TY - JOUR
T1 - Outcomes of epidemic models with general infection and removal rate functions at certain stopping times
AU - Clancy, Damian
PY - 1999
Y1 - 1999
N2 - This paper aims to show how certain known martingales for epidemic models may be derived using general techniques from the theory of stochastic integration, and hence to extend the allowable infection and removal rate functions of the model as far as possible. Denoting by x, y the numbers of susceptible and infective individuals in the population, then we assume that new infections occur at rate jxyxy and infectives are removed at rate Yxy y, where the ratio fxy lYxy can be written in the form q(x+y)/xp(x) for appropriate functions p, q. Under this condition, we find equations giving the distribution of the number of susceptibles remaining in the population at appropriately defined stopping times. Using results on Abel-Gontcharoff pseudopolynomials we also derive an expression for the expectation of any function of the number of susceptibles at these times, as well as considering certain integrals over the course of the epidemic. Finally, some simple examples are given to illustrate our results.
AB - This paper aims to show how certain known martingales for epidemic models may be derived using general techniques from the theory of stochastic integration, and hence to extend the allowable infection and removal rate functions of the model as far as possible. Denoting by x, y the numbers of susceptible and infective individuals in the population, then we assume that new infections occur at rate jxyxy and infectives are removed at rate Yxy y, where the ratio fxy lYxy can be written in the form q(x+y)/xp(x) for appropriate functions p, q. Under this condition, we find equations giving the distribution of the number of susceptibles remaining in the population at appropriately defined stopping times. Using results on Abel-Gontcharoff pseudopolynomials we also derive an expression for the expectation of any function of the number of susceptibles at these times, as well as considering certain integrals over the course of the epidemic. Finally, some simple examples are given to illustrate our results.
U2 - 10.1017/S0021900200017587
DO - 10.1017/S0021900200017587
M3 - Article
SN - 0021-9002
VL - 36
SP - 799
EP - 813
JO - Journal of Applied Probability
JF - Journal of Applied Probability
ER -