AARMS Homework Exercises

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1 1 For the gamma distribution, AARMS Homework Exercises (a) Show that the mgf is M(t) = (1 βt) α for t < 1/β (b) Use the mgf to find the mean and variance of the gamma distribution 2 A well-known inequality in probability theory is Markov s inequality The Markov inequality states that for a nonnegative random variable X with expectation µ, Verify Markov s inequality Prob{X c} µ c, c > 0 3 Another well-known inequality in probability theory is Chebyshev s inequality The Chebyshev inequality states that for any random variable X with finite expectation µ and positive variance σ 2, Prob{ X µ < kσ} 1 1 k 2, k > 0 Use the Markov inequality in Exercise 2 with Y 2 = (X µ) 2 and E(Y 2 ) = σ 2 to verify the Chebyshev inequality 4 Suppose the states for two different Markov chains are {1, 2, } and for a third Markov chain the states are {1, 2, 3} The corresponding transition matrices are a a 2 0 a a 1 a 1 a a a 1 a a 1 0 P 1 = , P 2 = 0 a 2 0, and P 3 = 1 0 1, a 2 0 a a 2 where 0 < a i < 1, i = 1, 2 and a 1 + a 2 = 1 (a) Draw a directed graph for each chain Determine whether the corresponding DTMC is reducible or irreducible (b) Identify the communicating classes, find their period (c) If the corresponding DTMC is irreducible and recurrent, compute lim n Pi n Hint: Compute Pi 2 and Pi 4 (d) For the recurrent DTMCs, use the Basic Limit Theorems (periodic or aperiodic) to determine the mean recurrence time for each state i Is the chain positive recurrent or null recurrent? 5 Suppose the transition matrix P of a finite Markov chain is doubly stochastic; that is, row and column sums equal one, p ij 0, N N p ij = 1, and p ij = 1 i=1 Prove the following: If an irreducible, aperiodic finite Markov chain (ergodic chain) has a doubly stochastic transition matrix, then all stationary probabilities are equal, π 1 = π 2 = = π N In particular π i = 1/N 1 j=1

2 6 Suppose the size of a population remains constant from generation to generation; the size equals N The dynamics of a particular gene in this population is modeled Suppose the gene has two alleles, A and a Therefore, individual genotypes are either AA, Aa, or aa Let the random variable X n denote the number of A alleles in the population in the nth generation, n = 0, 1, 2, Then X n {0, 1, 2,, 2N} Assume random mating of individuals so that the genes in generation n + 1 are found by sampling with replacement from the genes in generation n (Ewens, 1979) Then the one-step transition probability has a binomial probability distribution with the probability of success X n /(2N), ie, if X n = i, then the one-step transition probability is the binomial pdf b(2n, i/2n), p ji = ( ) ( ) 2N i j ( 1 i ) 2N j, j 2N 2N i, j = 0, 1, 2,, 2N (Ewens, 1979; Schinazi, 1999) This model is known as the Wright-Fisher model (a) Show that states 0 and 2N are absorbing and states {1, 2,, 2N 1} are transient (b) Given X n = k, show that the mean of X n+1 satisfies µ Xn+1 = E(X n+1 X n = k) = k A discretetime Markov process with the property E(X n+1 X n = k) = k is called a martingale (c) Show that the probability of fixation of allele A is k/2n, ie probability of absorption into state 2N 7 Normal cell division results in two identical daughter cells containing the same number of chromosomes as the original cell (2n for a diploid cell) Sometimes a mistake occurs and only one cell is produced having twice the number of chromosomes (4n chromosomes), referred to as omitosis When this abnormal cell divides again, it will produce two daughter cells with twice the number of chromosomes (4n chromosomes) Endomitosis can occur again for a cell having 4n chromosomes to produce a cell with 8n chromosomes and, in general, omitosis occurring in a cell with 2 i n chromosomes produces a cell with 2 i+1 n chromosomes Cells with more than two copies of the genes are known as polyploid cells The incidence of higher ploidies than four is small Therefore, it is reasonable to consider a cellular model with only two types: diploid cells and polyploid cells (Jagers, 1975) Let p be the probability of omitosis, 0 < p < 1/2, q the probability the cell dies, and 1 p q, the probability of cell division into two daughter cells of the same type The pgfs for the two cell types are f 1 (t 1, t 2 ) = (1 p q)t pt 2 + q and f 2 (t 1, t 2 ) = pt 2 + (1 p q)t q (a) Discuss why the pgfs have the given forms Calculate the expectation matrix M Show that matrix M is reducible (b) The pgf for the polyploid cells can be considered separately Beginning with one polyploid cell, what is the probability that the polyploid cell line will die out? f 2 (t) = pt + (1 p q)t 2 + q 8 Use the following MATLAB program to compute the expected population growth/year given the growth/year in a good year is m 1 = 4 and in a bad year is m 2 = 05 Let p 1 be the probability of a good year and p 2 be the probability of a bad year, p 1 + p 2 = 1 Complete the following table to compare the expected population growth/year in a random environment with the average of the growth rates Even though the average of the growth rates is greater than one, the expected population growth/year may be less than one clear all m1=4; % Mean growth in Good Year m2=05; % Mean growth in Bad Year 2

