Problem 41 - 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). We will assume that the habitat this population lives in is at its carrying capacity. If no individuals die then no reproduction occurs. If individuals die, then resources are freed up and more individuals will be born: one birth for every death that occurs. That is, the number of individuals born in one unit of time is equal to the number of individuals who die in that unit of time. In Problems 38–41 you will analyze models for lethal diseases. In Problems 38–40 you should assume that infants are initially uninfected by the disease but are also not immune to it, so new individuals added to the population are all in the susceptible class. Assume that all individuals in the population are equally likely to be parents to the \(m I\) offspring added to the population in each unit of time. If an offspring is born to an infected parent it will be born infected (i.e., into the infectious class). Similarly, offspring born to susceptible parents are susceptible, and offspring born to recovered parents are recovered. Derive an SIRS model to describe the spread of this disease. There is no need to analyze your model.

Step-by-Step Solution

Answer

The model equations are \( \frac{dS}{dt} = -\beta SI + mI(1 - m) \), \( \frac{dI}{dt} = \beta SI - \gamma I + mIm \), \( \frac{dR}{dt} = \gamma I - mIR \).

Step 1: Understand the Problem

We need to develop a model that describes how a lethal disease spreads in a population. The disease can cause death, after which resources for births are freed. The born individuals can become part of the Susceptible (S), Infected (I), or Recovered (R) classes based on their parents' status.

Step 2: Model the Population Dynamics

Understand that the population's changes relate only to births and deaths. Every death of an infected individual opens up resources for a birth. Birth rates, therefore, equal death rates, so the birth rate per unit time is given by the mortality rate \( m \) times the infected population \( I \).

Step 3: Analyze Transmission Dynamics

Assume all new individuals are susceptible unless they are born to infected parents, who make their offspring infected. The offspring from susceptible parents are susceptible, and similarly, offspring from recovered parents are born recovered.

Step 4: Formulate the SIRS Model Equations

Start by defining the changes in each class. \[\frac{dS}{dt} = -\beta SI + mI(1 - m) \] for susceptible individuals, where \( \beta \) is the transmission rate. \[\frac{dI}{dt} = \beta SI - \gamma I + mIm \] for infected individuals, where \( \gamma \) is the rate of recovery. \[\frac{dR}{dt} = \gamma I - \, mIR \] for recovered individuals, back to susceptible due to loss of immunity. Death rate equals birth rate; thus, death doesn’t change total population size.

Step 5: Justify the Components

The equations reflect birth equal to death; thus, a constant total population. \( \frac{dS}{dt} \) represents decrease due to infection and increase from births of offspring from susceptible and recovered parents not dying. \( \frac{dI}{dt} \) levels from infections and infected births minus recovery and deaths. \( \frac{dR}{dt} \), from recoveries, has no birth terms as assumed recovery status does not revert.

Key Concepts

Mortality Rate in SIRS Model

In the context of the SIRS model for a lethal disease, the mortality rate plays a crucial role. This rate, denoted by \( m \), defines the fraction of infected individuals who succumb to the disease in a given time unit. Understanding the mortality rate is essential as it directly influences the population's ability to recover and maintain equilibrium.
The mortality rate can be seen as a balancing factor in population dynamics. Here's how it works:

• When an infected individual dies, resources become available, allowing for a new birth; this keeps the population size constant.
• Every death is offset by a birth, which is crucial because it prevents unchecked population growth or decline.
• This balance ensures the model focuses solely on the disease's transmission and progression rather than changes in population size.

Mortality rate \( m \) is fundamental in equations, such as \( mI \), which indicates deaths among the infected. Overall, \( m \) is about understanding resource reallocation due to lethal outcomes.

Population Dynamics and Resource Allocation

Population dynamics in the SIRS model are tied closely to the habitat's carrying capacity. In this model, the total number of individuals remains constant, meaning the population does not grow or shrink over time. Here's how the model ensures this balance.

• No net change in population: The birth of new individuals matches exactly the number of deaths, dictated by the mortality rate \( m \).
• Resource availability: When individuals die, resources are freed, enabling the birth of new individuals. This allows the population to maintain its size despite the death and birth cycle.

The model posits that every individual in the population is equally likely to contribute to the birth of \( m I \) offspring. This means that:

• Infected parents will pass the infection to their newborns.
• Susceptible and recovered parents will birth offspring who are in the susceptible class.

Understanding these dynamics is essential for forming the SIRS equations that reflect birth and death balance, ensuring the constant total population size in the model.

Transmission Dynamics within the SIRS Model

Transmission dynamics are core to understanding the spread of diseases modeled by the SIRS system. These dynamics focus on how the disease is passed among individuals and how it influences the movement between Susceptible (S), Infected (I), and Recovered (R) compartments.
In this model, transmission is primarily driven by a few key points:

• The transmission rate, \( \beta \), determines the likelihood of the disease passing from one individual to another.
• Newly born individuals' status depends solely on their parents' condition, dictating whether they enter the susceptible, infected, or recovered class.
• An individual born to infected parents will automatically be infected, reflecting the transmission of the disease during birth.
• Individuals in the susceptible category may become infected based on the interaction with currently infected individuals.

The SIRS model captures these dynamics in its equations:

• The equation for \( \frac{dS}{dt} \) accounts for decreases in susceptible individuals due to infection contact and increases due to births.
• Similarly, \( \frac{dI}{dt} \) captures infection increase through transmission and decreases through recovery or death.

This framework captures how the transmission of the disease affects the population's structure and the flow between different health statuses.