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Chapter 6: Problem 27

Show that in an SIR model with disease fatality at rate \(\eta\) the disease will always eventually disappear.

Short Answer

Expert verified
Given \( \beta S < \gamma + \eta \), I will decrease to zero, leading to the disease's eventual disappearance.

Step by step solution

01

Define the SIR model with fatality

The SIR model is extended to account for disease fatality by including a term for rate \(\backslash eta \). The differential equations describing the model are: \begin{aligned}\frac{dS}{dt} &= -\beta I S \ \frac{dI}{dt} &= \beta I S - \gamma I - \eta I \ \frac{dR}{dt} &= \gamma I\end{aligned} where \(\beta\) is the infection rate, \(\backslash gamma\) is the recovery rate, and \(\backslash eta\) is the fatality rate.
02

Analyze the infected population

To determine if the disease will disappear, consider \(\frac{dI}{dt} = \beta I S - \gamma I - \eta I\). Factor out \(I\) from the right-hand side: \(\begin{aligned} \frac{dI}{dt} &= I (\beta S - \gamma - \eta) \end{aligned}\)
03

Determine the threshold

The term \(I (\beta S - \gamma - \eta)\) shows that the change in infected individuals depends on \(\beta S - \gamma - \eta\). If this term is negative, \(\frac{dI}{dt} < 0\), the number of infected individuals will decrease. The threshold condition is: \[ \beta S - \gamma - \eta < 0 \] Rewriting, we get: \[ \beta S < \gamma + \eta \]
04

Consider the long-term behavior

As time progresses, susceptible individuals ( S ) decrease due to infection, recovery, or fatality. Eventually, S will drop below \[ \frac{\gamma + \eta}{\beta} \]. When this happens, the inequality \(\beta S - \gamma - \eta < 0 \) holds, resulting in \(\frac{dI}{dt} < 0 \) and the number of infected individuals diminishing over time.
05

Conclusion

Since I decreases continuously when \(\beta S - \gamma - \eta < 0 \), and S will always drop below \[ \frac{\gamma + \eta}{\beta} \] eventually, I will approach zero. Thus, the disease will always eventually disappear.

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Key Concepts

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

disease modeling
Disease modeling is a method used to understand how diseases spread and to predict their outcomes. One common approach is the SIR model, which classifies the population into three groups: Susceptible (S), Infected (I), and Recovered (R). This model can be extended to include disease fatality by adding a term for the fatality rate, denoted as \(\backslash eta \). By incorporating this fatality rate, we can analyze how deaths impact the spread and eventual disappearance of the disease.
differential equations
Differential equations play a crucial role in disease modeling, including the SIR model with disease fatality. In this context, the following set of differential equations describe the system:

\(\begin{aligned}\frac{dS}{dt} &= -\beta I S \ \frac{dI}{dt} &= \beta I S - \gamma I - \eta I \ \frac{dR}{dt} &= \gamma I \end{aligned}\).

These equations help us understand how the populations of Susceptible (S), Infected (I), and Recovered (R) change over time. The parameters \(\beta\) (infection rate), \(\backslash gamma\) (recovery rate), and \(\backslash eta\) (fatality rate) determine the behavior of the model.
epidemiology
Epidemiology is the study of how diseases spread within populations. By using models like the SIR model, epidemiologists can predict the course of an outbreak and evaluate intervention strategies. For instance, adding a fatality rate term to the SIR model helps understand the impact of fatal diseases. It shows how factors like the infection rate (\(\beta\)), recovery rate (\(\backslash gamma\)), and fatality rate (\(\backslash eta\)) influence whether a disease will eventually disappear.
infection dynamics
Infection dynamics refer to how the number of infections changes over time. In our modified SIR model, this is described by the differential equation for \(\frac{dI}{dt}\):
\(\frac{dI}{dt} = \beta I S - \gamma I - \eta I\).

The term \(I (\beta S - \gamma - \eta)\) suggests that the change in the number of infected individuals depends on the value of \(\beta S - \gamma - \eta\). If this expression is negative, the number of infections will decrease over time, leading to the eventual disappearance of the disease.
disease transmission thresholds
A key concept in understanding whether a disease will die out is the disease transmission threshold. This threshold is determined by the inequality \(\beta S < \gamma + \eta\). When the number of susceptible individuals, \(S\), falls below the critical value of \(\frac{\beta}{\backslash eta + \gamma}\), the number of infected individuals, \(I\), starts to decrease. Over time, as \(S\) continues to decrease, the rate of new infections cannot sustain the epidemic, causing \(I\) to approach zero. This demonstrates that under these conditions, the disease will eventually disappear.

