Homework 6 Solutions

Transcription

1 Homework 6 Solutions Math 314, Fall 216 Problem 1 The HIV virus targets CD4+ T cells, sometimes called helper T cells During an infection, the total number T t) of helper T cells in the blood, the number T t) of infected helper T cells, and the number V t) of HIV virions free virus particles) in the blood can be modeled by the differential equations = kt V δt, where k, δ, π, and c are positive constants Part a) dv = πt cv, Briefly explain what each term in each of these two differential equations represents What assumptions about proportionality are being made in this model? What is the significance of the four constants k, δ, π, and c? We discuss each of the four terms: 1 In the first differential equation, kt V represents the rate at which helper T cells become infected by virions This is assumed to be proportional to both the number T of helper T cells and the number V of virions, with k being a constant of proportionality that represents the infectiousness of the virions in a given host 2 In the first differential equation, δt represents the rate at which infected helper T cells are cleared from the body, either by dying a natural death or being destroyed or cleared from the body by the immune system This rate is assumed to be proportional to the number of infected helper T cells, with δ being the clearance rate 3 In the second differential equation, πt represents the rate at which virions are produced by the infected helper T cells This is assumed to be proportional to the number of infected helper T cells, with π representing the rate at which a single infected T cell produces virions 4 In the second differential equation, cv represents the rate at which virions either die or are cleared from the body This is assumed to be proportional to the number of virions, with c being the clearance rate 1

2 Part b) A typical HIV infection spends many years in a quasi-steady state Find formulas for the ratio T /V and the total number T of helper T cells during this steady state in terms of the constants k, δ, π, and c Setting the derivatives equal to zero gives Solving the second equation for T /V gives = kt V δt and = πt cv T V = c π Substituting T = cv/π into the first equation gives = kt V δcv/π, and solving for T yields T = δc πk Part c) Administration of an RT inhibitor an antiretroviral drug) blocks the virus from infecting new cells, thereby decreasing k to zero Assuming we administer such a drug at t =, find a formula for T t) in this case, assuming the initial condition T ) = T Substituting k = into the first differential equation gives which has solution Part d) = δt T t) = T e δt Use your answer to part c) to find a general formula for V t) in the same scenario You will need to use the method of integrating factors to solve this part) Substituting T t) = T e δt into the second differential equation gives dv = πt e δt cv Adding cv to both sides and multiplying by e ct yields or equivalently ct dv e + cect V = πt e c δ)t, d [ ] e ct V = πt e c δ)t 2

3 Thus e ct V = where A is a constant Then V t) = πt e c δ)t = πt e c δ)t ) πt e δt + Ae ct + A Part e) Assuming the infection is in a quasi-steady state at t = with initial condition V ) = V, use your answers to parts b) and d) to show that V t) = ce δt δe ct) V Since the infection starts in a quasi-steady state, we know that T V = c π Substituting πt = cv into our solution from part c) yields ) cv V t) = e δt + Ae ct 1) We can solve for the constant A using the initial condition V ) = V Substituting in t = gives Solving for A and simplifying yields V = cv + A A = δv Substituting this into equation 1) above and simplifying gives us the desired equation: V t) = ce δt δe ct) V 3

4 Problem 2 A protease inhibitor is an antiviral drug that interferes with viral replication, causing newly produced virions to be non-infectious The effect of a protease inhibitor can be modeled by the differential equations = kt V I δt, dv NI = πt cv NI, dv I = cv I, where V I t) is the number of infectious HIV virions and V NI t) is the number of non-infections HIV virions Part a) Assuming a protease inhibitor is administered at t =, the number of infectious virions will decrease exponentially according to the equation V I = V e ct Find a general formula for T t) in this case, assuming that the total number T t) of helper T cells is constant Substituting V I = V e ct into the first differential equation gives = kt V e ct δt Adding δt to both sides and multiplying by e δt yields or equivalently Integrating gives where B is a constant Thus Part b) δt dt e + δeδt T = kt V e δ c)t, d ] [e δt T = kt V e δ c)t e δt T = kt V e δ c)t T = + B, ) kt V e ct + Be δt Assuming the HIV infection is in a quasi-steady state at t =, use your answer to part a) to show that δe ct T ce δt) T t) = Because the infection starts in a quasi-stead state, we know from question 1b) that T V = c π and T = δc πk 4

5 Substituting V = πt /c and kt = δc/π into our solution to part a) gives ) δ T T = e ct + Be δt To find the constant B, we substitute t = to get Solving for B and simplifying gives T = δ T + B so T = B = ct δe ct ce δt) T Part c) Use your answer to part b) to show that V NI t) = cv δte ct + c e δt e ct)) Substituting our answer from part b) into the differential equation for V NI gives dv NI δe ct ce δt) T = π cv NI Adding cv NI to both sides and multiplying by e ct yields or equivalently Integrating gives e ct dv NI + ce ct V NI = πt ) e c δ)t d ] [e ct V NI = πt ) e c δ)t e ct V NI = πt δt c ) ec δ)t + R where R is a constant Multiplying by e ct and substituting πt = cv gives V NI t) = cv δte ct + c ) e δt + Re ct 2) Now presumably V NI ) =, since there were no non-infectious virions before we administered to protease inhibitor Substituting t = gives us = cv ) c + R 5

6 so R = cv c ) Substituting this into equation 2) and simplifying yields the desired equation: V NI t) = cv δte ct + c e δt e ct)) 6