Question

In Equation (3.153), we saw a linear version of an epidemic model. The commonly used nonlinear SIR model is given by $$ \begin{aligned} \frac{d S}{d t} &=-\beta S I \\ \frac{d I}{d t} &=\beta S I-\gamma I \\ \frac{d R}{d t} &=\gamma I \end{aligned} $$ where \(S\) is the number of susceptible individuals, \(I\) is the number of infected individuals, and \(R\) is the number who have been removed from the other groups, either by recovering or dying. a. Let \(N=S+I+R\) be the total population. Prove that \(N=\) constant. Thus, one need only solve the first two equations and find \(R=N-S-I\) afterward. b. Find and classify the equilibria. Describe the equilibria in terms of the population behavior. c. Let \(\beta=0.05\) and \(\gamma=0.2\). Assume that in a population of 100 there is one infected person. Numerically solve the system of equations for \(S(t)\) and \(I(t)\) and describe the solution being careful to determine the units of population and the constants. d. The equations can be modified by adding constant birth and death rates. Assuming these rates are the same, one would have a new system. $$ \begin{aligned} \frac{d S}{d t} &=-\beta S I+\mu(N-S) \\ \frac{d I}{d t} &=\beta S I-\gamma I-\mu I \\ \frac{d R}{d t} &=-\gamma I-\mu R \end{aligned} $$ How does this affect any equilibrium solutions? e. Again, let \(\beta=0.05\) and \(\gamma=0.2\). Let \(\mu=0.1\) For a population of 100 with one infected person, numerically solve the system of equations for \(S(t)\) and \(I(t)\) and describe the solution being careful to determine the units of populations and the constants.

Short answer

There are two constant equilibria in the standard SIR model, but these can be modified by adding constant birth and death rates. Numerical solutions will depict the variation in susceptible and infected individuals over time for the given sets of constants in the standard and modified models.

Step-by-step solution

- Prove Constant Population

We differentiate \(N = S + I + R\) with respect to time to get: \(\frac{dN}{dt} = \frac{dS}{dt} + \frac{dI}{dt} + \frac{dR}{dt}\). From the equations in the problem, substituting \(\frac{dS}{dt} = -\beta SI\), \(\frac{dI}{dt} = \beta SI - \gamma I\), and \(\frac{dR}{dt} = \gamma I\), we get: \(\frac{dN}{dt} = -\beta SI + \beta SI - \gamma I + \gamma I = 0\). This proves that \(N = S + I + R\) is constant.

- Find and Classify Equilibria

For the equilibria, we set \(\frac{dS}{dt} = 0\) and \(\frac{dI}{dt} = 0\), from which we can find the conditions when the number of susceptibles and infected do not change in time.Solving these equations, we can find that there are two equilibria: (0,0) and \((\frac{\gamma}{\beta},N-\frac{\gamma}{\beta})\). In terms of the population behavior, the former equilibrium indicates the disease has been eradicated, while the latter signifies the disease persists in the population.

- Numerical Solution for Given Constants

To solve numerically, one would input the given values \(\beta = 0.05\), \(\gamma = 0.2\), initial susceptible individuals \(S(0) = 99\), infected individuals \(I(0) = 1\), and a system of differential equations into a numerical solver. The solution will describe the number of infected and susceptible individuals over time.

- Modifying System by Adding Constant Birth and Death rates

With addition of constant birth and death rates, the system equations have been given in the problem. It can affect the equilibrium solutions and result in a new set of equilibria. For an in-depth analysis, one would need to find new equilibrium points by solving \(\frac{dS}{dt} = 0\), \(\frac{dI}{dt} = 0\), and \(\frac{dR}{dt} = 0\).

- Numerical Solution for Modified System with Given Constants

Given the constants \(\beta = 0.05\), \(\gamma = 0.2\), \(\mu = 0.1\), initial susceptible individuals \(S(0) = 99\), infected individuals \(I(0) = 1\), input these values into the system of differential equations and calculate numerically. The solution will be a description of the number of infected and susceptible individuals over time for the modified system.

Key concepts

Nonlinear Differential Equations

In the context of the SIR epidemic model, we encounter a system of nonlinear differential equations. These are equations where the unknown function and its derivatives appear in a non-linear way, meaning they are not simply multiplied by constants or added together. In our case, \( \frac{dS}{dt} = -\beta SI \) and \( \frac{dI}{dt} = \beta SI - \gamma I \) involve terms like \( SI \) which makes the system nonlinear.

Nonlinear systems are particularly challenging because they don't always have straightforward, closed-form solutions. Instead, they often require numerical methods for approximation, as the behavior of the solution can be significantly complex. These dynamics are what make the study of epidemics interesting, as the interactions between susceptible (S) and infected (I) individuals give rise to various possible outcomes for the disease's spread.

When we analyze these equations, we look for patterns, seek equilibrium points, and understand the long-term behavior of the system. This kind of analysis helps us predict the course of an outbreak and potentially how to control it.

Equilibrium Solutions Analysis

The analysis of equilibrium solutions is a critical aspect of understanding mathematical models like the SIR model. Equilibrium solutions are points where the system does not change, meaning the derivative of each variable with respect to time is zero. For the SIR model, we seek values of S, I, and R such that \(\frac{dS}{dt} = \frac{dI}{dt} = \frac{dR}{dt} = 0\).

Once we find these points, classifying them helps us comprehend the stability and the nature of the disease's progression. Do small deviations away from this point cause the system to return to equilibrium, or do they drive the system further away? In the SIR model, one equilibrium could mean that the disease has been eradicated, while another could indicate an endemic state where the disease continues to exist in the population.

Analyzing the equilibrium solutions can lead to insights into possible outcomes of an epidemic and the effectiveness of interventions. For instance, knowing the conditions for disease eradication can guide public health policies towards achieving that state.

Numerical Methods for Differential Equations

Numerical methods for differential equations are computational approaches used to find approximate solutions to differential equations that do not have explicit solutions. For the SIR model, when we input specific values for \( \beta \) and \( \gamma \), and initial conditions for S(t) and I(t), we must resort to numerical solvers to understand how the disease will spread over time.

Numerical methods, such as Euler's method, Runge-Kutta methods, and others, enable us to approximate the paths of S(t) and I(t). They work by dissecting time into small increments and predicting the behavior of the system step-by-step within these increments. The solutions they provide give us graphs that depict the number of susceptibles and infected over time.

These methods are particularly invaluable in epidemiology, as they allow for the simulation of complex biological phenomena that can inform real-world decisions. Accurate numerical solutions can provide quantitative predictions of disease spread, peak infection rates, and the potential duration of an epidemic, all of which are essential for planning and response strategies in public health.