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Most popular questions from this chapter

Populations of lemmings, voles, and other small rodents are known to fluctuate from year to year. Early Scandinavians believed the lemmings to fall down from heaven during stormy weather. Later in history, the legend developed that they migrate periodically into the sea for suicide in order to reduce their numbers.... None of these theories, however, was supported by any accurate observations. (H. Dekker, 1975) An alternate hypothesis was suggested by Dekker to account for rodent population cycles. His theory is based on the idea that the rodents fall into two genotypic classes (Myers and Krebs, 1971) that interact. Type 1 reproduces rapidly, but migrates in response to overcrowding; type 2 is less sensitive to high densities but has a lower reproductive capacity. The following simple mathematical model was given by Dekker to demonstrate that oscillations could be produced when types 1 and 2 rodents were both present in the population: $$ \begin{aligned} &\frac{d n_{1}}{d t}=n_{1}\left[a_{1}-\left(b_{1}-c_{1}\right) n_{2}-c_{1}\left(n_{1}+n_{2}\right)\right] \\ &\frac{d n_{2}}{d t}=n_{2}\left[-a_{2}+b_{2} n_{1}\right] \end{aligned} $$ where $$ \begin{aligned} &n_{1}=\text { density per acre of type } 1 \\ &n_{2}=\text { density per acre of type } 2 . \end{aligned} $$ The term \(b_{1}-c_{1}\) was chosen for convenience in the mathematical calculations rather than for particular biological reasons. (a) On the basis of the information, give an interpretation of the individual terms in the equations. (b) Using phase-plane methods, determine the qualitative behavior of solutions to Dekker's equations. If there is more than one case, pay particular attention to the case in which oscillations are present. Give conditions on the parameters \(a_{j}, b_{j}\), and \(c\), for which oscillatory behavior will be seen. (c) Give a short critique of Dekker's model, indicating whether you would change his assumptions and/or equations. Dekker's article has received a somewhat critical peer review by Nichols et. al. (1979). You may wish to comment on their specific points of contention.

For single-species populations, which of the following density-dependent growth rates would lead to a decelerating rate of growth as the population increases? Which would result in a stable population size? (a) \(g(N)=\frac{\beta}{1+N}, \beta>0\). (c) \(g(N)=N-e^{\mathrm{aN}}, \alpha>0\). (b) \(g(N)=\beta-N, \beta>0\). (d) \(g(N)=\log N\).

The factor \(g(N)=r(1-N / K)\) in equation (2a) is a per capita growth rate. Smith (1963) observed that in cultures of the unicellular alga Daphnia magna \(g\) decreases at a nonlinear rate as \(N\) increases. To account for this fact, Smith suggested that the growth rate depends on the rate at which food is utilized: $$ g(N)=r \frac{T-F}{T} $$ where \(F\) is the rate of utilization when the population size is \(N\), and \(T\) is the maximal rate, when the population has reached a saturated level. He further assumed that $$ F=c_{1} N+c_{2} \frac{d N}{d t}, \quad\left(c_{1}, c_{2}>0\right) $$ as long as \(d N / d t>0\). (a) Explain this assumption for \(F\). (b) Show that the modified logistic equation is then $$ \frac{d N}{d t}=r N\left[\frac{K-N}{K+(\gamma N)}\right] $$ where \(\gamma=r c_{2} / c_{1}\) and \(K=T / c_{1}\). (c) Sketch the expression in square brackets as a function of \(N\). (d) What would be the qualitative behavior of this population growth? (For a deeper analysis of this problem see Pielou, 1977.)

(a) Suppose a one-time fishing expedition reduced the prey population by \(10 \%\) of its current level. What does the Lotka-Volterra model predict about the subsequent behavior of the system? (Note: this prediction is one of the most objectionable features of the model and will be dealt with in a. later chapter.) (b) Now consider the situation in which there is a constant level of fishing in which both prey and predatory fish are caught and removed at rates proportional to their densities, \(\phi x\) and \(\phi y\). Compare this to the situation in the absence of fishing, and show what Volterra concluded about d'Ancona's observation. (For one treatment of this problem see Braun, 1979 ; a more advanced mathematical treatment can be found in Brauer and Soudack, 1979.)

Models that are commonly used in fisheries are $$ \frac{d N}{d t}=N g(N) \text {, } $$ where \(g(N)\) is given by $$ \begin{array}{ll} \text { Ricker model: } & g(N)=r e^{-\beta N} . \\ \text { Beverton-Holt: } & g(N)=\frac{r}{\alpha+N} . \end{array} $$ Analyze the behavior of the solutions to these questions. (Assume \(\alpha, \beta\), \(r>0\) ).
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