Problem 10 - Calculus for the Life Sciences: A Modeling Approach Volume II

Question

a. The units on \(S^{\prime}, I^{\prime}\) and \(R^{\prime}\) in the SIR model are people/day. In order for the units to balance on the equations, what must be the units on \(\beta\) and \(\gamma ?\) b. Suppose the infection typically lasts seven days. What is an appropriate value of \(\gamma ?\) Note: The answer we would expect you to give for \(\gamma\) is \(1 / 7 \doteq 0.143 .\) We will find that a better answer is \(-\ln (1-1 / 7) \doteq 0.154\)

Step-by-Step Solution

Answer

a. β and γ must have units of 1/day. b. γ is approximately 0.143 or slightly better, 0.154.

Step 1: Identifying Units in the SIR Model

The SIR model consists of the equations for the susceptible ( S' ), infectious ( I' ), and recovered ( R' ) populations. The units of S' , I' , and R' are given in people per day. The equations also include parameters β (beta) and γ (gamma). To determine the units of β and γ , we must ensure dimensional consistency within these differential equations.

Step 2: Balancing Units for β and γ

In the equation I' = βIS - γI , both terms on the right side should have the same units as the left side, which is people/day. For βIS to have units of people/day, if I and S are in people, then β must have units of 1/day. Similarly, for γI to have units of people/day, where I has units of people, γ must also have units of 1/day.

Step 3: Determine Value of γ based on Infection Duration

Given that an infection typically lasts seven days, we use this to determine γ . The simplest assumption is that γ is the reciprocal of the duration: γ = 1/7 or approximately 0.143. This is because γ represents the rate of recovery or removal per day from the infectious class.

Step 4: Revisiting γ using Better Approximation

To refine our estimate of γ , we consider that γ represents the daily rate at which individuals recover. Using a continuous decay model, we use the formula - ln(1-1/7) to get a more precise estimate. By calculation, γ becomes approximately 0.154.

Key Concepts

Dimensional Analysis

Dimensional analysis is a fundamental technique used to ensure that mathematical equations, especially those involving physical quantities, are dimensionally consistent. In the context of the SIR model, dimensional analysis involves ensuring that all terms in the differential equations describing susceptible, infectious, and recovered populations have consistent units.

For example, the change in the infected population over time is represented by the equation:

• \(I' = \beta IS - \gamma I\)

To have dimensional consistency, the units on each term must be the same as the units for \(I'\), which are people per day. This requires careful analysis of the units of \(\beta\) and \(\gamma\).

In this case, if \(I\) and \(S\) are both measured in people, then for \(\beta IS\) and \(\gamma I\) to also be in people per day, both \(\beta\) and \(\gamma\) must have units of 1/day. This analysis is crucial as it helps to ensure that the mathematical model is correctly scaled and interpretable.

Differential Equations

Differential equations form the core of the SIR model, as they describe how populations of susceptible, infectious, and recovered individuals change over time. In the SIR model, these populations are expressed through three differential equations:

• \(S' = -\beta IS\)
• \(I' = \beta IS - \gamma I\)
• \(R' = \gamma I\)

These equations are interpreted as follows:- \(S'\) represents the rate at which susceptible people become infected.- \(I'\) models the rate of change in the infectious population, accounting for both new infections and recoveries.- \(R'\) denotes the rate at which people recover from the infection.

Differential equations allow us to predict how the disease spreads and eventually declines, by understanding how these three groups interact with parameters like \(\beta\) and \(\gamma\). They are fundamental in epidemiological modeling since they help form predictions and strategies for controlling the spread of diseases.

Epidemiological Modeling

Epidemiological modeling is the scientific process of using mathematical models to study the spread of infectious diseases within populations. The SIR model is a classic example of such a model, which divides the population into compartments: susceptible (S), infectious (I), and recovered (R).

The model employs parameters like \(\beta\), the transmission rate, and \(\gamma\), the recovery rate. These parameters are used to simulate real-world scenarios, providing insights into how a disease might spread and impact a community.

By adjusting \(\beta\) and \(\gamma\), public health officials can model different control measures, predict outcomes, and perhaps contain outbreaks. It's a vital tool for understanding potential healthcare needs and planning interventions to minimize disease impact.

Infection Duration

Understanding the duration of infection is critical in accurately modeling disease spread and management. In the SIR model, the duration for which individuals remain infectious is inverse to the recovery rate, \(\gamma\).

Considering an infection typically lasts seven days, a basic calculation suggests \(\gamma = 1/7\) or approximately 0.143. This implies, on average, 1/7th of the infected individuals recover each day.

A more precise approach uses continuous decay to model recovery, leading to a slightly different estimate. This method calculates \(\gamma\) as:

• \(-\ln(1-1/7)\)

Using this calculation provides a refined \(\gamma\) value of about 0.154. Such precision in calculating \(\gamma\) enhances the model's accuracy and reliability, making it a better tool for anticipating the course of an outbreak.