Draw conclusions about the mode of transmission of the disease based on the analysis of SARS and population in Epiville
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Epidemics. The use of mathematical methods to study the spread of contagious diseases goes back at least to some work by Daniel Bernoulli in 1760 on smallpox. In more recent years many mathematical models have been proposed and studied for many different diseases. Deal with a few of the simpler models and the conclusions that can be drawn from them. Similar models have also been used to describe the spread of rumors and of consumer products. Some diseases (such as typhoid fever) are spread largely by carriers, individuals who can transmit the disease, but who exhibit no overt symptoms. Let $x$ and $y,$ respectively, denote the proportion of susceptibles and carriers in the population. Suppose that carriers are identified and removed from the population at a rate $eta,$ so $$ d y / d t=-eta y $$ Suppose also that the disease spreads at a rate proportional to the product of $x$ and $y$; thus $$ d x / d t=-alpha x y $$ (a) Determine $y$ at any time $t$ by solving Eq. (i) subject to the initial condition $y(0)=y_{0}$. (b) Use the result of part (a) to find $x$ at any time $t$ by solving Eq. (ii) subject to the initial condition $x(0)=x_{0}$. (c) Find the proportion of the population that escapes the epidemic by finding the limiting value of $x$ as $t ightarrow infty$.
Adi S.
Question 1: Consider the SIR model of host-pathogen dynamics, presented in Chapter 13 as: dI/dt = βSI - mI. Where dI/dt is the rate of growth of the infected population, β is the transmission coefficient, S is the number of uninfected hosts, and m is the death and recovery (with immunity) rate. This simple model captures key elements of the combined interaction between hosts and pathogens. Notice that we model the number of infected hosts (I). Use this model to answer the following questions: 1a: Assume you lived in a small town within one day's walking distance of a large port city in Europe in the mid-1500s. You have heard rumors that a terrible plague has befallen the inhabitants of the city. One day, a mother and two children straggle into your town. All three are sick with early symptoms of a bacterial disease. Your town has a population of 330 people, none of whom have been exposed to the disease previously. For this strain of the disease, the incidence of death or recovery (with immunity), m, is about 0.7 or 70%. The rate of transmission, β, is 0.002. Using the above equation, calculate the population size of potential hosts (S) needed for the disease to spread (S = m/β). Assuming that the bacterium is only spread by human-to-human contact, what is likely to happen to the people of your town if these people stayed in the town? Why? 1b: Now suppose that at about the same time as the three infected individuals arrived at the gate of your town, three others arrived. The others had a less virulent strain of the same bacterial disease, and the incidence of death or recovery (with immunity), m, is about 0.2 or 20%. The rate of transmission, β, is still 0.002. What is likely to happen to the people of your town if these three people stayed? Why? 1c: Now suppose that the transmission rate (β) of both groups is different than we thought in Questions 1 and 2. The transmission rate of the first group (where m = 0.7) is really 0.003 (higher than in Questions 1 and 2). The transmission rate of the second group (where m = 0.2) is really 0.0005 (lower than in Questions 1 and 2). Does this change the outcome for the people of the town if either of these two groups of sick people were to stay? Why? 1d: Three organisms were involved in the Black Death (besides humans) - the bacterium, Yersinia pestis; the flea, Xenopsylla cheopis; and the black rat, Rattus rattus. Of these, which would you vote for as the organism that changed the world? Why?
Madhur L.
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