A stochastic distributed-delay model of disease processes in dynamic populations

Hurd, H.S., Kaneene, J.B. and Lloyd, J.W., 1993. A stochastic distributed-delay model of disease processes in dynamic populations. Prev. Vet. Med., 16:21-29. 1A simulation model that is applicable to infectious and non-infectious disease is proposed. This paper describes a model for simulation of infectious and non-infectious disease processes in dynamic populations, and compares its behavior to a stochastic version of the Reed-Frost model for a hypothetical infectious disease. A distributed-delay model is applied. Monte-Carlo simulations of both modeling approaches produced epidemics of randomly determined sizes. Although both models demonstrated the characteristic bimodal distributions of total number of cases per epidemic, the shape of the distributions was slightly different. Separation between the two peaks was not as great with the distributed-delay model as with the Reed-Frost model. The tail was slightly more extended than the Reed-Frost, and there were more epidemics in the 50100 case range. Both models produced similar average attack rates. I N T R O D U C T I O N Simulation modeling has been used in epidemiology since the nineteenth century (Susser, 1985 ). It is useful for generating hypotheses regarding disease processes, for quantitatively describing the behavior of a complex system, and for determining sensitive areas in the system for management input. Virtually all of the work in epidemiologic simulation modeling has taken place in the area of infectious diseases (Bailey, 1975; King and Soskolne, 1988 ). Infectious-disease models have been built on the premise that disease can be viewed as the movement from the state of susceptibility, to latency, to incubation, to infection, to recovery, death, or immunity (Nokes and Anderson, 1988 ). Using this perspective it was straightforward to the development Correspondence to: J.B. Kaneene, Population Medicine Center and Department of Large Animal Clinical Sciences, College of Veterinary Medicine, Michigan State University, East Lansing, MI 48824, USA. © 1993 Elsevier Science Publishers B.V. All rights reserved 0167-5877/93/$06.00 2 "9 H.S. HURD ET AL. of various state-transition models that would simulate the movement of individuals from one state to the next. The driving force for the critical transition from susceptible to infected was the interaction between current infectious and susceptible individuals (Abbey, 1952; Ackerman et al., 1984). Non-infectious disease processes move through a similar series of states. For example, cancer development can be seen in terms of the onset of early cellular changes followed by undiagnosed disease which progresses to clinical manifestations, followed by death or recovery (Morrison, 1979). However, very few models of non-infectious disease have been developed. The two general modeling techniques used for infection-disease modeling (mass action and chain binomial) have certain limitations. Some of the limiting assumptions of mass-action models are that they assume random and homogeneous mixing of susceptible individuals (Fine, 1982 ), and that there is a linear relationship between the incidence rate and the number of cases. They can be deterministic or stochastic (Bartlett, 1953 ), and they can be discrete time (Soper, 1929 ) or continuous t ime (Bailey, 1955 ). However, massaction models are always continuous entity. Also, the exit rate from these firstorder differential equations has an exponential distribution over time. In order to overcome these limiting assumptions, the chain-binomial models were developed (Greenwood, 1946). In these models, new cases of disease occur in a series of stages. The number of cases at any stage will have a binomial distribution depending on the number of infectious and susceptible individuals at the previous stage (Bailey, 1975 ). These models are fully stochastic, discrete time and discrete entity. These models assume that the period of infectiousness is relatively short and of constant duration, and that there is a constant probability of infection in each serial interval (Fine, 1982). There are at least four types of chain-binomial models; the Greenwood type (Greenwood, 1946 ), Reed-Frost , the Elveback type, and Markov models. The objectives of this paper are to describe a distributed-delay model (DDEL) for the simulation of infectious and non-infectious disease. The behavior of the model will be compared with that of a Reed-Frost model. The output distributions and average attack rates for a hypothetical infectious model will be evaluated. M A T E R I A L S A N D M E T H O D S The distributed delay used in this paper is a Euler numerical integration (Hamming, 1962) of the kth order differe~tial equation shown in eqn. ( 1 ). (The Quick Basic (Microsoft, 1988) subroutine for simulating this equation is available from the senior author. ) d~y(t) ' d~-lY(t)~+ay(t)=x(t) (1) a~ dt--~:---e a~_ ~ dt ~~ "'" MODELING DISEASE PROCESSES IN DYNAMIC POPULATIONS 23 where x(t) is the input at t ime t, i.e. new susceptible individuals, y(t) is the output at t ime t, i.e. new cases, k is the order of the defining differential equation, ak is the kth specific parameter defining the response ofy (t) to x ( t ), e.g. 1/k of the delay E(z) .