3 p 1 Expected Growth Average p 1 m 1 + p 2 m p1=08; % Probability of Good Year sumx=0; n=10^7; % Total number of years for i=1:n u=rand; if u<p1 x=log(m1); else x=log(m2); sumx=x+sumx; ExpectedGRand=exp(sumx/n) Average=m1*p1+m2*(1-p1) 9 Suppose the generator matrix of a finite CTMC is Q = (a) Compute the corresponding transition matrix T of the embedded Markov chain Show that the CTMC is irreducible (b) Find the stationary distribution of the CTMC: lim t P (t)p(0) 10 Use the backward Kolmogorov differential equations dp/ = P Q for the Poisson process and write differential equations to find the probability of hitting state 3, ie, p i (t) = probability of hitting state 3 at time t beginning from state i, i = 0, 1, 2, p i (0) = 0, i = 0, 1, 2 and p 3 (t) = 1 Solve these three differential equations to find explicit expressions for p i (t), i = 0, 1, 2 11 Consider the simple birth and death process with immigration (a) Use the mgf M(θ, t) to find expressions for the mean µ(t) and variance σ 2 (t) of the process (b) Find the limit of the mean and variance, lim t µ(t) and lim t σ 2 (t), when λ < µ (c) What are the mean and variance of the process if there is no immigration? 12 Suppose a general birth and death process has birth and death rates given by λ i = b 0 + b 1 i + b 2 i 2, and µ i = d 1 i + d 2 i 2, for i = 0, 1, 2, (a) Write the forward Kolmogorov differential equations Then write differential equations satisfied by the mgf (b) Write the partial differential equation for the mgf in the more general case, where λ i = n k=0 b ki k and µ i = n k=1 d ki k 3

4 13 The mgf M(θ, t) for a simple birth process is a solution of the following first order partial differential equation: M t = λ(e θ 1) M θ, M(θ, 0) = enθ (a) Differentiate the partial differential equation with respect to θ and evaluate at θ = 0 to write M(θ, t) a differential equation for the mean, m(t) = E(X(t)) = θ Solve the differential θ=0 equation for m(t) (b) Differentiate the partial differential equation twice with respect to θ and evaluate at θ = 0 to write a differential equation for the second moment E(X 2 (t)) Solve the differential equation for E(X 2 (t)) (c) Use parts (a) and (b) to find the variance σ 2 (t) of the process 14 Consider a model for a lytic viral population After successful attachment and entry into a host cell, the virus uses the host cell for its own reproduction, killing the cell to release the new virus particles or virions The number of virions produced per host cell is referred to as the burst size We describe a burst-death process studied by Hubbarde et al (2007) Let µ be the death rate, λ be the birth rate, and β the burst size, 0 < µ < λ, and β is some positive integer Then the pgf for the next generation of virions is f(z) = µ µ + λ + λ µ + λ zβ (a) For β = 2, this model is the same as the simple birth and death process and the probability of extinction is µ/λ Show for β > 2 that the probability of extinction for the burst-death model is less than µ/λ (b) Show that if β and µ and λ are fixed constants, then the probability of extinction approaches µ/(µ + λ) This means the probability of survival is λ/(µ + λ) Even if the burst size is extremely large, the probability of viral survival is less than one (provided the birth and death rates remain constant) 15 The following MATLAB program is for the SIR epidemic model Graph three sample paths for the case that N = 500, I(0) = 1 and R 0 = β/γ = 2 Modify the program and compute the following based on 10, 000 sample paths (1) probability histogram for time the epidemic s (duration of the epidemic) (2) estimate the probability of an outbreak (run until time t when I(t) = 10 or I(t) = 0, then count number of times of hitting 10 infectives before hitting zero) The estimated value should be close to 1 (1/R 0 ) I(0) How does the duration of an epidemic and probability of an outbreak change if I(0) = 2 or I(0) = 3 or if R 0 = 3? %Matlab Program: Three sample paths for the SIR epidemic clear all beta=1; g=05; N=500; % Parameters for k=1:3 clear t s i t(1)=0; i(1)=1; s(1)=n-i(1); j=1; while i(j)>0 & t(j)<100; 4

