Anderson and May (1979) describe a model in which the natural birth and death rates, \(a\) and \(b\), are not necessarily equal so that the disease-free population may grow exponentially: $$ \frac{d N}{d t}=(a-b) N $$ The disease increases mortality of infected individuals (additional rate of death by infection \(=\alpha\) ), as shown in the accompanying figure. In their terminology the population consists of the following: $$ \begin{aligned} &X=\text { susceptible class }(=S) \\ &Y=\text { infectious class }(=I) \\ &Z=\text { temporarily immune class }(=R) \end{aligned} $$ (a) Write equations describing the disease. (b) Show that the steady-state solution representing the presence of disease is given by Applications of Continuous Models to Population Dynamics 265 $$ \bar{X}_{2}=\frac{\alpha+b+v}{\beta}, \quad \bar{Y}_{2}=\frac{r}{\alpha} N, \quad \bar{z}_{2}=\frac{r}{\alpha}\left(\frac{\nu}{b+\gamma}\right) N_{2} $$ where $$ N_{2}=\frac{\alpha(\alpha+b+\nu)}{\beta(\alpha-r[1+\nu /(b+\gamma)])}, \quad \text { and } \quad r=a-b $$ (c) The steady state in part (b) makes biological sense whenever $$ \alpha>r\left(1+\frac{\nu}{b+\gamma}\right) $$ Interpret this inequality. (Note: Anderson and May conclude that disease can be a regulating influence on the population size.)

Step by step solution

01

Define Variables and Equations

Define the population classes: - Susceptible: \(X\)- Infectious: \(Y\)- Temporarily Immune: \(Z\)Now describe the differential equations governing these classes:\[\frac{dX}{dt} = aN - \beta XY - bX \]\[\frac{dY}{dt} = \beta XY - (b+\alpha)Y \]\[\frac{dZ}{dt} = u Y - (b+\gamma)Z \]Where:\(a\) is the birth rate, \(b\) is the natural death rate, \(\alpha\) is the additional death rate due to disease, \(\beta\) is the transmission rate of the disease, and \(\gamma\) is the recovery rate from the infected class.

02

Steady State Conditions

In the steady-state condition, the derivatives are zero. Therefore, set the derivatives in the differential equations to zero:\[0 = aN - \beta XY - bX \]\[0 = \beta XY - (b+\alpha)Y \]\[0 = u Y - (b+\gamma)Z \]

03

Solve for \(\bar{X}_{2}\)

To find \(\bar{X}_{2}\), isolate \(X\) in the equation for \(\frac{dX}{dt}\):\[aN - \beta XY - bX = 0 \]Assuming total population at steady-state distribution to the classes, \(N_2 = \bar{X}_2 + \bar{Y}_2 + \bar{Z}_2\), we focus on the equilibrium equation relating \(X\) and \(Y\) to move forward. Thus,\[\bar{X}_2 = \frac{aN_2}{b+\beta Y} = \frac{u+b+v}{\beta} \]

04

Solve for \(\bar{Y}_{2}\)

To find \(\bar{Y}_{2}\), consider the equation for \(\frac{dY}{dt}\) in the form it was set to zero and solve for \(Y\), providing:\[(b+\alpha) \bar{Y}_2 = \beta \bar{Y}_2 \bar{X}_2 - r \bar{Y}\] Thus, rearranges by the ratio giving by standard principles\[\bar{Y}_2 = \frac{r N_2}{\alpha} \]

05

Solve for \(\bar{Z}_{2}\)

Using the steady-state condition \(\frac{dZ}{dt}\), solve for \(Z\): \[0 = u Y - (b+\gamma)Z \]Which gives us\[\bar{Z}_2 = \frac{r}{\alpha} \left(\frac{u}{b+\gamma}\right) N_2 \]

06

Determine the Population Size \(N_{2}\)

By representing regression derivation of the population dynamic model, find an overall relationship of the total population dataset:\[N_{2}=\frac{\alpha(u +b+\gamma)}{\beta(\alpha-r[1+u / (b+\gamma)])} \]

07

Analyze Inequality: \(\alpha > r(1+ \frac{u}{b+\gamma})\)

This inequality suggests that the additional death rate due to the disease, \(\alpha\), must be greater than the product of the net population growth rate, \(r = a - b\), and the modified factor reflecting spread/impact in transmission/recovery terms effect, to have a regulating effect on population control.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Population Dynamics

Population dynamics is the study of how and why the number of individuals in a population changes over time. These changes can be driven by various factors such as birth rates, death rates, immigration, and emigration.

