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Chapter 5: Problem 23

With the disease malaria in humans, the disease is carried by mosquitoes that cannot infect each other. Infectious mosquitoes can only infect susceptible humans and infected humans can only infect susceptible mosquitoes when they are bitten by a susceptible mosquito. Assume the rate of transmission is proportional to both numbers of mosquitoes and number of humans for transmission in both directions, and assume (for this model) once infected, both humans and mosquitoes never recover. Ignoring any births and deaths, develop a mathematical model for susceptible and infected humans \(S_{h}(t), I_{h}(t)\), and susceptible and infected mosquitoes \(S_{m}(t), I_{m}(t)\).

Short Answer

Expert verified
The model consists of four differential equations describing the changes in susceptible and infected humans and mosquitoes, with no recovery terms.

Step by step solution

01

Understand the Variables

First, identify the key components of the problem: You have susceptible and infected humans, denoted as \(S_h(t)\) and \(I_h(t)\), respectively. For mosquitoes, use \(S_m(t)\) for susceptible and \(I_m(t)\) for infected mosquitoes. The goal is to express how these populations change over time.
02

Identify the Interaction

Recognize that the transmission of the disease depends on interactions between mosquitoes and humans. The problem states that no recovery occurs once infected, so focus only on the process of infection. There are two main interactions: infected mosquitoes can infect susceptible humans, and infected humans can infect susceptible mosquitoes if bitten.
03

Model the Interaction

Since the rate of transmission is proportional to the number of contacts between susceptible and infected individuals (both humans and mosquitoes), let \(\beta_1\) be the transmission rate constant from mosquitoes to humans, and \(\beta_2\) be the rate from humans to mosquitoes. You'll create rate equations that describe these interactions.
04

Develop the Mathematical Model

Write down the differential equations that represent the changes in each population over time. The rate of change in susceptible humans, \(S_h(t)\), is affected by the number of infected mosquitoes \(I_m(t)\), and is given by:\[\frac{dS_h}{dt} = -\beta_1 S_h I_m\]The rate of change in infected humans, \(I_h(t)\), is:\[\frac{dI_h}{dt} = \beta_1 S_h I_m\]Similarly, for mosquitoes, the rate of change in susceptible mosquitoes, \(S_m(t)\), is:\[\frac{dS_m}{dt} = -\beta_2 S_m I_h\]And for infected mosquitoes, \(I_m(t)\), is:\[\frac{dI_m}{dt} = \beta_2 S_m I_h\]
05

Explanation of Non-recovery Condition

Since there is no recovery, the model does not include terms for transitions from infected back to susceptible states. This simplifies the model to only accounting for the increase in infected individuals based on initial susceptible and infected interactions.

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Key Concepts

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

Differential Equations
Differential equations form the backbone of mathematical modeling for various phenomena, including disease transmission. In the context of malaria transmission, these equations describe how the proportions of susceptible and infected individuals change over time. The power of differential equations lies in their ability to model dynamic processes. By observing changes in quantities and rates, they provide a continuous view of the interactions between mosquitoes and humans. To put it simply, they help predict how an entire system evolves given specific initial conditions.For our exercise, the differential equations capture the rates at which humans and mosquitoes either become susceptible or infected. These changes are influenced by constants that characterize the transmission process. For example, the constants \(\beta_1\) and \(\beta_2\) depict how fast the disease is transmitted between species. By solving these equations, we can predict the future number of susceptible and infected individuals.
Susceptible-Infected Model
The susceptible-infected (SI) model is a simple yet powerful framework used to understand how infections spread in a population. This model categorizes the population into two primary groups: susceptible and infected. In other models, there might be recovered categories, but in our specific case of malaria, individuals do not recover. Here鈥檚 a breakdown of the SI Model:
  • Susceptible (S): These are individuals that can potentially become infected.
  • Infected (I): Individuals that carry the disease and can pass it on to others.
The progression from susceptible to infected is modeled using differential equations. They quantify how quickly the disease is transmitted based on effective contact rates represented by the rate of change equations. The SI model assumes no immunity from the disease and no transition from being infected back to susceptible, aligning with our malaria context where recovery does not occur. This makes the equations linear and easier to solve compared to models incorporating recovery.
Mosquito-Human Interaction
The interaction between mosquitoes and humans is central to understanding the spread of malaria. Mosquitoes act as vectors, transferring the malaria pathogen between humans. In mathematical modeling, it's essential to establish interaction dynamics to predict outcomes accurately.Key features of mosquito-human interaction include:
  • Unidirectional Transmission: Infected mosquitoes spread the disease to humans when they bite, and infected humans transmit the disease to mosquitoes when bitten.
  • Proportional Transmission Rate: The rate at which the disease spreads is proportional to the number of contacts between susceptible and infected individuals, quantified by our constants \(\beta_1\) and \(\beta_2\).
The modeling reflects these interactions through differential equations that account for the increase in infected populations without considering birth, death, or recovery. This focus on infection dynamics captures the critical interaction points where the disease lifecycle perpetuates.By understanding these interactions, we gain insights into controlling and possibly reducing the spread of malaria through strategic interventions aimed at breaking the transmission cycle.

