which have a strong influence on fertility will be different from those diseases that do not. The basic components of the models are the patterns of influence of disease on mortality and the fertility of the hosts, and the manner in which the force of infection varies with population size and composition. The models incorporate a saturating force of infection and additional mortality and reduced fertility due to the disease. 3 models are introduced and methodology explained: 1) Model I explains microparasitic diseases, 2) Model II explains macroparasitic diseases, and 3) Model III a transformation and unification of Model I and II applied to 4 different propositions. For example, the 1st proposition is that the set M is positively invariant and there exists a bounded region Q which absorbs all orbits. When "g" has no zero in the right "y," all orbits converge to the segment of the y-axis between 0 and y. When "g" has no zero at all, (0, 0) is globally asymptotically stable. Substantiation was given for the idea that disease gains a stronger hold on the population as the contact parameter "k" (the least upper bound for the force of infection) is increased. There is the possibility of unstable behavior so that the disease may or may not reduce the growth rate of the population. Also substantiated was that a multiplicative negative influence of the disease on the fertility of the infectives can lead to sustained and sometimes large oscillations in the sizes of the population, but does not occur if the influence on fertility is modeled additively. The way in which interaction occurs between infectives and susceptible is also important. Model I is unique in that it contradicts prior assumptions. It presents phenomena as oscillating solutions. This opens up the question of whether oscillatory behavior in real biological host/parasite systems may be an attribute of the influence of the parasite on the fertility of the host.