Katie Kruzan

Mathematician

Emily Detmer

Math Intern

Chronic Disease Susceptibility, Infection, and Recovery Analysis

The SIR Model is a compartmental model that tracks the growth and decay of the three main stages of a patient’s disease Susceptibility, Infection, and Recovery. At Perception Health, we leveraged an application of the SIR model to conduct a study on chronic diseases. The analysis catalyzed multiple discoveries regarding the behavior of the disease. As a patient is treated, the SIR model calculates the rate at which infection and recovery occur on average across a specified population.

A generalized diagram of the SIR Model adapted for this analysis.


The Model

To apply the mathematical model, the first assumption is that population growth/decay occur at equal proportions (g) for Susceptibility, Infection, and Recovery. It is also reasonable to assume that a certain percentage (β) of the Susceptible population will become Infected at each time step, or at the given point of time in the date range we are working with - i.e., days, months, or years. Additional assumptions are that there will be a percentage of the population (d) who will die from the disease at each time step, and that a percentage (α) of the Infected population will Recover from the disease at every step.

With these assumptions, it can be concluded that -1 ≤ g ≤1 and β, d, and α range from 0 to 1. The model is then defined using a system of differential equations - a set of equations that displays the rates of change that occur for each compartment in the model. The following system of differential equations was used for this application of the SIR model:

These equations define the rates of change for each group, adding and subtracting the factors in the model influencing the group.

The Analysis

Reaching an Equilibrium

One question mathematicians seek to answer in differential equation models is “When will this system be at equilibrium?” The system is in equilibrium when there is no movement, or change, occurring. In this case, equilibrium is the point at which people are no longer diagnosed with the disease and there are no patients recovering from the disease. To find equilibrium, it must be determined when the system is no longer changing. This happens when the derivatives (rates of change) are equal to 0.

In applying this to the model, the derivatives are equal to zero when S = I = R = 0. However, this is not the only time the system is at equilibrium. When looking for another equilibrium, we used the original assumption that population growth/decay happens in equal proportions (g) for Susceptibility, Infection, and Recovery. Solving for equation (4) at equilibrium S’ = I’ = R’ = 0, produces:

From the original assumption, β cannot be less than 0, because people move from susceptible to infected in one way. Using these two findings, two cases can be considered when thinking about the rest of the system:

Case 1: g > 0
From a previous assumption, it is known that α  0, because people are moving from Susceptible to Infected in one way. By this assumption, it can be concluded that equation (6), 0 = α I (t) + g R (t), will never be true because α is non-negative and g will always be positive in this case. Therefore, there is no non-zero equilibrium in this case. (Note: Equilibrium is  a point at which the system of equations is satisfied.)

Case 2: g = 0
Since β equals g, β = 0 as well. Knowing this, it is easily concluded that equation (4) is true using substitution, meaning equation (1) is at equilibrium. Looking at the equations (5) and (6) and setting β = g = 0 on equation (6), the results are:

The above is true when α = 0 and is also true when I(t) = 0. These two subcases will be considered separately.

Subcase 1: g = 0, α = 0  and  I > 0
In this subcase, the following has been established: g = 0, α = 0, and β = g. This means β = 0. Using these findings in equations (4) - (6) yields the following results:

It is seen that all three of these equations are true only if d = 0 (since I ≠ 0).

If there is no population growth (g = 0), no one getting diagnosed with the disease (β = 0), no recovery from the disease (α = 0), yet there are still infected people in the population (I > 0), then there must also be no people dying from the disease (d = 0) for the system to be at equilibrium. Note that in this case, there can still be people in all three groups (Susceptible, Infected, and Recovered).

Subcase 2: g = 0  and  I = 0
In this subcase the following has been established: g = 0, I = 0, and β = g. This means β = 0. Using these findings in equations (4)-(6) results in the following:

Here it is shown that this condition (g = 0 and I = 0) alone is enough for these equations to be true.

This means if there is no population growth (g = 0), no one getting the disease (β = 0), and no one who has the disease (I = 0), then the system will be in equilibrium.

Chronic Diseases Aren’t Going Away Anytime Soon

Once the equilibriums have been determined, one question mathematicians want to answer in differential equation models is “How stable are the equilibriums of the system?” To find the answer to this question,  the eigenvalues of the Jacobian matrix of the system must be found. The Jacobian matrix of the system is as follows.

This matrix is not dependent on the values of S, I, and R so it can be concluded that this Jacobian matrix is not dependent on the location of the equilibrium point. Knowing this, it can be concluded that the matrix will be the same for any values of S, I, and R. This means any conclusions from the Jacobian matrix of this system apply for any equilibrium of the system.

