In an outbreak of tuberculosis in a population of total size N, starting from one infected...

     

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In an outbreak of tuberculosis in a population of total size N, starting from one infected person on day 0, the number of infected individuals T(t) after t days can be modelled by the function T(t) = NeBNt N-1+ eBNt where is the transmission rate, and we assume a zero recovery rate r. (a) Determine the long-time behaviour of this outbreak, by calculating lim T(t). t [You should find that T(t) N as t; that is, in this simplified model of an outbreak without recovery, eventually everyone will be infected...] (b) For short times (small t), we can approximate the function T(t) using a Taylor polynomial. Find the first-order Taylor approximation (that is, the linearization) of T(t) at t = 0. (c) Suppose that in a population of size N = 1000, there is a tuberculosis outbreak with transmission rate = 0.00017 per person per day. Use the first-order Taylor approximation you found in (b) to estimate the number of days until 5 people are infected with tuberculosis. (d) It turns out that for this function T(t), the linearization (= first-degree Taylor polynomial) is not a good approximation beyond about 4-5 days, which is much shorter than the estimated time you should have found in (c) above (optional: confirm this for yourself by plotting the function and its linearization on the same axes). To find actual number of days until 5 people are infected (according to our model), we really need to find the solution t* of the equation T(t) = 5. We can find a more accurate value for t* by rewriting this equation as a root-finding problem: define the function f(t) = T(t) - 5, and seek a numerical approximation for the root of f, that is, the solution of f(t) = 0 (equivalent to T(t) = 5). Beginning at to = 0, perform two iterations of Newton's method (also called the Newton- Raphson method) applied to the function f(t) = T(t) - 5, to find an improved approximation to to the number of days t* for 5 people to be infected. [Notes: (i) since f'(t) = T'(t), you don't need to recompute the derivative; (ii) your first iterate t should be the same as your estimate in (c) above.] In an outbreak of tuberculosis in a population of total size N, starting from one infected person on day 0, the number of infected individuals T(t) after t days can be modelled by the function T(t) = NeBNt N-1+ eBNt where is the transmission rate, and we assume a zero recovery rate r. (a) Determine the long-time behaviour of this outbreak, by calculating lim T(t). t [You should find that T(t) N as t; that is, in this simplified model of an outbreak without recovery, eventually everyone will be infected...] (b) For short times (small t), we can approximate the function T(t) using a Taylor polynomial. Find the first-order Taylor approximation (that is, the linearization) of T(t) at t = 0. (c) Suppose that in a population of size N = 1000, there is a tuberculosis outbreak with transmission rate = 0.00017 per person per day. Use the first-order Taylor approximation you found in (b) to estimate the number of days until 5 people are infected with tuberculosis. (d) It turns out that for this function T(t), the linearization (= first-degree Taylor polynomial) is not a good approximation beyond about 4-5 days, which is much shorter than the estimated time you should have found in (c) above (optional: confirm this for yourself by plotting the function and its linearization on the same axes). To find actual number of days until 5 people are infected (according to our model), we really need to find the solution t* of the equation T(t) = 5. We can find a more accurate value for t* by rewriting this equation as a root-finding problem: define the function f(t) = T(t) - 5, and seek a numerical approximation for the root of f, that is, the solution of f(t) = 0 (equivalent to T(t) = 5). Beginning at to = 0, perform two iterations of Newton's method (also called the Newton- Raphson method) applied to the function f(t) = T(t) - 5, to find an improved approximation to to the number of days t* for 5 people to be infected. [Notes: (i) since f'(t) = T'(t), you don't need to recompute the derivative; (ii) your first iterate t should be the same as your estimate in (c) above.]