Computational methods for stochastic epidemics

Abstract

Compartmental models constructed for stochastic epidemics are usually difficult to analyze mathematically or computationally. Researchers have mostly resorted to deterministic approximations or simulation to investigate these models. This dissertation describes three original computational methods for analyzing compartmental models of stochastic epidemics. The first method is the Markov Process Method which computes the probability law for the epidemic by solving the Chapman-Kolmogorov ordinary differential equations as an initial value problem using standard numerical analysis techniques. It is limited to models with small populations and few compartments and requires sophisticated numerical analysis tools and relatively extensive computer resources. The second method is the Probability Vector Method which can estimate the first few moments of a discrete time epidemic model over a limited time period (i.e. if Y(t) is the number of individuals in a given compartment then this method can estimate E[ Yr for small positive integers r. Size restrictions limit the maximum order of the moment that can be computed. For compartmental models with a constant, homogeneous population, this method requires modest computational resources to estimate the first two moments of Y(t). The third method is the Linear Extrapolation Method, which computes the moments of a compartmental model with a large population by extrapolating from the given moments of the same model with smaller populations. This method is limited to models that have some alternate way of calculating the moments for small populations. These moments should be computed exactly from probabilistic principles. When this is not practical, any method that can produce accurate estimates of these moments for small populations can be used. Two compartmental epidemic models are analyzed using these three methods. First, the simple susceptible/infective epidemic is used to illustrate each method and serves as a benchmark for accuracy and performance. These computations show that each algorithm is capable of producing acceptably accurate solutions (at least for the specific parameters that were used). Next, an HIV/AIDS model is analyzed and the numerical results are presented and compared with the deterministic and simulation solutions. Only the probability vector method could compete with simulation on the larger (i.e. more compartments) HIV/AIDS model.