Compartment Analysis of Disease Progression in Time. II. Restricted Disease Increase

Abstract

AbstractThree‐compartmenl analysis was performed to describe the disease motion through presporulating (latent), sporulating (infectious) and postsporulating (removal) stages in the general case, where total disease increase is density dependent. A second‐order Runge‐Kutta method for numerical integration of the system of difterential equations was used to solve the equations. Total and postsporulating disease progress curves are S‐shaped while latent and infectious disease progress in the form of optimum curves. The curve of a composite variable defined as total (latent and infectious) inoculum reservoir of the host progresses also in the form of an optimum curve showing a maxi at the total disease level yt/K = 1 − 1/iR, where K is the total population of infection sites, i is the infectious period, R the intrinsic infection rate and iR ≥ 1/(1 − Vo/K). The size of this maxi is a monotone increasing function of the product iR by given initial disease level Vo. Condition iR =1/(1−yo/K) is a threshold condition for total inoculum to increase over yo, or alternatively a threshold condition for a rapid disease increase at the start resulting, possibly, in a large epidemic. Condition iR = O is a threshold condition for total disease to increase over initial disease level. Total disease reaches an asymptotic value less than unity if and only if infectious period is linite (existence of removals). In the compartment system there is a consistency regarding the threshold conditions for total disease to increase over initial disease level in the cases with and without a density‐dependent factor. Conversely, in the Vanderplankian system of differential‐difference equation the threshold conditions are iR = 0 and iR = 1 respectively due to the assumption of an exponential increase of total disease early in the epidemic. The particular cases without latent period and without removals are treated separately. The implications derived from the compartment analysis are compared with those derived from the Vanderplankian system of epidemiological analysis.