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. Daniel Bemoulli's work in 1760 had the goal of appraising the effectiveness of a controversial inoculation program against smallpox, which at that time was a major threat to public health. His model applies equally well to any other disease that, once contracted and survived, confers a lifetime immunity. Consider the cohort of individuals born in a given year \((t=0),\) and let \(n(t)\) be the number of these individuals surviving \(l\) years later. Let \(x(t)\) be the number of members of this cohort who have not had smallpox by year \(t,\) and who are therefore still susceptible. Let \(\beta\) be the rate at which susceptibles contract smallpox, and let \(v\) be the rate at which people who contract smallpox die from the disease. Finally, let \(\mu(t)\) be the death rate from all causes other than smallpox. Then \(d x / d t,\) the rate at which the number of susceptibles declines, is given by $$ d x / d t=-[\beta+\mu(t)] x $$ the first term on the right side of Eq. (i) is the rate at which susceptibles contract smallpox, while the second term is the rate at which they die from all other causes. Also $$ d n / d t=-v \beta x-\mu(t) n $$ where \(d n / d t\) is the death rate of the entire cohort, and the two terms on the right side are the death rates duc to smallpox and to all other causes, respectively. (a) Let \(z=x / n\) and show that \(z\) satisfics the initial value problem $$ d z / d t=-\beta z(1-v z), \quad z(0)=1 $$ Observe that the initial value problem (iii) does not depend on \(\mu(t) .\) (b) Find \(z(t)\) by solving Eq. (iii). (c) Bernoulli estimated that \(v=\beta=\frac{1}{8} .\) Using these values, determine the proportion of 20 -year-olds who have not had smallpox.

Short answer

Answer: Approximately 3.74% of 20-year-olds have not had smallpox.

Step-by-step solution

Rewrite the z(tn) equation

Let's rewrite the equation in terms of x(t) and n(t): $$ z(t) = \frac{x(t)}{n(t)} $$ Differentiate both sides with respect to t to get: $$ \frac{dz}{dt} = \frac{dx/dt \cdot n(t) - x(t) \cdot dn/dt}{[n(t)]^2} $$ We are given that \(dx/dt = -[\beta + \mu(t)] x\) and \(dn/dt = -v\beta x - \mu(t)n\). Substitute the values of \(dx/dt\) and \(dn/dt\) into the equation above.

Apply the initial value problem conditions

Now, plug in the equations for \(dx/dt\) and \(dn/dt\): $$ \frac{dz}{dt} = \frac{-[\beta + \mu(t)] x \cdot n(t) - x(t) \cdot (-v\beta x - \mu(t)n)}{[n(t)]^2} $$ Simplify the equation: $$ \frac{dz}{dt} = \frac{-\beta x +v\beta x^2}{n(t)} $$ Next, we will divide both sides of the equation by \(x(t)\): $$ \frac{1}{x(t)}\frac{dz}{dt} = -\beta +v\beta x(t) $$ We can observe that: $$ -\beta +v\beta x(t) = -\beta (1 - vz(t)) $$ Now, substitute \(z(t)\): $$ \frac{1}{x(t)}\frac{dz}{dt} = -\beta (1 - vz(t)) $$ Multiply both sides by \(n(t)\) and x(t): $$ \frac{dz}{dt} = -\beta z(t)(1-vz(t)), \quad z(0) = 1 $$ Thus, our initial value problem is as required.

Solve the initial value problem

To solve the initial value problem, we will use separation of variables for the differential equation: $$ \frac{dz}{dt} = -\beta z(t)(1-vz(t)) $$ Rearrange the terms: $$ \frac{dz}{z(t)(1-vz(t))} = -\beta dt $$ Integrate both sides: $$ \int\frac{dz}{z(t)(1-vz(t))} = -\beta \int dt $$ The left side can be integrated using partial fraction decomposition. The result will be: $$ \frac{1}{v}\ln\frac{z(t)-1}{z(t)} = -\beta t + C $$ Where C is the constant of integration.

Solve for z(t)

Solve for z(t) by considering the initial condition \(z(0)=1\): $$ \frac{1}{v}\ln\frac{1-1}{1} = -\beta(0) + C \\ C = 0 $$ Now we have: $$ \frac{1}{v}\ln\frac{z(t)-1}{z(t)} = -\beta t $$ Exponentiate both sides: $$ \frac{z(t)-1}{z(t)} = e^{-v\beta t} $$ Rearrange the terms to isolate \(z(t)\): $$ z(t) = \frac{1}{1+ve^{-\beta t}} $$

Calculate the proportion of 20-year-olds without smallpox

We will now use the above equation with Bernoulli's estimated values of \(v=\beta=\frac{1}{8}\), and we want to find \(z(20)\): $$ z(20) = \frac{1}{1+\frac{1}{8}e^{-\frac{1}{8} \cdot 20}} \approx 0.0374 $$ Therefore, approximately 3.74% of 20-year-olds have not had smallpox.

Key concepts

Differential Equations

At the heart of modeling phenomena such as the spread of diseases is a mathematical tool known as a differential equation. A differential equation is an equation that relates a function with its derivatives. It describes the rate of change of a variable with respect to another variable. For instance, when studying epidemics, we often want to know how the number of infected individuals changes over time. This change is not usually constant; it can speed up or slow down based on various factors like the rate of contact between individuals and recovery rates.

In the context of an epidemic, the differential equation will incorporate terms representing these factors, leading to equations that predict how the disease spreads. By solving these equations, we can forecast the number of people who will be infected, recovered, or susceptible at any given time. Understanding and solving differential equations allow health organizations and governments to implement strategies to control the spread of diseases effectively.

The solution to a differential equation isn't always straightforward because it depends on the initial conditions of the system being modeled. This is where the concept of an initial value problem becomes crucial, as it provides the starting point from which we can integrate the differential equation to find a unique solution.

Initial Value Problem

An initial value problem combines a differential equation with a specified value of the function at a starting point. This initial condition allows us to solve for a specific solution that satisfies both the equation and the condition. Think of it as plotting a path on a map; the differential equation outlines all possible routes you can take, while the initial condition marks your starting point.

For our epidemiological example, setting an initial condition such as 'all individuals are susceptible at the beginning of the study' is key to predicting the disease's course. It tells us where to start plotting the path of the disease spread on our mathematical 'map'. In practice, the initial value problem helps turn a general equation with many potential solutions into a specific, practical answer for the scenario at hand.

Why are initial conditions so important? Without them, we might have an infinite number of solutions, and we wouldn't be able to accurately predict or model the behavior of complex systems like those found in epidemiology.

Bernoulli's Epidemiological Model

Daniel Bernoulli's model from 1760 is one of the earliest mathematical approaches to understanding epidemiology, the study of how diseases spread. Bernoulli's model sought to evaluate the effectiveness of inoculation against smallpox, a significant health crisis at the time. In the model, individuals are born susceptible to the disease, then become immune upon recovery, assuming they don't succumb to the illness or other causes of death first.

In Bernoulli's epidemiological model, he used differential equations to represent the dynamics of the disease spread, including the rate of infection and mortality due to smallpox compared to other causes. The key insight from his model was the identification of the rate at which people contract the disease and the rate at which they die from it. By using the concept of an initial value problem, Bernoulli's model could make predictions about the future spread of the disease.

This model is a specific example of a broader category of models used in mathematical epidemiology. These models offer critical insights that inform public health decisions, such as vaccination strategies and forecasts of how a disease will affect a population over time. In essence, Bernoulli's work laid the foundation for the complex models that epidemiologists use today to tackle diseases much like smallpox from centuries ago.