Question

The Kermack-McKendrick model was developed to explain the rapid rise and fall in the number of infected people during epidemics. This model involves the interaction of susceptible (S), infected (I), and recovered (R) people through the following mechanism: \\[ \begin{array}{l} \mathrm{S}+\mathrm{I} \stackrel{k_{1}}{\longrightarrow} \mathrm{I}+\mathrm{I} \\\ \mathrm{I} \stackrel{k_{2}}{\longrightarrow} \mathrm{R} \end{array} \\] a. Write down the differential rate expressions for \(S,\) I, and \(R\) b. The key quantity in this mechanism is called the epidemiological threshold defined as the ratio of \([\mathrm{S}] k_{1} / k_{2}\). When this ratio is greater than 1 the epidemic will spread; however, when the threshold is less than 1 the epidemic will die out. Based on the mechanism, explain why this behavior is observed.

Short answer

Short Answer Question: Using the Kermack-McKendrick model, explain the behavior of an epidemic based on the epidemiological threshold.

Step-by-step solution

Interpret the model

The Kermack-McKendrick model has two reactions happening: susceptible people (S) getting infected (I) with a rate constant \(k_1\), and infected people (I) recovering (R) with a rate constant \(k_2\). Write down the reactions: 1. \(\mathrm{S}+\mathrm{I} \stackrel{k_{1}}{\longrightarrow} \mathrm{I}+\mathrm{I}\) 2. \(\mathrm{I} \stackrel{k_{2}}{\longrightarrow} \mathrm{R}\)

Write the differential rate expressions

Now, we will derive the differential rate expressions for S, I, and R using the reactions from Step 1. The rate of change for each population depends on the respective reaction rates. 1. For S: \(-\frac{d[\mathrm{S}]}{dt} = k_1[\mathrm{S}][\mathrm{I}]\) 2. For I: \(\frac{d[\mathrm{I}]}{dt} = k_1[\mathrm{S}][\mathrm{I}] - k_2[\mathrm{I}]\) 3. For R: \(\frac{d[\mathrm{R}]}{dt} = k_2[\mathrm{I}]\) #b. Analyzing the Epidemiological Threshold#

Define the epidemiological threshold

The epidemiological threshold is defined as the ratio of \([\mathrm{S}] k_{1} / k_{2}\). If this threshold is greater than 1, the epidemic will spread. If it is less than 1, the epidemic will die out.

Compare the threshold to the rate expressions

To explain the behavior of the epidemic based on the epidemiological threshold, let's look at the rate expression for the infected population. \(\frac{d[\mathrm{I}]}{dt} = k_1[\mathrm{S}][\mathrm{I}] - k_2[\mathrm{I}]\) Let's analyze the situation when the epidemiological threshold is greater than 1: \([\mathrm{S}] k_{1} > k_{2}\) In this case, the \(k_1[\mathrm{S}][\mathrm{I}]\) term is greater than the \(k_2[\mathrm{I}]\) term. It means that the number of new infections (represented by \(k_1[\mathrm{S}][\mathrm{I}]\)) is larger than the number of recoveries (represented by \(k_2[\mathrm{I}]\)). Therefore, the infected population increases over time, which shows that the epidemic is spreading. Now let's analyze the situation when the epidemiological threshold is less than 1: \([\mathrm{S}] k_{1} < k_{2}\) In this case, the \(k_1[\mathrm{S}][\mathrm{I}]\) term is smaller than the \(k_2[\mathrm{I}]\) term. It means that the number of new infections (represented by \(k_1[\mathrm{S}][\mathrm{I}]\)) is smaller than the number of recoveries (represented by \(k_2[\mathrm{I}]\)). Therefore, the infected population decreases over time, which shows that the epidemic is dying out. In conclusion, the epidemiological threshold determines the behavior of the epidemic based on the rate of new infections and recoveries. It helps to predict whether the epidemic will continue to spread or eventually die out.

Key concepts

Kermack-McKendrick Model

The Kermack-McKendrick model is a foundational framework in the study of epidemiology. It is designed to capture the dynamics of infectious diseases as they spread through a population. The key idea behind the model is to observe changes in three compartments: susceptible (S), infected (I), and recovered (R) individuals. This model assumes a closed population where individuals transition from the susceptible category to the infected category and finally to the recovered category.
This model simplifies complexity by using two main reactions:

• Susceptibles becoming infected when they encounter an infected person, at a rate defined by \(k_1\).
• Infected individuals recovering at a rate determined by \(k_2\).

In mathematical terms, this is represented as:- \(\mathrm{S} + \mathrm{I} \xrightarrow[]{k_1} \mathrm{I} + \mathrm{I}\)- \(\mathrm{I} \xrightarrow[]{k_2} \mathrm{R}\)
The Kermack-McKendrick model is instrumental because it provides insights into how infectious diseases progress within a community. By tracking these transitions, researchers can predict the duration and severity of an epidemic.

Differential Rate Expressions

Differential rate expressions are crucial for understanding how the concentrations of susceptible, infected, and recovered individuals change over time in the Kermack-McKendrick model. These expressions are derived from the reactions and represent the rates at which individuals move from one compartment to another.
For susceptible individuals (S), the expression is:- \(-\frac{d[S]}{dt} = k_1[S][I]\)Here, \(k_1[S][I]\) reflects the rate at which susceptible individuals become infected.
For infected individuals (I), the expression is:- \(\frac{d[I]}{dt} = k_1[S][I] - k_2[I]\)This captures the balance between new infections and recoveries. The \(k_1[S][I]\) term indicates the rate of new infections, while \(k_2[I]\) accounts for recoveries.
Lastly, for recovered individuals (R), the expression is:- \(\frac{d[R]}{dt} = k_2[I]\)This shows that the recovery rate is directly proportional to the number of infected individuals.
These differential equations provide a mathematical framework for predicting the course of an epidemic by analyzing how quickly people transition between these states.

Epidemiological Threshold

The concept of the epidemiological threshold is a pivotal element in epidemic modeling. It acts as a predictive marker that indicates whether an infectious disease will spread through a population or die out. The threshold is specifically defined as the ratio \( \frac{[S]k_1}{k_2} \).

• If this ratio is greater than 1, it implies that each infected individual is causing more than one new infection, leading to the spread of the disease. In this scenario, the infection rate \(k_1[S][I]\) surpasses the recovery rate \(k_2[I]\), thus increasing the infected population.
• Conversely, if the ratio is less than 1, the disease will eventually die out. This is because recoveries \(k_2[I]\) occur faster than new infections \(k_1[S][I]\), reducing the number of infected individuals.

Understanding the epidemiological threshold helps public health officials in assessing the potential spread of the disease. It is invaluable for implementing timely interventions to control outbreaks before they spiral out of control. By identifying this threshold, strategies can be tailored to adjust variables, such as vaccination levels or social behavior changes, to shift the ratio below the critical value of 1, thereby halting the epidemic.