Problem 37 - Calculus for Biology and Medicine

Question

Some diseases are lethal; not every individual infected by the disease will recover; some will die. Assume that in one unit of time a fraction \(m\) of infected individuals will die \((m\) is called the mortality rate \() .\) (a) Explain how the SIRS model equations should be modified to incorporate deaths. In particular you should write down a new differential equation for \(\frac{d I}{d t}\). (b) Explain why it is no longer possible to eliminate \(R(t)\) from the SIRS model equations.

Step-by-Step Solution

Answer

Incorporate death by adding \(-mI\) to the \(\frac{dI}{dt}\) equation. Elimination of \(R(t)\) is not possible due to loss of individuals to death.

Step 1: Understanding SIRS Model

The SIRS model is a compartmental model in epidemiology. It consists of three compartments: Susceptible \((S)\), Infected \((I)\), and Recovered \((R)\). The typical equations without considering mortality are: \[ \frac{dS}{dt} = -\beta SI + \gamma R \] \[ \frac{dI}{dt} = \beta SI - u I \] \[ \frac{dR}{dt} = u I - \gamma R \] where \(\beta\) is the transmission rate, \(u\) is the recovery rate, and \(\gamma\) is the rate at which immunity is lost.

Step 2: Incorporating Mortality Rate

To incorporate deaths into the SIRS model, we need to adjust the equations to account for individuals that die due to the disease. This affects the \(I\) compartment. The new equation for \(\frac{dI}{dt}\) considering the mortality rate \(m\) is: \[ \frac{dI}{dt} = \beta SI - u I - m I \] where \(m I\) represents the fraction of infected individuals dying each unit of time.

Step 3: Impact on the Recovered Compartment

Since individuals who die do not recover, the presence of mortality means individuals are leaving the \(I\) compartment without entering the \(R\) compartment. This affects the relationship between \(I\) and \(R\), making it complicated to eliminate \(R(t)\) from the equations, as the usual conservation of individuals does not hold due to the losses from death.

Key Concepts

Mortality Rate

In the context of diseases, the mortality rate is a crucial concept. It measures the fraction of individuals who die from a disease over a specific period. This rate helps understand how deadly a disease is, which is essential for assessing its impact on a population.
The mortality rate is especially relevant when modeling the spread of infectious diseases using systems like the SIRS model. In this model, the mortality rate is symbolized by the variable \(m\), representing the fraction of infected individuals who die in each unit of time.
When we incorporate the mortality rate into the infected compartment (\(I\)) of the model, it influences the dynamics by reducing the number of people recovering or returning to the susceptible state. Thus, the adjusted differential equation for \(\frac{dI}{dt}\) reflects this additional exit from the infectious stage due to death:

• \( \frac{dI}{dt} = \beta SI - u I - m I \)

This equation highlights how mortality directly reduces the number of infected individuals over time.

Differential Equations

Differential equations play a fundamental role in modeling how quantities change over time. In epidemiology, these equations help describe how diseases spread, recover, or become fatal, as seen in the SIRS model.
In the SIRS model, each compartment—Susceptible \((S)\), Infected \((I)\), and Recovered \((R)\)—has its own differential equation. These equations are interconnected, reflecting the transitions between compartments. For example:

• \( \frac{dS}{dt} = -\beta SI + \gamma R \)
• \( \frac{dI}{dt} = \beta SI - u I - m I \)
• \( \frac{dR}{dt} = u I - \gamma R \)

Here, \(\beta\) is the transmission rate, \(u\) is the recovery rate, \(m\) is the mortality rate, and \(\gamma\) is the rate of losing immunity. Each term in the equations represents a flow of individuals between compartments or out of the system (in the case of fatalities).
Understanding these equations is vital to predict the impacts of different factors on disease progression and to develop effective strategies to control disease outbreaks.

Epidemiology

Epidemiology is the science that studies the spread, control, and effects of diseases in populations. It uses models like the SIRS model to understand how diseases impact different groups.
In an SIRS model:

• \(S\): The number of susceptible individuals who can become infected.
• \(I\): The number of infected individuals who can spread the disease further.
• \(R\): The number of recovered individuals who may gain temporary immunity.

The addition of a mortality rate impacts these dynamics, emphasizing the need to track fatal outcomes. With such data, epidemiologists can develop interventions such as improved healthcare strategies or vaccination programs that aim to reduce both infection and mortality rates.
By utilizing epidemiological models, scientists and public health officials can make informed decisions, ideally leading to better health outcomes and managing disease outbreaks effectively.