Question

Let \(p\) denote the population density, and let it depend on time \(t\) and age \(r\). The number of people of ages between \(r\) and \(r+d r\) at a certain time \(t\) is given by \(p(t, r) d r\). Define the mortality rate \(\mu(t, r)\) in the following way: \(\mu(t, r) d r d t\) is the fraction of people in the age class \([r, r+d r]\) who die in the time interval \([t, t+d t]\). Based on the infinitesimal equality $$ p(t+d t, r+d t) d r-p(t, r) d r=-\mu p d r d t . $$ Show that \(p\) satisfies the following partial differential equation $$ \frac{\partial p}{\partial r}+\frac{\partial p}{\partial t}=-\mu p $$ Let the initial age distribution be given as $$ p(0, r)=p_{0}(r), \quad 0 \leq r \leq 1 $$ and the birth rate function as the boundary condition $$ p(t, 0)=u(t), \quad t \geq 0 . $$ Here it assumed that the age \(r\) is scaled in such a way that nobody reaches an age \(r>1\). One can consider \(u(t)\) as the input to the system and as output \(y(t)\) for instance the number of people in the working age, say between the ages \(a\) and \(b\), \(0

Short answer

The PDE \(\frac{\partial p}{\partial r} + \frac{\partial p}{\partial t} = -\mu p\) describes how population changes over time and age. Initial and boundary conditions define population distribution and birth rates.

Step-by-step solution

Interpret the Given Equation

Start by examining the given infinitesimal equality: \[ p(t+d t, r+d t) d r - p(t, r) d r = -\mu p \, d r \, d t \]This states how the population changes due to age progression and mortality. On the left, the population after a time \(d t\) and age change \(d t\) is compared to the initial state.

Utilize Taylor Series Expansion

Using a Taylor series expansion for \(p(t+d t, r+d t)\), we express it as:\[ p(t+d t, r+d t) \approx p(t, r) + \frac{\partial p}{\partial t} d t + \frac{\partial p}{\partial r} d t \].Substituting this into the left-hand side of the equality gives: \[ \left(p(t, r) + \frac{\partial p}{\partial t} d t + \frac{\partial p}{\partial r} d t\right) d r - p(t, r) d r \].This simplifies to:\[ \frac{\partial p}{\partial t} d t \, d r + \frac{\partial p}{\partial r} d t \, d r \].

Set the Terms Equal

Equating the right-hand side of the expression from the Taylor expansion with the infinitesimal change due to mortality, we get:\[ \frac{\partial p}{\partial r} d t \, d r + \frac{\partial p}{\partial t} d t \, d r = -\mu p \, d r \, d t \].

Factor Out Common Terms

Cancel out \(d r \, d t\) from both sides of the equation (assuming \(d r, d t > 0\)): \[ \frac{\partial p}{\partial r} + \frac{\partial p}{\partial t} = -\mu p \].This is the required partial differential equation.

Apply Initial Conditions and Boundary Conditions

The initial condition \(p(0, r)=p_{0}(r)\) sets the distribution of the population at time \(t=0\). The boundary condition \(p(t, 0)=u(t)\) provides the birth rate for new individuals entering the population. These conditions are essential for solving the PDE uniquely.

Evaluate the Output Function

Use the output function to evaluate the number of people in a specific age range at any time \(t\):\[ y(t) = \int_{a}^{b} p(t, r) \, d r \].This requires solving the PDE and integrating over the age range to find the population size between ages \(a\) and \(b\).

Key concepts

Population Dynamics

Population dynamics is a field that studies how populations of organisms, particularly humans in this context, change over time. It involves the use of mathematical models such as partial differential equations to understand these changes.
In the exercise we tackled, we look at how a human population varies with age, represented as a continuous variable, and over time. The population density function, denoted as \(p(t, r)\), helps us understand how many individuals are of a given age \(r\) at any specific time \(t\).

• Age and Time Interest: The model is interested in two main variables, age \(r\) and time \(t\). These two factors affect the population's behavior and structure.
• Infinitesimal Changes: The change in population due to individuals aging or dying within an infinitesimal time interval or age bracket is critical in modeling population dynamics.

Understanding this allows policymakers and researchers to plan for resources needed over time.

Mortality Rate

Mortality rate is a key factor in determining how a population evolves since it measures the fraction of individuals that die within a certain time frame, given a particular age range.
In our exercise, mortality is expressed as \(\mu(t, r)\), representing how age and time influence the likelihood of individuals dying. \(\mu(t, r) \cdot d r \cdot d t\) gives the fraction of individuals in a specific age bracket that die during a specific time period.

• Influence of Age: Mortality often increases with age, though this isn't always linear. Different age groups have varying susceptibility to mortality risks.
• Mortality and Dynamics: By understanding \(\mu(t, r)\), we get insights into how mortality influences overall population dynamics, allowing adjustments in policy for healthcare, pensions, or other needs.

Researchers use these mortality rates as parameters to predict and influence future population structures.

Age-Structured Models

Age-structured models are essential in population dynamics as they allow us to consider how age-specific factors affect the overall population. They provide a framework to account for various characteristics that differ by age, such as mortality and birth rates.
In our example, the partial differential equation \(\frac{\partial p}{\partial r} + \frac{\partial p}{\partial t} = -\mu p\) describes how the population density changes with age and time, taking into account both aging and mortality.

• Initial Conditions: These models often start with a specified initial distribution \(p(0, r) = p_0(r)\), which sets the stage for how the population evolves.
• Boundary Conditions: Birth rates \(p(t, 0) = u(t)\) are a fundamental boundary condition that influence how new individuals enter the population.

Age-structured models are thus invaluable in planning for and managing population-related needs, taking into account both current and future age distributions.