Question

Epidemics. The use of mathematical methods to study the spread of contagious diseases goes back at least to some work by Daniel Bernoulli in 1760 on smallpox. In more recent years many mathematical models have been proposed and studied for many different diseases. Deal with a few of the simpler models and the conclusions that can be drawn from them. Similar models have also been used to describe the spread of rumors and of consumer products. Some diseases (such as typhoid fever) are spread largely by carriers, individuals who can transmit the disease, but who exhibit no overt symptoms. Let \(x\) and \(y,\) respectively, denote the proportion of susceptibles and carriers in the population. Suppose that carriers are identified and removed from the population at a rate \(\beta,\) so $$ d y / d t=-\beta y $$ Suppose also that the disease spreads at a rate proportional to the product of \(x\) and \(y\); thus $$ d x / d t=\alpha x y $$ (a) Determine \(y\) at any time \(t\) by solving Eq. (i) subject to the initial condition \(y(0)=y_{0}\). (b) Use the result of part (a) to find \(x\) at any time \(t\) by solving Eq. (ii) subject to the initial condition \(x(0)=x_{0}\). (c) Find the proportion of the population that escapes the epidemic by finding the limiting value of \(x\) as \(t \rightarrow \infty\).

Short answer

Answer: The proportion of the population that escapes the epidemic in the long run is equal to the initial proportion of susceptibles, x₀.

Step-by-step solution

Solve the differential equation for y(t)

First, let's integrate the differential equation for y with respect to time: $$ \dfrac{d y}{d t}=-\beta y $$ Separate variables and integrate: $$ \int \dfrac{d y}{y}=-\beta \int dt $$ Evaluating the integrals, we get $$ \ln{y}=-\beta t + C_1 $$ Now, we solve for y(t) by exponentiating both sides, and then apply the initial condition y(0) = y_₀: $$ y(t)=e^{-\beta t + C_1}\\ y(0) = y_0\\ e^{C_1} = y_0\\ y(t)=y_0 e^{-\beta t} $$

Solve the differential equation for x(t) using y(t)

Now that we have an expression for y(t), we use it to solve the second differential equation for x(t): $$ \dfrac{d x}{d t}=\alpha x y $$ Replace y(t) with y_0 e^{-βt}: $$ \dfrac{d x}{d t}=\alpha x (y_0 e^{-\beta t}) $$ Again, we separate variables and integrate: $$ \int \dfrac{d x}{x}=\alpha y_0 \int e^{-\beta t} dt $$ Evaluating the integrals, we get $$ \ln{x}=-\frac{\alpha y_0}{\beta} (e^{-\beta t}) + C_2 $$ Now, we solve for x(t) by exponentiating both sides, and then apply the initial condition x(0) = x_₀: $$ x(t)=e^{-\frac{\alpha y_0}{\beta} (e^{-\beta t}) + C_2}\\ x(0) = x_0\\ e^{C_2} = x_0\\ x(t)=x_0 e^{-\frac{\alpha y_0}{\beta} (e^{-\beta t})} $$

Find the proportion of the population that escapes the epidemic

Finally, we find the limiting value of x(t) as t approaches infinity: $$ \lim_{t \to \infty} x(t) = x_0 e^{-\frac{\alpha y_0}{\beta} (e^{-\beta t})} $$ As t approaches infinity, e^{-βt} approaches 0. This implies: $$ \lim_{t \to \infty} x(t) = x_0 e^{-\frac{\alpha y_0}{\beta} (0)} = x_0 $$ Thus, the proportion of the population that escapes the epidemic is equal to the initial proportion of susceptibles, x₀.

Key concepts

Differential Equations

Differential equations are mathematical tools used to describe how a certain quantity changes over time. They play a vital role in modeling real-world phenomena, such as the spread of infectious diseases. In this context, a differential equation establishes a relationship between a function and its derivatives, providing a way to understand the rate of change within a system.

For instance, in the problem approach above, we encounter the differential equation \( \frac{dy}{dt} = -\beta y \). This equation tells us how the proportion of carriers, \( y \), declines over time due to their removal at rate \( \beta \).

Differential equations can be solved using various methods, such as separation of variables or integrating factors. By solving these equations, we can predict the future behavior of dynamic systems, helping us understand not just current trends but also potential future scenarios in the context of epidemics.

Mathematical Modeling

Mathematical modeling is the process of using mathematics to represent, analyze, and predict real-world systems. In the context of epidemics, models can help us simulate the spread of diseases and assess the effectiveness of different intervention strategies. Such models are crucial for public health planning and response.

The model in the exercise uses mathematical equations to describe the dynamics between susceptible individuals and carriers. The equations \( \frac{dy}{dt} = -\beta y \) and \( \frac{dx}{dt} = \alpha xy \) illustrate how the interaction between these groups influences the spread of the disease and how interventions like removing carriers (\( \beta \)) can alter the course of an outbreak.

Mathematical modeling often requires simplifying assumptions to make problems tractable while retaining essential features of the disease dynamics. These models, despite simplifications, provide a framework to understand complex systems, helping guide decision-making in infectious disease management.

Population Dynamics

Population dynamics refers to the study of how and why populations change over time. It considers factors like birth rates, death rates, and migration patterns. When applied to diseases, it involves understanding how infections influence the size and composition of populations.

In our exercise, the population is divided into different compartments: susceptibles \( x \) and carriers \( y \). Each compartment has its dynamics governed by differential equations. Carriers are reduced by a constant rate \( \beta \), while new infections occur at a rate proportional to the product of \( x \) and \( y \).

This approach helps us explore key questions, such as how quickly a disease spreads, the impact of removing carriers, and the proportion of the population likely to get infected. Such insights into population dynamics help in creating effective control measures and strategies to mitigate the effects of an epidemic.

Infectious Disease Spread

Understanding infectious disease spread is crucial for control and prevention. It looks at how diseases transmit from person to person and factors affecting transmission rates.

In the mathematical model, the spread of disease is captured by the differential equation \( \frac{dx}{dt} = \alpha xy \). This represents the rate at which susceptibles become carriers. The term \( \alpha xy \) indicates that the spread is dependent on the interaction between the susceptibles and carriers. The more carriers there are, the faster the disease can spread through the susceptible population.

Knowing how diseases spread allows us to evaluate various intervention strategies, such as vaccination or quarantine measures. Mathematical modeling provides a quantitative framework to simulate different scenarios, thus informing policymakers about potential outcomes and responses to control the spread of infectious diseases effectively.