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Most popular questions from this chapter

We formulate two models for continuous vaccination of the susceptible population; the first assumes the vaccine gives perfect life-long protection and the second assumes the vaccine is only temporary. Ignore any births or deaths. (a) Formulate 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 where a proportion \(\nu\) per unit time of susceptibles are vaccinated. We want to track the number vaccinated so you need to have an additional variable \(V(t) .\) (b) Assume the vaccine is only partially effective, where vaccinated individuals return to the susceptible state after being protected by vaccination for an average time \(\mu^{-1}\).

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 $$ c(N)=\frac{c_{m} N}{K(1-\epsilon)+\epsilon N} $$ where \(\epsilon\) is a positive constant between 0 and 1, and \(K\) is a positive constant. Consider \(\epsilon=0.5\), and using Maple or MATLAB with \(p c_{m}=1.62, r=0.44, K=1,000\), and initial values \(i_{0}=1, s_{0}=762\) (time is measured in days), graph the number of susceptibles over time and determine when the number of infectives is at a maximum. Compare this model to those with \(\epsilon=1\) and \(\epsilon=0\), and discuss the differences.

In the fermentation of wine, yeast cells digest sugar from the grapes and produce alcohol as a waste product, which is toxic to the yeast cells. Develop a model consisting of three coupled differential equations for the density of yeast cells, the amount of alcohol and the amount of sugar. In the model assume the yeast cells have a per-capita birth rate proportional to the amount of sugar, and a per-capita death rate proportional to the amount of alcohol present. Assume the rate of alcohol produced is proportional to the density of yeast cells, and the rate of sugar consumed is proportional to the density of yeast cells.

Consider an infectious disease model based on the standard SIR model, where an individual passes through two infectious states with different degrees of infectivity, with transmission coefficients \(\beta_{1}\) and \(\beta_{2}\) (this could be due to different rates of virus shedding or to different contact rates, for example). Ignoring any births or deaths, formulate a model as three differential equations, for \(S\) the number of susceptible, \(I_{1}\) the first infectious state, and \(I_{2}\) the second infectious state. Recovery (with lifelong immunity) can only occur when the individual has passed through the second infectious state.

Formulate a model for an infectious disease where there is immunity for only some of those who recover; others 'recover' to become permanent carriers, who can still cause infections. Thus susceptibles, \(S(t)\), may be infected by either infectious individuals, \(I(t)\), or carriers, \(C(t) .\) A carrier can infect others at a reduced rate compared to infectious individuals but shows no symptoms. (a) Assume there is a fixed proportion \(q\) of those recovering from the infection become carriers. Assume transmission rates \(\beta_{1}\) for normal infectives and \(\beta_{2}\) for carriers and assume that individuals remain infective for a mean time \(\gamma^{-1}\). Give equations for the number of susceptibles \(S(t)\), the number of infectives \(I(t)\), the number of carriers \(C(t)\), and the number of recovered who are immune \(R(t)\). (c) Give at least one example of an infectious disease that could be modelled by the equation you have developed.
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