PartIII: The SIS Model : Listening for the Cry of the Earth

Introduction

The SIS model of an epidemic refers to a situation where people are susceptible to being infected (S), become infected (I), and when they recover return to the population that are susceptible to being infected (S). It is the recovery and removal from the infectious population (I) that is different from the SI model.

The SIS model adds one more parameter, the rate of recovery from the illness, which we will label with the Greek lower case letter gamma $\gamma$ . The basic infection rate $\lambda$ is the same as the SI model. The size of these two parameters determine how fast things happen. The ratio determine what happens.

This section describes the traditional SIS model, sets up the differential equations formulation, and gives the solution to the differential equations. The characteristics of the solution are examined and compared with the SI model for identical situations. The Hubei data will allow some insight into the parameters of the SI model.

Most sane people modify their behavior in an epidemic or pandemic. This behavior change is usually one of isolation, changing the number of potential infectious encounters in a day: $\lambda$ starts to drop. Doctors will work to try to come up with therapeutic approaches to speed up recovery: $\gamma$ increases. We will consider the effects of the change in $\lambda$ in the model, but not the potential changes in $\gamma$ .

Approaching the SIS model: 3 views.

The SIS model itself describes a situation.. This section will explore three different approaches to implementing the SIS model.

Normally the SIS model is described by a set of differential equations. The advantage of this approach is that the equations can be solved exactly, as long as $\lambda$ and $\gamma$ don’t change. The following link will take you to a description of this approach and the python codes that implement the model. https://davidalarrabee.com/?page_id=1011

Like the SI model, the SIS model can be set up as a probability problem. If an individual is a member of those who are susceptible, they have a probability of becoming infected, depending on $\lambda$ . If they are infected they have a probability of recovery, depending on $\gamma$ . This allow for the SI model to be solved with a Monte-Carol approach, just like the SI model.

There is a third way to set up this model that is closely related to the Monte-Carlo approach. a transition matrix approach. This approach is easily generalization to models where the individual can be in many situations (states); susceptible, exposed, infectious, recovered, immune, dead, …. The ability to introduce time delays is also rather easy. This approach is related to the Mathematical analysis of Markov-Chains. We will set up the simple case here and extend it later.

The plan is to have all three approaches illustrated with this SIS model.

Estimating the parameters.

The parameters of models should flow from actual data. Sometimes the data is preliminary at best. In those cases we can try and fit the model to the reality. If there are enough parameters this approach can sometimes make a bad model reproduce the results.

We can generate an estimate for γ if we have an estimate of how long it takes 1/2 of the population to recover from the disease. Using COVID-19 as a test case, the best data I have at the time of this writing is from the WHO china joint missions https://www.who.int/docs/default-source/coronaviruse/who-china-joint-mission-on-covid-19-final-report.pdf (see pages 11-14). The report suggests that 80% of the cases have an average recovery time of about 14 days. The remaining 20% have average recovery times (or death) that is longer then 3 weeks, so we will take 3 weeks. If we take a weighted average of these two numbers the average recovery time is about 2.2 weeks of 15.4 days. Using this number we can estimate γ to be $\frac{ln(2)}{15.4} = 0.045$ .

The infectious parameter λ can be estimated at the start of the epidemic, if it starts with a period of exponential growth. For example, previously we found the Hubei initial exponent to be 0.2137. In the SIS model this is actually λ-γ. Therefore we have a revised estimate for λ =0.2137 + 0.045 = 0.2587. This does not take into account the actual time delay between an individual becoming infected and recovery.

These two numbers allow us to predict the peak number of infectious people (if there is no intervention).

$\frac{\lambda -\gamma}{\lambda} = \frac{0.2137}{0.2587} = 82.6\%$

For the details see the mathematics in the section on the differential equation model. This is significantly higher than the fraction of people that were actually infected. Thus we still have more work to do….

Consequences and Observations

The SIS model deals with diseases for which we do not develop immunity. One example might be the common cold. There are a few points about the model that warrant consideration.

The number of infected people is not the same as the number of people who have been effected. First, some have recovered. Second, since the recovered are returned to being susceptible, people can be infected 2, 3, 4 or an arbitrary number of times. The number of infections can exceed the population!

At equilibrium the number of people being infected is equal to the number of people that are recovering. This means that ultimately every gets the infection!

Lets assume an epidemic gets started in the population and we want to change something to end the epidemic. The SIS model suggests that this requires that $\lambda < \gamma$ or the infection rate has to be less than the recovery rate. This is the basis of a social isolation policy.

Both $\lambda$ and $\gamma$ are determined both by the nature of the disease and the social response. The willingness to socially isolate can lower $\lambda$ the quality of medical care (plus any effective therapeutic responses) can influence $\gamma$ .

What’s Next

What is still missing is the possibility that someone is in a state that is neither susceptible nor infectious. One addition would be someone that is removed from both populations. This could be due to death, or some immunity. This is the SIR model which we will consider next.

Implementation

In what follows we will develop three implementation strategies for an SIS model. We have been adding features to our simulations as we proceed. One feature we will now add is the ability to change parameters at specific time times. This will allow us to simulate the initial spread of the disease and a social response.

We can solve the differential equation. Here is a link https://davidalarrabee.com/?page_id=1011
We can perform a Monte Carlo simulation. Here is a link (TBD)
We develop a method motivated by by Markov Chains. Here is a link (TBD)