Problem 45 - Calculus for Biology and Medicine

Question

In some diseases (such as herpes simplex), an individual may apparently recover from the disease, and in fact gain immunity to it, but the disease continues to be harbored in the person’s body, breaking out some time after they recover from the initial infection. To model this process we will modify our SIRS model as follows: Since there is no loss of immunity, a = 0. However, in each unit of time a fraction r (r is a constant called the rate of relapse) of the individuals from the recovered class become infected with the disease. You will analyze models for relapsing infections. Assume that the following model can be used to represent the spread of a relapsing infection in a population of size \(N=100\) and with relapse rate \(r=\frac{1}{100}:\) $$ \begin{array}{l} \frac{d S}{d t}=-\frac{1}{50} S I \\ \frac{d I}{d t}=\frac{1}{50} S I-\frac{1}{10} I+\frac{1}{100} R \\ \frac{d R}{d t}=\frac{1}{10} I-\frac{1}{100} R \end{array} $$ (a) What is the domain for this differential equation system? (b) Find all of the possible equilibria for this system of differential equations. (c) Use the fact that \(S+I+R=100\) to eliminate \(S\) from the system, and to write it as a pair of differential equations with \(I\) and \(R\) as dependent variables. (d) By linearizing the differential equation system near each of the equilibria that you discovered in part (b), classify these equilibria (e.g., as a stable node, spiral, or saddle).

Step-by-Step Solution

Answer

Domain: \(S+I+R=100\), \(S, I, R \geq 0\). Equilibria: \((0, 0, 100)\). Use \(I\) and \(R\) equations to classify stability.

Step 1: Understanding the Differential Equation Domain

In the problem statement, it is mentioned that the population size is fixed, meaning \(S + I + R = 100\). Since these are proportions of the population, we have \(S, I, R \geq 0\) and \(S + I + R = 100\). The domain consists of all possible values for \(S, I, R\) where these conditions hold true.

Step 2: Finding Equilibria Points

To find equilibria, solve \(\frac{dS}{dt} = 0\), \(\frac{dI}{dt} = 0\), and \(\frac{dR}{dt} = 0\).1. From \(\frac{dS}{dt} = 0\), \(-\frac{1}{50}SI = 0\). Thus, \(S = 0\) or \(I = 0\).2. Given \(\frac{dI}{dt} = 0\), substitute \(S = 0\) or \(I = 0\) into \(\frac{1}{50}SI - \frac{1}{10}I + \frac{1}{100}R = 0\) to find possible equilibria.3. Similarly, use \(\frac{dR}{dt} = 0\), which translates to \(\frac{1}{10}I - \frac{1}{100}R = 0\).4. Solve the set of equations to find all equilibria: \((S, I, R) = (0, 0, 100)\).

Step 3: Eliminating S Using Population Constraint

Since \(S + I + R = 100\), substitute \(S = 100 - I - R\) into the differential equations:1. \(\frac{dI}{dt} = \frac{1}{50}(100 - I - R)I - \frac{1}{10}I + \frac{1}{100}R\).2. \(\frac{dR}{dt} = \frac{1}{10}I - \frac{1}{100}R\).Express the system only in terms of \(I\) and \(R\).

Step 4: Linearizing Near Equilibrium Points

For the equilibrium \((S, I, R) = (0, 0, 100)\), compute the Jacobian of the system at this point:1. Partial derivatives of \(\frac{dI}{dt}\) and \(\frac{dR}{dt}\) w.r.t. \(I\) and \(R\).2. Evaluate the Jacobian at the equilibrium to determine stability. Compute eigenvalues from the Jacobian: If all eigenvalues have negative real parts, the equilibrium is stable; otherwise, it may be unstable or a saddle point.

Key Concepts

Differential Equations

Differential equations are mathematical tools used to describe how something changes over time. In the context of the SIRS model for relapsing infections, they help determine how the number of susceptible (S), infected (I), and recovered (R) individuals in a population evolve. The SIRS model is particularly useful for understanding diseases where immunity is temporary, allowing for individuals to relapse.
The differential equations allow us to map how quickly people move from one group to another over time, influenced by various factors: infection rate, recovery rate, and relapse rate. Here are the equations in this model:

• For susceptible individuals: \[\frac{dS}{dt} = -\frac{1}{50} SI\]This equation shows that the number of susceptible individuals decreases as they become infected.
• For infected individuals: \[\frac{dI}{dt} = \frac{1}{50} SI - \frac{1}{10} I + \frac{1}{100} R\]The equation indicates that infections rise due to susceptible individuals catching the disease, but decrease as they recover or relapse.
• For recovered individuals: \[\frac{dR}{dt} = \frac{1}{10} I - \frac{1}{100} R\]This describes how recovered numbers adjust as people get better or relapse.

These equations provide a pathway to model disease progressions accurately and see the effects of different rates in the population's health state dynamics.

Relapsing Infections

Relapsing infections are characterized by a disease reappearing after a recovery period. This phenomenon can occur even when an individual has developed temporary immunity. In the context of the SIRS model, the 'R' stands for 'Recovered', which ideally should denote immunity, albeit not permanent in relapsing situations.
Understanding relapsing infections requires considering that the recovered class isn't entirely safe from future infections. This is modeled by the relapse rate \(r\) within our differential equations:

• The relapse rate \(r\) is a proportion of recovered individuals that transition back into the infected class over time.
• In the example provided, with a population size of 100, a relapse rate \(r=\frac{1}{100}\) signifies that every time unit, some recovered individuals will once again become infected.

Relapsing infections challenge the notion of disease elimination, emphasizing continuous monitoring and treatment, as immunity isn't a guarantee of long-term protection.

Stability Analysis

Stability analysis is essential for understanding the long-term behavior of differential equations systems, like the SIRS model. It helps identify whether an equilibrium point is stable—meaning the system will return to this state after a disturbance—or unstable.
To perform a stability analysis, mathematicians use techniques like linearization. This involves approximating the nonlinear differential equations by a linear system at equilibrium points:

• Finding Equilibria Points: Solve for conditions where all differentials are zero, meaning the state won't change without external influence.
• Linearizing Around Equilibrium: Simplify the system near these points by computing the Jacobian matrix, which helps analyze the system locally.
• Eigenvalue Evaluation: Analyze the Jacobian's eigenvalues to determine stability. If all have negative real parts, the equilibrium is stable.

In this SIRS model, stability helps predict whether an outbreak will die out or persist, aiding in public health decision-making by illustrating the potential need for interventions or modifications to existing strategies.