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Chapter 5: Problem 17

5.17. Battle loss due to disease. Develop a model (a pair of differential equations) for a battle between two armies where both groups use aimed fire. Assume that the red army has a significant loss due to disease, where the associated death rate (from disease) is proportional to the number of soldiers in that army.

Short Answer

Expert verified
The model for the battle is \( \frac{dR}{dt} = -k_1 B - k_2 R \) and \( \frac{dB}{dt} = -m R \).

Step by step solution

01

Identify the Variables

Let \( R(t) \) be the number of soldiers in the Red army at time \( t \), and \( B(t) \) be the number in the Blue army. We need to account for battle losses and disease in the Red army.
02

Model the Red Army's Losses

The rate of change for the Red army is affected by losses from battle and disease. The battle losses are proportional to \( B(t) \) and the disease losses are proportional to \( R(t) \). These can be written as \( \frac{dR}{dt} = -k_1 B(t) - k_2 R(t) \), where \( k_1 \) and \( k_2 \) are constants.
03

Model the Blue Army's Losses

Assume the Blue army loses soldiers in a manner proportional to the number of Red army soldiers. Thus, we write \( \frac{dB}{dt} = -m R(t) \), where \( m \) is a constant.
04

Formulate the Differential Equations

Now, combining the models from steps 2 and 3, we have the following system of differential equations: \[ \frac{dR}{dt} = -k_1 B(t) - k_2 R(t) \] \[ \frac{dB}{dt} = -m R(t) \]. These equations describe the battle with losses from fighting and disease in the Red army.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Mathematical Modelling
Mathematical modelling is a powerful way to represent real-world systems using mathematical concepts and equations. It helps us understand complex situations by breaking them down into manageable parts. In this exercise, we develop a model to illustrate how two armies, Red and Blue, lose soldiers due to both battle and disease. We use differential equations to represent this scenario because they allow us to describe how quantities such as the number of soldiers change over time.
  • Conceptualization: Identify the relevant components of the system, such as the number of soldiers in each army and the factors contributing to their loss.
  • Translation into Mathematics: Convert these components into mathematical expressions, such as differential equations, that can describe the relationships and changes within the system.
By translating the problem into a set of mathematical equations, we can predict and analyze the behavior of the two armies over time, considering the effects of actions like battle and disease.
Epidemiology in Battle
Epidemiology is the study of how diseases spread and affect populations. In a battle context, diseases can significantly impact the fighting capacity of an army. The exercise takes this into account by introducing the concept of disease-related loss. The death rate from disease is assumed to be proportional to the number of soldiers in the Red army.In this scenario:
  • Impact of Disease: For the Red army, disease acts as an additional factor that decreases their numbers alongside battle losses.
  • Modelling Disease Spread: The use of a proportionality constant, denoted as \( k_2 \), helps model this loss stemming from disease. The larger \( k_2 \) is, the more severe the disease impact is on the army.
This understanding reflects real-life military struggles, where besides skirmishes, health issues can drastically alter the outcome of a battle.
System of Equations
A system of equations is a set of equations with multiple variables that are solved together. In this problem, we have two differential equations that represent the dynamics between the Red and Blue armies. The purpose of setting up a system is to capture the interaction between these two groups more comprehensively.The system of differential equations is:\[\frac{dR}{dt} = -k_1 B(t) - k_2 R(t)\]\[\frac{dB}{dt} = -m R(t)\]
  • Interdependence: Each equation not only describes the dynamics of one army but also considers the influence of the other army.
  • Simultaneous Resolution: By solving these equations together, we gain insights into the overall interaction and potential outcomes of the battle.
This approach allows us to study how the size of each army changes over time, offering a more nuanced understanding of their fates.
Proportionality Constants
Proportionality constants are values that define the strength of relationships between variables in mathematical models. They help quantify how changes in one variable affect another. In our given model, the constants play a crucial role in determining how battle and disease losses affect the Red and Blue armies.Consider the constants in the equations:
  • \( k_1 \): Represents the rate at which the Red army loses soldiers due to battle engagements with the Blue army. Higher values indicate more significant losses per encounter.
  • \( k_2 \): Denotes the rate of loss due to disease within the Red army. A higher \( k_2 \) signifies a more severe epidemic.
  • \( m \): Captures the battle-related losses of the Blue army due to the Red army's presence. Like \( k_1 \), higher values imply more substantial battlefield effectiveness of the Red army against the Blue.
These constants are not just abstract values; they are vital in accurately reflecting the dynamics and outcome predictions of the model.

