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Chapter 5: Problem 2

I model, Contagious for life. Consider a disease where all those infected remain contagious for life. Ignore all births and deaths. (a) Write down suitable word equations for the rate of change of numbers of susceptibles and infectives. Hence develop a pair of differential equations. (Define any notation you introduce.) (b) With a transmission coefficient of 0.002, and initial numbers of susceptibles 500 and infectives 1, use Maple or MATLAB to sketch time- dependent plots for the sub-populations (susceptibles and infectives) over time.

Short Answer

Expert verified
The differential equations are \( \frac{dS}{dt} = -0.002 S I \) and \( \frac{dI}{dt} = 0.002 S I \); plots show the dynamics over time with initial conditions \( S(0) = 500 \) and \( I(0) = 1 \).

Step by step solution

01

Define Variables and Parameters

Let's define the variables and parameters for the problem. Let \( S(t) \) be the number of susceptibles at time \( t \), and \( I(t) \) be the number of infectives at time \( t \). Additionally, the transmission coefficient, \( \beta \), is given as 0.002.
02

Word Equations for Susceptibles

The rate of change of susceptibles, \( \frac{dS}{dt} \), is equal to the negative number of new infections. New infections occur when susceptibles come into contact with infectives, which is given by the term \( \beta S I \). Therefore, the word equation is: 'Change in susceptibles = - (Transmission coefficient)  (Susceptibles)  (Infectives)'. So, \( \frac{dS}{dt} = -\beta S I \).
03

Word Equations for Infectives

The rate of change of infectives, \( \frac{dI}{dt} \), is equal to the number of new infections, as individuals remain infectious for life and there are no recoveries or deaths. This is again given by the term \( \beta S I \). Thus, the word equation is: 'Change in infectives = + (Transmission coefficient)  (Susceptibles)  (Infectives)'. Therefore, \( \frac{dI}{dt} = \beta S I \).
04

Write Differential Equations

From the word equations, we have: \[ \frac{dS}{dt} = -\beta S I \] \[ \frac{dI}{dt} = \beta S I \] Substituting \( \beta = 0.002 \), we have the differential equations: \[ \frac{dS}{dt} = -0.002 S I \] \[ \frac{dI}{dt} = 0.002 S I \]
05

Use Software to Sketch Plots

Use software such as MATLAB or Maple to solve the differential equations and plot \( S(t) \) and \( I(t) \) over time given the initial conditions \( S(0) = 500 \) and \( I(0) = 1 \). The plots will show how the populations of susceptibles and infectives change over time.

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Key Concepts

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

Differential Equations
Differential equations are a powerful mathematical tool used to describe the dynamics of various systems. They depict how a variable changes over time, indicating the relationship between the rate of change of these variables and the current state of the system. In the context of disease modeling, differential equations help us understand how a disease spreads through a population over time. By setting up equations that represent the rate of change of different groups, such as susceptibles and infectives, we can predict the future state of an epidemic. For example, in the problem given, the differential equation \( \frac{dS}{dt} = -\beta S I \) tells us how the number of susceptibles decreases as they get infected. Similarly, \( \frac{dI}{dt} = \beta S I \) speaks to the increase in infectives, as new individuals become infected.
Disease Transmission
Understanding disease transmission is vital for modeling how infections spread through populations. Disease transmission describes how an infectious agent is passed from one individual to another. Factors affecting transmission include contact rate, the infectiousness of the disease, and population density. In mathematical models, these aspects are often captured by a transmission coefficient, denoted by \( \beta \). This coefficient represents the efficiency with which a disease spreads in a population. In the provided exercise, \( \beta \) was given a value of 0.002, signifying a specific rate of transmission. The transmission coefficient, in conjunction with the number of susceptibles and infectives, directly influences the rate at which a disease spreads within the population.
SIR Model
The SIR model is a common framework in epidemic modeling, used to simulate how diseases propagate in a population. The acronym SIR stands for Susceptible, Infective, and Recovered - the three compartments into which individuals in a population are classified. Each compartment represents a stage of the disease:
  • Susceptible (S): Individuals who can contract the disease.
  • Infective (I): Individuals currently infected and capable of spreading the disease.
  • Recovered (R): Individuals who have recovered and gained immunity.
In this exercise, however, the model simplifies the SIR model to a SI model, as individuals remain infectives for life and never enter the Recovered stage. Understanding this model helps us to derive the differential equations which predict how the number of susceptibles and infectives change over time.
Epidemic Modeling
Epidemic modeling is a significant application of mathematical models to predict and analyze the spread of diseases. These models help public health officials and policymakers understand potential future scenarios and make informed decisions. Models like the SIR and its variations, are used to estimate how diseases may spread, peak, and eventually decline. Epidemic models can also evaluate the effects of intervention strategies such as vaccination, social distancing, or quarantine. By using differential equations and initial conditions, such models can produce simulations that depict possible outcomes of epidemic situations. This offers valuable insights into controlling and preventing outbreaks, as well as assessing the economic and social impacts of infectious diseases.

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Most popular questions from this chapter

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.)

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?

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.

SEIR model, disease with a latent period. Many diseases have a latent period, which is when there is a period of time between infection and when an infected individual becomes infectious. One example is measles, where the latent period is approximately 5 days. Extend the basic epidemic model to one with an additional population class \(E(t)\), corresponding to individuals who have been exposed to the disease, so they are no longer susceptibles, but are not yet infectious. You may assume the per-capita rate at which an individual in the exposed class becomes an infective is constant. Also the infectious recover in a mean time \(\gamma^{-1}\) and have lifelong immunity. (Give a suitable compartmental diagram or a set of word equations and define any new parameters you introduce.) Note that the latent period is not the same as the incubation period (the time from infection to when symptoms appear).

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 $$ c(N)=\frac{c_{m} N}{K(1-\epsilon)+\epsilon N} $$
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