5 u1=rand; u2=rand; t(j+1)=-log(u1)/((beta/n)*i(j)*s(j)+g*i(j))+t(j); if u2<=(beta/n)*s(j)/(beta/n*s(j)+g) i(j+1)=i(j)+1; s(j+1)=s(j)-1; else i(j+1)=i(j)-1; s(j+1)=s(j); j=j+1; t(k)=t(j);% Time epidemic s l1=stairs(t,i); set(l1, LineWih,2); hold on ylabel( I(t) ); xlabel( Time ); hold off 16 Group Project 1: The following SEIR epidemic model includes births and deaths and an additional class of exposed or latent individuals, E, individuals that are not yet infectious The differential equations are ds de di dr = Λ β S N I µs = β S I σe µe N = σe γi µi = γi µr, where S + E + I + R = N The parameter Λ is the birth/immigration rate µ is the natural death rate, and σ is the rate exposed individuals become infectious The basic reproduction number of the SEIR epidemic model is βσ R 0 = (γ + µ)(σ + µ) (Hethcote, 2000) (a) Show that the total population size approaches Λ/µ (b) Set up a table of the infinitesimal transition probabilities for the eight events in a CTMC SEIR model (c) Assume Λ = 1, d = 0002, β = 1, σ = 025, γ = 025 If N = Λ/µ, S(0) = N 1, E(0) = 0, I(0) = 1 and R(0) = 0, use the table of transition probabilities to write a MATLAB program to simulate sample paths of the SEIR model Compute the probability of an outbreak by computing the probability of hitting E(t) + I(t) = 20 before hitting E(t) + I(t) = 0 (epidemic s) based on 10,000 sample paths What is the probability of an outbreak? (d) If time permits, change the parameters and the initial conditions to graph several sample paths Compute the probability of an outbreak given N = Λ/µ, S(0) = N 1, E(0) = 1, I(0) = 0 and R(0) = 0 and hypothesize a formula for the probability of an outbreak 5

6 17 Group Project 2: Suppose one population moves between two habitats (patches) The population size in each patch changes and can be modeled by the following differential equation dx ( i = rx i 1 x i K ) + m(x j x i ), i, j = 1, 2, j i (a) Without movement or migration, m = 0, the population grows logistically What is the stable positive equilibrium in each patch when m = 0 (b) Let X i (t) i = 1, 2 denote random variables for the stochastic process Assume the birth and death rates in each patch are λ i = ri(1 i 2 /(2K)) and µ i = ri 2 /(2K), i = 0, 1, 2,, 2K Consider only a single patch Let r = 2, K = 10, and m = 0 The maximum population size in patch 1 with no migration is N = 2K = 20 Find the infinitesimal generator matrix Q (of size 21 21) Compute the expected time to extinction, τ: τ Q = 1, where Q is the infinitesimal generator matrix with the first row and first column deleted, 1 = (1, 1,, 1) and τ = (τ 1,, τ N ) (d) Set up a table with six transition probabilities for the births and deaths in each patch and movement between the two patches Use the transition probabilities to write a MATLAB computer program to graph sample paths for the preceding model with m = 0 and m = 01 Estimate the probability of extinction from the MATLAB program with m = 01 and compare your results to part (b) References (e) If time permits, modify this patch model, to investigate the impact of patch depent parameters, r i, K i, m i x i on population survival Discuss your results Allen, LJS 2011 An Introduction to Stochastic Processes with Applications to Biology and Hall/CRC, Boca Raton, FL Chapman Ewens, WJ 1979 Mathematical Population Genetics Springer-Verlag, Berlin, Heidelberg, New York Hethcote, HW 2000 The mathematics of infectious diseases SIAM Review 42: Hubbarde, JE, G Wild, and LM Wahl 2007 Fixation probabilities when generation times are variable: the burst-death model Genetics 176: Jagers, P 1975 Branching Processes with Biological Applications John Wiley & Sons, London Schinazi, RB 1999 Classical and Spatial Stochastic Processes Birkhäuser, Boston 6