Question

Logistic equation for an epidemic When an infected person is introduced into a closed and otherwise healthy community, the number of people who contract the disease (in the absence of any intervention) may be modeled by the logistic equation $$\frac{d P}{d t}=k P\left(1-\frac{P}{A}\right), P(0)=P_{0}$$ where \(k\) is a positive infection rate, \(A\) is the number of people in the community, and \(P_{0}\) is the number of infected people at \(t=0\) The model also assumes no recovery. a. Find the solution of the initial value problem, for \(t \geq 0\), in terms of \(k, A,\) and \(P_{0}.\) b. Graph the solution in the case that \(k=0.025, A=300,\) and \(P_{0}=1.\) c. For a fixed value of \(k\) and \(A\), describe the long-term behavior of the solutions, for any \(P_{0}\) with \(0

Short answer

The general equation is: $$P(t) = \frac{AP_{0}}{(A-P_{0})e^{-kt}+P_{0}}$$ 2. What is the long-term behavior of the logistic model as time goes to infinity? As time goes to infinity, the number of infected individuals approaches the total population (A) of the community.

Step-by-step solution

Solving the Differential Equation

We are given the logistic differential equation $$\frac{dP}{dt} = k P\left(1-\frac{P}{A}\right).$$ This is a first-order, separable differential equation. We can rewrite it as $$\frac{dP}{P\left(1-\frac{P}{A}\right)} = k \,dt.$$ Now, we integrate both sides: $$\int \frac{1}{P}+\frac{1}{A-P} \, dP = \int k \, dt.$$

Integrating Both Sides and Solving for P

To solve the left-hand side integral, we can make the substitution \(u = A - P\). The integral then becomes $$-\int\frac{1}{A-u}+\frac{1}{u} \, du = \int k \, dt.$$ We can now integrate both sides: \(-\ln\frac{A-u}{u} = kt + C\), where \(C\) is the constant of integration.

Applying the Initial Condition

Now we apply the initial condition \(P(0) = P_{0}\). Plugging these values into the equation from step 2, we get $$-\ln\frac{A-P_{0}}{P_{0}} = k(0) + C.$$ So, \(C = -\ln\frac{A-P_{0}}{P_{0}}\). We can now substitute this back into the expression for \(P(t)\): $$-\ln\frac{A-u}{u} = kt - \ln\frac{A-P_{0}}{P_{0}}.$$

Isolating P(t)

To isolate \(P(t)\), do the following steps: 1. Take the exponent of both sides. 2. Invert the right-hand side. 3. Substituting \(u = A - P\) back in. This results in $$P(t) = \frac{AP_{0}}{(A-P_{0})e^{-kt}+P_{0}}.$$

Graph for Specific Values

Now, we are asked to graph the solution for the specific values of \(k=0.025\), \(A=300\), and \(P_{0}=1\). Plugging these values into the expression for \(P(t)\) gives $$P(t) = \frac{300}{299e^{-0.025t}+1}.$$ To graph this function, plot \(P(t)\) on the vertical axis and \(t\) on the horizontal axis. The graph will show the number of infected individuals over time.

Analyzing Long-term Behavior

Lastly, we will analyze the long-term behavior of the solution for fixed values of \(k\) and \(A\), and any \(P_{0}\) with \(0<P_{0}<A\). As \(t\) approaches infinity, the exponential term \(e^{-kt}\) approaches 0. Then, the logistic equation becomes $$P(t) \approx \frac{AP_{0}}{P_{0}} = A.$$ Thus, in the long-term, the number of infected individuals approaches the total population \(A\) of the community.

Key concepts

Epidemic Modeling

When we hear about an outbreak of a disease, scientists and public health officials use mathematical models to predict its spread and impact. One commonly used model is the logistic model, particularly potent for representing the spread of an infection within a closed population.

The logistic model takes into account the fact that as more individuals become infected, fewer are left to be infected, slowing down the spread rate; this is well-known as the 'saturation effect'. In mathematical terms, the logistic differential equation for such an epidemic can be expressed as

\[\begin{equation}\frac{dP}{dt} = kP\bigg(1 - \frac{P}{A}\bigg),\text{where } P(0) = P_0.\end{equation}\]
Here, \(P(t)\) represents the number of infected individuals at a time \(t\), \(k\) is the infection rate, \(A\) is the total population, and \(P_0\) is the initial number of infected individuals. This equation strives to incorporate real-world constraints, such as limited resources and interaction rates, into the dynamics of disease spread.

Initial Value Problem

The logistical equation provided in the problem represents what mathematicians call an 'Initial Value Problem' (IVP). An IVP is a differential equation that comes with a set of initial condition(s). These conditions give us specific information about the value of the unknown function at a particular point. For our epidemic problem, the initial condition is \(P(0)= P_0\), which tells us the number of infected individuals at the start, \(t=0\).

Establishing this initial condition is critical because it allows us to solve the logistic differential equation uniquely. Without it, we might end up with a family of solutions, but not the particular one that describes the evolution of the epidemic accurately.

Separable Differential Equations

The equation given is a classic example of a separable differential equation. In these equations, the variables can be separated on different sides of the equation - typically one side with the dependent variable and the other with the independent variable.

The solution involves integrating both sides, often after some algebraic manipulation. For the logistic equation in our context, the separation of variables looks like this:
\[\begin{equation}\int \frac{1}{P(1 - P/A)}\,dP = \int k \,dt.\end{equation}\]
As the name suggests, separable equations are 'separable' because we can move all instances of the dependent variable (and its derivatives) to one side and all instances of the independent variable to the other. Once separated, integration helps find the solution to the differential equation, giving us the function that predicts the number of infected individuals over time.

Long-Term Behavior of Solutions

Understanding the long-term behavior of solutions to differential equations is fundamental in many fields, including epidemiology. For logistic equations, the long-term behavior indicates what happens as time goes toward infinity - will the number of infected individuals grow boundlessly, stabilize, or decline?

For our specific logistic model, as time increases indefinitely, it forecasts that the number of infected individuals approaches the total population size \(A\), assuming there's no recovery or death altering the conditions. Mathematically, this is seen when the term \(e^{-kt}\) diminishes, effectively making the fraction \(P_0/(A - P_0)e^{-kt} + P_0\) approach zero, resulting in the solution \(P(t)\) approaching \(A\). This long-term projection is an essential piece of information for policy-makers and health officials in planning and deploying resources for an epidemic response.