Moment Equations and Dynamics of a Household SIS Epidemiological Model

An SIS epidemiological model of individuals partitioned into households is studied, where infections take place either within or between households, the latter generally happening much less frequently. The model is explored using stochastic spatial simulations, as well as mathematical models which consist of an infinite system of ordinary differential equations for the moments of the distribution describing the proportions of individuals who are infectious among households. Various moment-closure approximations are used to truncate the system of ODEs to finite systems of equations. These approximations can sometimes lead to a system of ill-behaved ODEs which predict moments which become negative or unbounded. A reparametrization of the ODEs is then developed, which forces all moments to satisfy necessary constraints.Changing the proportion of contacts within and between households does not change the endemic equilibrium, but does affect the amount of time it takes to approach the fixed point; increasing the proportion of contacts within households slows the spread of the infection toward endemic equilibrium. The system of moment equations does describe this phenomenon, although less accurately in the limit as the proportion of between-household contacts approaches zero. The results indicate that although controlling the movement of individuals does not affect the long-term frequency of an infection with SIS dynamics, it can have a large effect on the time-scale of the dynamics, which may provide an opportunity for other controls such as immunizations to be applied.