Population Modeling
MatLab, Engineering Analysis
For a class project, I worked with a group to create an SIR (Susceptible, Infected, Recovered) and SIS(Susceptible, Infected, Susceptible) model for population modeling of infectious diseases, tracking the percent of the population that falls under the following categories of susceptibile, infected, and recovered. In this project I provided an analytical solution for the model and compared it to numerical solutions generated using MATLAB’s Runge-Kutta 4 (RK4) and the ODE45 solver.
This section walks through some derivations of both the SIR and SIS models, along variable meanings:
β is the number of individuals, on average, that an individual makes clinically significant contact with per unit of time (e.g., weeks). Thus, one infected individual will cause β new infections in a population of N susceptible individuals. β could be made into a function accounting for varying contact rates with social distancing guidelines in different areas of a region. However, in our experiment, β was kept constant.
𝑆(𝑡) is the total number of susceptible people, which could be lowered by government constraints such as event capacity limits or vaccination efforts. Since 𝑆(𝑡) represents the total number of susceptible people, 𝑆/N is the fraction of the susceptible population. A single infectious person will, on average, infect about (β)(𝑆/N) = β𝑆/N individuals.
Multiplying this effect by the number of infected individuals, I(t), we find that there should be a total of βSI/N new infections per unit of time. The model assumes that, on average, a fraction α of the infected individuals will have recovered in a given unit of time. In the SIS model, these individuals are again susceptible. In the SIR model, these individuals cannot be infected again.
The system of equations for the SIR model:
SIS ODE: Similarly, if recovered individuals are again susceptible, we find just two coupled ordinary differential equations:
Normalization: Both models can be simplified by dividing through by the constant population size, N. Define s = S/N, i = I/N, and r = R/N. The system becomes:
This section walks through deriving the general equation i(t)
1. Given that S + I = N and s + i = 1, in system (4):
Substituting s = 1 - i into the equation for di/dt and distributing yields:
2. The equation di/dt = βi(1 − α/β − i) shows that one point where di/dt = 0 occurs when i* = 0. The endemic state can be found by setting (1 − α/β − i) = 0. Isolating i gives the second point i* = 1 − α/β.
3. β must be greater than α, as α represents the fraction of infected individuals who recover over a given time, and β represents the new infections caused by an infected individual. R0 must be greater than 1 for an endemic to exist, as more people must be infected than recovered.
4. The equation i∞ = 1 − α/β gives the steady state. Substituting into the original equation gives:
After taking the integral of each side, the expression becomes:
5. Rewriiting the equation using the given equations, we derive the general solution for i(t):
6. SIR Analysis: The general solution for i(t) is:
This section walks through deriving the general equation s(t):
7. Using the given equations in the system:
We can derive d𝑠/d𝑟 as:
Defining 𝑠(0) = 𝑠o, we have:
Thus, 𝑙𝑛|𝑠|= −𝑅𝑜r + 𝐶 and 𝑠 = 𝐴𝑒−𝑅o𝑟 where 𝑠(0) = 𝑠𝑜. Therefore:
8. First Order ODE and Rescaling
Given that 𝑆 + 𝐼 + 𝑅 = 𝑁, and thus 𝑠 + 𝑖 + 𝑟 = 1, we can substitute 𝑖 = 1 − 𝑟 − 𝑠 into d𝑟 = α from the system. This satisfies the first-order ODE:
9. Nondimensionalization
We can rescale as 𝑢 = 𝑘1𝑟 and τ = 𝑘2𝑡. The transformed equation becomes:
10. Effect of Social Distancing
To account for social distancing, both β and 𝑆(𝑡) must be modified. The parameter β represents the number of clinically significant contacts an individual makes per unit of time, which can change based on social distancing guidelines. Thus, β could become a function of social behavior and guidelines, and 𝑆(𝑡) could reflect the use of masks and other interventions to reduce susceptibility.
The SIR and SIS analytics solutions were utilized to describe susceptibility, infection, and recovery rates during an isolated epidemic, and the RK4 technique and ODE45 solver were evaluated. The initial step in this experiment were to demonstrate that the total of susceptibility, infection, and recovery rates remained constant. This information was then combined with the provided normalizations to arrive at an analytical solution for the SIS model. The SIR analysis then solved for various parameters during the epidemic, such as when the ratio of recovered individuals to new infections caused by one individual is greater than the initially susceptible population, no epidemic occurs, or when social distancing and the effectiveness of masking in the spread of the disease are considered. When the analytical solution was compared to RK4 and ODE45, ODE45 was determined to be closer to the analytical solution than RK4. The RK4 error and the analytical solution were then displayed in a log-log graph, with the slope of the graph being 4 and RK4 being a fourth order technique, indicating that our code worked correctly. Finally, using a SIR model, it was discovered that the susceptible and recovered populations both fall and rise, with the same inflection point. The infected population grew until it was equal to the susceptible population, at which point it declined proportionally to the susceptible population.