Problem 50 - Calculus for Biology and Medicine

Question

Parasites live by stealing resources from hosts. When parasites reproduce their offspring must find new hosts. However, if a potential host is already infected by parasites, then new parasites will not be able to infect it. This leads to interference between parasites, and we will build a model for these effects in this Problem. We assume that \(N\) is the number of hosts in a given area, and \(P\) is the number of parasites. A frequently used model for host- parasite interactions is the Nicholson-Bailey model (see Nicholson and Bailey, 1935 ), in which it is assumed that the number of parasitized hosts, denoted by \(N_{a}\), is given by $$N_{a}=N\left[1-e^{-b P}\right]$$ where \(b\) is the searching efficiency. (a) Let's treat \(N\) and \(P\) as independent variables and \(N_{n}\) as a function of \(N\) and \(P .\) By calculating the appropriate partial derivatives investigate how: (i) an increase in the number of hosts \(N\) affects the number of parasitized hosts \(N_{a}(N, P)\) (ii) an increase in the number of parasites affects the number of parasitized hosts \(N_{a}(N, P)\) (b) Show that $$b=\frac{1}{P} \ln \frac{N}{N-N_{a}}$$ by solving \((10.6)\) for \(b\). (c) Consider $$b=f\left(P, N, N_{\alpha}\right)=\frac{1}{P} \ln \frac{N}{N-N_{a}}$$ That is, we regard searching efficiency as a function of \(P, N\), and \(N_{n} .\) How is the searching efficiency \(b\) affected when the, number of parasites increases?

Step-by-Step Solution

Answer

Increasing \(N\) or \(P\) increases \(N_a\), \(b\) is expressed as \(b = \frac{1}{P} \ln \frac{N}{N-N_a}\), and \(b\) decreases as \(P\) increases.

Step 1: Differentiate with respect to N

To analyze how the number of hosts, \(N\), affects the number of parasitized hosts, \(N_a\), we take the partial derivative of \(N_a\) with respect to \(N\). Given \(N_a = N \left( 1 - e^{-bP} \right) \), the partial derivative is \(\frac{\partial N_a}{\partial N} = 1 - e^{-bP} \). Since \(e^{-bP}\) is a positive value less than 1, the derivative is positive, meaning \(N_a\) increases with \(N\).

Step 2: Differentiate with respect to P

Next, we check the effect of an increase in \(P\) on \(N_a\) by taking the partial derivative with respect to \(P\). Start from \(N_a = N (1 - e^{-bP})\), giving \(\frac{\partial N_a}{\partial P} = N b e^{-bP} \). Since \(N\), \(b\), and \(e^{-bP}\) are positive, \(\frac{\partial N_a}{\partial P}\) is also positive, showing \(N_a\) increases with \(P\).

Step 3: Solve for b in terms of N, P, Na

To solve for \(b\) in terms of \(N\), \(P\), and \(N_a\), start from \(N_a = N \left( 1 - e^{-bP} \right)\). Rearrange: \(e^{-bP} = 1 - \frac{N_a}{N}\). Take the natural log: \(-bP = \ln \left( 1 - \frac{N_a}{N} \right)\). Rearrange to find \(b\): \(b = \frac{-1}{P} \ln \left( 1 - \frac{N_a}{N} \right)\). Use identity \(\ln(1 - x) = -\ln(\frac{1}{1-x})\): \(b = \frac{1}{P} \ln \frac{N}{N-N_a}\).

Step 4: Analyze the effect of increasing P on b

In \(b = \frac{1}{P} \ln \frac{N}{N-N_a}\), consider \(b\) as a function of \(P\). As \(P\) increases, the term \(\frac{1}{P}\) decreases, suggesting that \(b\) decreases if \(N\) and \(N_a\) remain constant. Hence, the searching efficiency \(b\) decreases as the number of parasites \(P\) increases.

Key Concepts

Host-Parasite Interactions

In ecosystems, host-parasite interactions play a significant role. Hosts are organisms that provide resources for parasites, which can lead to a complex balance between benefit and harm. The Nicholson-Bailey model is a mathematical representation used to study these interactions. This model helps scientists understand how parasites affect host populations.
In the Nicholson-Bailey model, the number of parasitized hosts, denoted as \(N_a\), is dependent on both the number of hosts \(N\) and the number of parasites \(P\). It is expressed through the equation:

• \(N_a = N (1 - e^{-bP})\)

where \(b\) is the searching efficiency of the parasite. This interaction is not merely about existence; the mathematical model illustrates how an increase or decrease in hosts or parasites affects overall dynamics.
Understanding these relations can aid in predicting potential outbreaks of parasitic infections, and thereby, helping in the formation of control strategies.

Partial Derivatives

Partial derivatives are a crucial concept in multivariable calculus, especially in analyzing models with more than one independent variable. In the Nicholson-Bailey model, \(N_a\) depends on two independent variables: \(N\) and \(P\). Hence, we use partial derivatives to explore these dependencies in detail.
When you take the partial derivative of \(N_a\) with respect to \(N\), you get

• \(\frac{\partial N_a}{\partial N} = 1 - e^{-bP}\)

This derivative is positive, meaning that as the number of hosts increases, the number of parasitized hosts also increases.
Similarly, when you take the partial derivative of \(N_a\) with respect to \(P\), it is given by:

• \(\frac{\partial N_a}{\partial P} = N b e^{-bP}\)

Since this derivative is also positive, it implies that the number of parasitized hosts increases with an increase in the number of parasites. These derivatives enhance our understanding by illustrating how slight changes in one variable affect \(N_a\).

Searching Efficiency

Searching efficiency \(b\) is a fundamental parameter in the Nicholson-Bailey model, determining how effectively parasites can find and parasitize hosts. It is crucial because it affects the dynamics of parasitism significantly.
The formula for \(b\) in terms of the number of hosts, parasites, and parasitized hosts is:

• \(b = \frac{1}{P} \ln \frac{N}{N-N_a}\)

This relationship shows that the searching efficiency depends inversely on the number of parasites. In simpler terms, as the number of parasites increases, the searching efficiency \(b\) decreases. This is because as \(P\) increases, \(\frac{1}{P}\) decreases, leading to lesser value of \(b\).
This insight can help in effectively managing and forecasting the spread of parasite infections by adjusting for changes in host and parasite populations.