Logistic Model for Propagation of a Disease - Continuous Version

As an alternate model, which we will discuss in class later, consider the following model, which is continuous in time. Let $p$ be the fraction of a mixing populace which is infected with a disease. Then $1-p$ is the fraction of the population which is susceptible, and a model for $p$ changes in time is

$$\frac{dp}{dt} = \lambda p (1-p) - \gamma p ,\hspace{3in} ( \star )$$

where $\lambda$ captures the rate of transmission of the disease in the population and $\gamma$ reflects the recovery of the infectious population. For different diseases and different mixing populations, $\lambda$ and $\gamma$ will vary.