A Density–Dependent Epidemiological Model for the Spread of Infectious Diseases

A population consists of healthy (H), infected (I) and recovered (R) individuals at any time (t). The infected individuals I(t) are capable of infecting the healthy individuals (H(t)) but not the recovered ones. A density dependent population model is used for the healthy individuals within the environment with maximum carrying capacity M. The first model developed and used in this paper consists of a coupled system of differential equations. The model can be used to predict an outbreak or massive epidemic through its infection rate b. The second model is a simplification of the first model assuming a density-dependence structure in all the compartments or states of the disease. It is shown that with such simplification, it is possible to predict the trajectory of the infectious disease over time given the rate of population growth of the susceptible or healthy population, the rate of infection and the rate of recovery from the infectious disease. Due to the density-dependence structure of both models, the birth and death rates are already implicitly factored into the epidemic model. Data for the spread of HIV across different countries of the world were used to illustrate the usefulness of the model.

Keywords - epidemic, non-linear differential equations, logistic curve, variation of parameters, chaotic dynamical systems, human immune-deficiency virus (HIV)