Simulations

First, we can explore the impact Beta has on this model. Beta represents the probability of transmitting disease between a susceptible and an infectious individual.

As Beta increases, moving left from the blue lines to the red lines, the equilibrium is achieved much faster, and the equilibrium number of infected people is higher. If we relate this back to R0, the blue set of lines represents a scenario with R0 = 3, and the red set of lines represents a scenario with R0 = 4. This shows two things: 1) the relationship between beta and R0 is proportional 2) as probability of transmission increases, R0 also increases. Put into context, the approximated value for the COVIDs R0 is ~2.5.

Second, we can use sensitivity of the gamma variable to understand how the average duration of infection influences the amount of people that are infected. (Gamma is the inverse of the average duration of infection)

As Gamma increases, moving right from the blue lines to the red lines, the equilibrium number of infected people decreases very drastically relative to the increase in time it takes to reach equilibrium. Intuitively this makes sense. Gamma increases when the average duration of infection decreases. When this occurs, there is lower risk of contacting other susceptible individuals. Conducting a similar analysis on R0 values, the R0 of the blue set of lines is 2.5, and the R0 of the red set is 1.67. This also exemplifies that as R0 approaches 1, the environment gets closer towards a state of being infection-free in the long run.

As mentioned at the top of this page, while Gamma and Beta are important in identifying the equilibrium state of the SIS model and the progression of infection within a system, what really drives this model is the interaction of these two variables. The interaction between these variables is R0. Using the SIS equilibrium definition from the SIS page, we know that R0 = beta/gamma. In the above examples, we've seen the R0 value range from 1.667, where at equilibrium, there were more susceptible people than infected, to 3, where the intersection between susceptible and infected cases occurred in as fast as 60 days. As in the figure to the right, when R0 = 1, regardless of the beta and gamma values, widespread infection is prevented. We know that R0 is the average number of people that a single individual infects. Intuitively, if on average a person isn't infecting anybody else, then the disease will not be transmitted further.