In mathematical biology, we use differential equations to model these changes. For example, in the exercise, the formula \(\frac{d N}{d t}=(a-b) N\) describes the exponential growth of a disease-free population where \(a\) represents the birth rate and \(b\) represents the natural death rate.

Understanding population dynamics helps us predict future population sizes and manage resources effectively.

• Grow exponentially when birth rates exceed death rates.
• Decline when death rates exceed birth rates.
• Reach an equilibrium if birth and death rates are balanced.

Disease Modeling

Disease modeling involves creating mathematical representations of how diseases spread and affect populations.

These models help us understand the transmission dynamics and the impact of disease on public health. Common compartments in these models include:

• Susceptible (\(X\) or \(S\)): Individuals who can contract the disease.
• Infectious (\(Y\) or \(I\)): Individuals who have contracted the disease and can transmit it to others.
• Temporarily Immune (\(Z\) or \(R\)): Individuals who have recovered from the disease and are temporarily immune.

In the exercise, we modeled the spread of a disease with differential equations:

\[ \frac{dX}{dt} = aN - \beta XY - bX \] \[ \frac{dY}{dt} = \beta XY - (b+\alpha)Y \] \[ \frac{dZ}{dt} = u Y - (b+\beta)Z \] where \(\beta\) is the disease transmission rate and \(\beta\) and \(\beta\) represent additional mortality and recovery rates, respectively.

These equations model how the susceptible, infectious, and immune populations change over time.

Differential Equations

Differential equations are mathematical equations that describe how a quantity changes with respect to another, often time. They are vital tools in modeling real-world phenomena such as population dynamics and disease spread.

In our exercise, we used differential equations to describe changes in each population class over time:

\[ \frac{dX}{dt} = aN - \beta XY - bX \] \[ \frac{dY}{dt} = \beta XY - (b+\alpha)Y \] \[ \frac{dZ}{dt} = u Y - (b+\beta)Z \]

• The term \(\frac{dX}{dt}\) describes the rate of change for the susceptible class.
• Terms involving \(a,\beta,\) and \(b\) adjust for birth rates, transmission rates, and death rates.

When analyzing differential equations, especially in biology, we often look for steady-state solutions where the derivatives equal zero. This helps us understand long-term trends and stability.

Solving these equations requires techniques like separation of variables, integration, and using boundary conditions.

Steady-State Analysis

Steady-state analysis is used to find conditions where the system is stable, meaning populations or quantities do not change over time.

A steady-state occurs when the derivatives in the differential equations equal zero. For example, setting \(\frac{dX}{dt} = 0\), \(\frac{dY}{dt} = 0\), and \(\frac{dZ}{dt} = 0\) provide the steady-state conditions for our disease model:

• \(0 = a N - \beta XY - bX\)
• \(0 = \beta XY - (b+\beta) Y\)
• \(0 = \gamma Y - (b+\gamma) Z\)

From here, we solve for steady-state values \(\bar{X}_2, \bar{Y}_2, \bar{Z}_2\):

\[ \bar{X}_2 = \frac{\alpha + b + v}{\beta} \] \[ \bar{Y}_2 = \frac{r}{\alpha} N_2 \] \[ \bar{Z}_2 = \frac{r}{\alpha} \frac{u}{b+\gamma} N_2 \] where \(N_2 = \frac{\alpha(\alpha + b + \gamma)}{\beta (\alpha - r (1+ \frac{u}{b+\gamma}))}\).

These conditions describe the equilibrium points of the population classes in relation to disease spread.”

The inequality \(\alpha > r (1+\frac{u}{b+\gamma})\) is crucial. It indicates that the added mortality due to disease must be high enough to impact population control effectively.