Given that this matrix is a triangular matrix, the eigenvalues (represented by λ) are just the entries along the diagonal of the matrix. In this case λ₁ = -β + g, λ₂ = -α – d + g, λ₃ = g. Therefore, λ₁ = -β + g corresponds to the Susceptible population, λ₂ = -α – d + g corresponds to the Infected population, and λ₃ = g corresponds to the Recovered population. If the eigenvalue is negative, the axis is stable. If the eigenvalue is positive, the eigenvalue is negative For an axis to be stable, the eigenvalue needs to be negative. This leads to the following conclusions:

  • For the Susceptible population to be at stable equilibrium, the rate at which people are diagnosed with the disease (β) must be greater than the population growth rate (g).
  • For the Infected population to be at stable equilibrium, the sum of the recovery rate (α) and the death rate (d) must be greater than the population growth rate (g).
  • For the Recovered population to be at stable equilibrium, the population growth rate (g) must be negative.

From these conclusions, it can be shown that this system will cycle through periods of stability and instability. When the system is stable, the system will be headed towards the equilibrium and when it is unstable, the system will be headed away from the equilibrium. The system has multiple cases of equilibrium. In the case that the equilibrium is not at the origin, it has been shown that β = g . Using this finding when looking at the eigenvalue corresponding to the Recovered group we can see that the eigenvalue will always be positive in this case, leading to an unstable axis. For the Recovered group to be in stable equilibrium, the population growth rate must be negative and stay that way, eventually leading towards the equilibrium at the origin (S = I = R = 0). Historically, we can see that when the growth rate becomes negative, it returns to a positive state within the next 10 years.

In the case that the equilibrium is not at the origin, the eigenvalue corresponding to the Susceptible group is equal to 0, meaning further analysis must be done to determine the stability of this equilibrium. For the Susceptible group to be at a stable equilibrium, the rate at which people are diagnosed with the disease must be higher than the growth rate. If this condition sticks, there will be no more Susceptible people in the population. A higher diagnosis rate can come from increased public awareness, prevalence of the disease, or other factors. However, unless every person is expected to be diagnosed with the disease, these increased rates are usually temporary, eventually lowering back down and making the equilibrium unstable.

For the Infected group to be at a stable equilibrium, people must be recovering or dying from the disease at a higher rate than the population growth rate. This condition can happen when the disease wipes out the Infected population. It can also happen when a cure is found for the disease, leading to massive recovery. However, if either of these cases happens, the rates would eventually become lower than the population growth rate. If there are no Infected people left to die or recover from the disease, then both the death rate (d) and recovery rate (α) would be at zero. However, we have previously stated that the growth rate fluctuates, but has historically always returned to a positive state. That means there would once again be a point where the axis would be unstable, and people would become Infected once again.

Therefore, chronic diseases are not likely to go away anytime soon.

We Can Model the Past and the Future

Using the SIR model, the system of equations shown below to solve for the previous time period can be used. Therefore, the historical behavior for the disease can be modeled. The system of differential equations (equations (1) - (3)) are equations that calculate the changes in each group at any given time t. If the values of S, I, and R at time t are known,  those values can be found at the next time step t+1. Given any function ƒ(t), the value of the function at ƒ(t+1) is equal to the value of the function at time t added to the change in that function at time t. In other words,

This knowledge can be used to find what is happening in the future given the present.

Below is an example using S(t):

This result can also be used to go back in time. Given a time in the future, t+1, the values at the previous step can be calculated. Solving for the previous time period, S(t), the equation gives the following results:

Repeating this process for I produces the following results:

Solving for I(t) gives:

This process for R produces the following results:

Solving for R(t) results in:

With these equations, as long as the population growth/decay rate (g), the rate at which Susceptible people are getting the disease (β), the rate at which people are recovering from the disease (α), the rate at which people are dying from the disease (d) at a time t, and how many people are in each group (Susceptible, Infected, and Recovered) are known at time t+1, then it can be calculated how many people are in each group at time t. Historical data for each of these values exist. Therefore, knowing the present, the future can be predicted, and the past discovered.

The Conclusion

Following the application of the SIR Model, Perception Health has made multiple discoveries about the growth and decay of chronic diseases throughout the three model states: Susceptibility, Infection, and Recovery. Using the system of equations derived from the model and various implementations of linear algebra, the behavior of the disease can be modeled. Specific analyses can be done for particular chronic diseases over a specified population using this generalized model, resulting in a better understanding of disease patterns for a given population.

While chronic diseases are an on-going epidemic, providers and healthcare systems can utilize the various applications of the SIR Model displayed throughout this text, to effectively predict fluctuations of chronic disease states. Using the present to model both the past and future states of chronic disease will allow for proper preparation, treatment, and overall care for each disease.

Photo by Ishan @seefromthesky on Unsplash