An Application of the Kermack-McKendrick Theory to the Epidemiology of Schistosomiasis

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  • Center for Documentation and Communication Research and Department of Preventive Medicine, Case Western Reserve University, Cleveland, Ohio 44106

A mathematical representation of the epidemiology of schistosomiasis was developed from the classic Kermack-McKendrick model of an epidemic process involving an intermediate host. The Kermack-McKendrick equations, which are not generally solvable for two-stage diseases such as malaria, are shown to be solvable for the four-stage disease schistosomiasis. Consequently, it was deduced from this model that 1) there exists a threshold condition stating that the susceptible snail population must not exceed a certain number to ensure against an epidemic outbreak of schistosomiasis among human beings, 2) this number represents the break point if it is not exceeded by the total active snail population (susceptible plus infective snails) because, under these circumstances, the disease in the human population will decline to the point of extinction, 3) in the event of an epidemic outbreak among human beings, the process will converge to an endemic state that is precisely at the threshold, and 4) the outcome of the epidemic process is independent of any variation above the threshold in the susceptible snail population. Hence, control of the disease may be achieved by maintaining the snail population at or below a certain critical level. That is, if the density of susceptible snails is above the critical level, an epidemic will occur, if the density of snails (susceptible plus infective) is above the critical level but the number of susceptible snails is at or below it, the process will be stable (endemic state), and if the density of snails falls to or below the critical level, then decline in the disease will ensue.