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Most popular questions from this chapter

Density dependent contact rate. For a fatal disease, if the basic epidemic model of Section \(5.2\) is modified to include density dependent disease transmission, the resulting differential equations are $$ \frac{d S}{d t}=-p \frac{c(N)}{N} S I, \quad \frac{d I}{d t}=p \frac{c(N)}{N} S I-\gamma I $$ where \(N=S+I, p\) is a constant (the probability of infection) and the contact rate function \(c(N)\) is given by $$ c(N)=\frac{c_{m} N}{K(1-\epsilon)+\epsilon N} $$

Competing species with density dependence. Consider the following model for two competing species, with densities, \(X(t)\) and \(Y(t)\), given by the differential equations $$ \frac{d X}{d t}=X\left(\beta_{1}-c_{1} Y-d_{1} X\right), \quad \frac{d Y}{d t}=Y\left(\beta_{2}-c_{2} X-d_{2} Y\right) $$ with parameter values \(\beta_{1}=3, \beta_{2}=3, c_{1}=2, c_{2}=1, d_{1}=2\) and \(d_{2}=2.5\). (a) What is the carrying capacity for each of the species, evaluated for the given parameter values? (Hint: Compare with equations (5.15) in Section 5.5.) (b) With the above parameter values, and the initial values \(X=2\) and \(Y=2\), use MATLAB or Maple to draw time-dependent plots for these populations. Over a period of time what population densities do you estimate they will approach?

Continuous vaccination. Consider a model for the spread of a disease where lifelong immunity is attained after catching the disease. The susceptibles are continuously vaccinated against the disease at a rate proportional to their number. Write down suitable word equations to describe the process, and hence obtain a pair of differential equations.

Simple age-based model. Consider a population split into two groups: adults and juveniles, where the adults give birth to juveniles but juveniles are not yet fertile. Eventually juveniles mature into adults. You may assume constant per-capita birth and death rates for the population, and also assume that the young mature into adults at a constant per-capita rate \(\sigma\). Starting from suitable word equations or a compartment diagram formulate a pair of differential equations describing the density of adults, \(A(t)\), and the density of juveniles, \(J(t) .\) Define all variables and parameters used.

Exact solution for battle model. Consider the aimed fire battle model developed in the text $$ \frac{d R}{d t}=-a_{1} B, \quad \frac{d B}{d t}=-a_{2} R $$ The exact solution can be found using theoretical techniques as follows: (a) Take the derivative of the first equation to get a second-order differential equation, and then eliminate \(d B / d t\) from this equation by substituting the second equation (given above) into this second-order equation. (b) Now assume the solution to be an exponential of the form \(e^{\lambda t}\). Substitute it into the secondorder equation and solve for the two possible values of \(\lambda .\) The general solution for \(R\) will be of the form $$ R(t)=c_{1} e^{\lambda_{1}}+c_{2} e^{\lambda_{2}} $$ where \(c_{1}\) and \(c_{2}\) are the arbitrary constants of integration. The solution for \(B\) is then found using the equation \(d R / d t=-a_{1} B\) (c) Now find the arbitrary constants by solving the simultaneous equations for \(R(0)=r_{0}\) and \(B(0)=b_{0}\), when \(t=0 .\) The final solution is given in the text in Section 5.7. (d) Using Maple or MATLAB (with symbolic toolbox) check the solution above. Use the dsolve command or just substitute back into the original differential equations. (Further details about methods for solving second-order differential equations, in particular for differential equations with constant coefficients, as used here, can be found in Appendix A.5.)
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