Modeling Epidemics: A Primer and Numerus Software Implementation

Wayne M. Getz; Richard Salter; Oliver Muellerklein; Hyun S. Yoon; Krti Tallam

In the continuous-time formulation, λ is the rate at which new individuals are recruited to the susceptible population (births or emigration), µ and α are natural and disease-induced mortality rates respectively. In the discrete-time formulation, a competing rates approach is used to derive transition proportions p, with identifying subscripts, as described in the text. out, we use the roman font S, E, I, V and D to name the class itself and the italic font S, E, I, V and D to refer to the variables representing the number of individuals in the corresponding classes. The assumption of homogeneity implies that age and sex structure are ignored. We incorporate population spatial structure-as would be found in countries comprising of a network of cities, towns, and villages-into a metapopulation framework [15, 18] , if we assume that a set of homogeneous subpopulations can be organized into a network of subpopulations, among which individuals move in a fashion that reflects appropriate movement rates (e.g., propensity to move as a function of age and sex [19] )and geographical factors (e.g., distances, geographical barriers, desirability of possible destinations). If the time scales of the epidemic and movement processes among subpopulations, including disease induced mortality, are much faster than the time scale of the background population demography (births, recruitment, natural mortality and population level migration) then we can ignore the demography; otherwise we cannot. For example, in the case of influenza, epidemiological and local movement processes involve noticeable changes at the scale of weeks, while demographic changes in the underlying population itself (beyond epidemic disease induced death rates) are obvious only at the scale of years. In this case, we can ignore natural births and deaths, and focus on epidemic processes alone.