A Simple Interpretation Of Two Stochastic Processes Subject To An Independent Death Process

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1 Applied Mathematics E-Notes, 92009, c ISSN Available free at mirror sites of amen/ A Simple Interpretation Of Two Stochastic Processes Subject To An Independent Death Process Joseph Gani, Randall J Swift Received 21 January 2008 Abstract In this explanatory note, we interpret two nown results for a death process and a birth process subject to an independent death process The first example is the carrier-borne epidemic, while the second is the polymerisation chain reaction The interpretations allow a more intuitive understanding of the resulting formulae for the probability generating functions of the processes 1 Introduction In some biological phenomena, the size of a population Xt at time t 0 is influenced by an independent process Y t and is often modeled as a continuous time bivariate Marov chain {Xt, Y t : t 0} One classic example is the carrier-borne epidemic process detailed by Weiss 1965 In this process the number of susceptibles Xt are modeled as a death process subject to the number of infectious carriers Y t, which themselves follow an independent death process The bivariate Marov chain {Xt, Y t : t 0} for the process Xt subject to the independent process Y t is often characterized by the probability generating function pgf for the transient probabilities The pgf which usually arises as a solution of a partial differential equation is often difficult to interpret conceptually In this brief article, we present a conceptual framewor for the pgf of two such processes 2 The Carrier-Borne Epidemic In the carrier-borne epidemic process, as outlined in Weiss 1965 and Daley & Gani 1999, infection spreads through contact between an infectious carrier and a susceptible The carriers are subject to a pure death process while an infected susceptible is directly removed from the population Mathematics Subject Classifications: 60J80, 92D20 Mathematical Sciences Institute, Australian National University, Canberra ACT 0200, Australia Department of Mathematics and Statistics, California State Polytechnic University Pomona, CA USA 89

2 90 Stochastic Processes Subject to an Independent Death Process If there are initially n susceptibles and b infectious carriers at time t 0, and we let the nonnegative integer-valued processes Xt and Y t represent the numbers of susceptibles and carriers of the disease, then {Xt, Y t : t 0} can be modeled as a continuous time bivariate Marov chain The transitions and rates for this chain in the interval t, t + δt are described as transition x, y x 1, y x, y x, y 1 rate βxyδt yδt, where β is the infection parameter and is the death parameter of the carriers The carrier process {Y t : t 0} is a pure death process with the well nown pgf ψ Y v, t Ev Y t ve t + 1 e t b, for 0 v 1, so that PY t Y 0 b b e t 1 e t b, 0,, b The susceptible process Xt is subject to the influence of the process Y t, which is itself independent of Xt If we let P ij t PrXt i, Y t j X0 n, Y 0 b, for i 0,, n, j 0,, b, then it can be shown that the pgf φz, v, t E z Xt v Y t j0 b P ij tz i v j of the process satisfies the partial differential equation pde φ t β1 zv 2 φ v 1 φ z v v 1 The solution of this pde is obtained using the separation of variables method and can be found in either Bailey 1975 or Daley & Gani 1999; the resulting pgf is φz, v, t [ n z 1 i i + βi + v e +βit 2 + βi Let us attempt to interpret the structure of this pgf Given that the number of susceptibles Xt i is fixed, a single carrier will have the partial pgf ve +βit + t 0 e +βiu du ve +βit + 1 e +βit, + βi

3 J Gani and R J Swift 91 so that the b independent carriers will have the partial pgf [ + βi + v e +βit + βi For v 1, this leads to the probability [ βi P {0 Y t b Xt i} e +βit + p i + βi + βi it, say, for the carriers when there are Xt i susceptibles Now the susceptibles follow a binomial death distribution with probability q i t of death, and p i t 1 q i t, so that the pgf is [zp i t + 1 p i t] n [z 1p i t + 1] n n z 1 p i t 0 Thus if one writes for the ith term of a series a, the indicator I i a a i then φz, 1, t I i [z 1p i t + 1] n n z 1 i p i i it [ n βi z 1 i e +βit +, i + βi + βi which would ensue if the pgf Ez Xt v Y t is [ n φz, v, t z 1 i i + βi + v e +βit + βi 3 The PCR Process Recently, Gani & Swift 2007 considered the polymerisation chain reaction PCR process of enzyme molecules DNA polymerase that have the property of causing the replication of DNA strands, while themselves degrading after a certain period They modeled this process as a DNA strand birth process subject to a death process for the enzymes Their model considers Xt DNA strands and Y t enzyme molecules at time t 0, with X0 n and Y 0 b with the transitions and rates in t, t + δt given by

4 92 Stochastic Processes Subject to an Independent Death Process transition x, y x + 1, y x, y x, y 1 rate βxyδt yδt, The forward Kolmogorov equations for the probabilities p i,j t P {Xt i, Y t j X0 n, Y 0 b} are d dt p i,jt βi + jp i,j t + j + 1p i,j+1 t + βji 1p i 1,j t, for n i <, 0 j b The pgf ψu, v, t in j0 b p i,j tu i v j, 0 u, v 1, satisfies the pde ψ 1 v ψ t v + βuvu 1 2 ψ u v The solution, obtained by separation of variables, is n u n + i 1 u ψu, v, t 1 i 1 u i 1 u [ v e βi+n+t + + βi + n n u n + i 1 1 u i [ v + βi + n u u 1 i e βi+n+t + i + βi + n + βi + n The pgf for the number of DNA strands Xt is found as n+i n + i 1 u ψu, 1, t 1 i i 1 u [ βi + n e βj+n+t + + βi + n + βi + n To interpret the structure of this pgf, we follow a similar reasoning to that in Section 2 Given that Xt i + n is fixed, a single enzyme will have the partial pgf t [ ve βi+n+t + e βi+n+u du v e βi+n+t + 0 βi + n + so that for the b independent enzymes, the partial pgf will be [ v e βi+n+t + βi + n + βi + n + βi + n + ]

5 J Gani and R J Swift 93 For v 1, this leads to the probability P {Xt Xt i + n} [ βi + n + βi + n p n+i n+i t, say e βi+n+t + βi + n + Now the DNA strands follow a birth process with probability p i+n t of birth when Xt i + n so that the pgf is n p i+n tu p n u i+nt 1 u + p i+n tu 1 + p i+n t u n u p n i+n t i n + i 1 u 1 i p i i+n 1 u i 1 u t Writing once again I i a a i then ψu, 1, t I i p i+nt 1 + p i+n t u u n u n + i 1 u 1 i 1 u i 1 u n+i n + i 1 u 1 i i 1 u [ βi + n e βj+n+t + + βi + n n i p i+n i+n t + βi + n as required The ensuing pgf for the bivariate process {Xt, Y t : t 0} would then be n u i n + i 1 u φz, v, t 1 u i u 1 [ v e βi+n+t + + βi + n + βi + n 4 Concluding Remars We have attempted to provide a more intuitive approach to the rather complex formulae for the pgfs of a death process and a birth process, each subject to an independent death process Much yet remains to be done to mae such formulae more accessible to worers in applied probability

6 94 Stochastic Processes Subject to an Independent Death Process References [1] N T J Bailey, The Mathematical Theory of Infectious Disease, Griffin, London, 1975 [2] D J Daley and J Gani, Epidemic Modelling, Cambridge University Press, 1999 [3] J Gani and R J Swift, A PCR birth process subject to an enzyme death process, preprint [4] G H Weiss, On the spread of epidemics by carriers, Biometrics, ,