Error Bound for Hybrid Models of Two-Scaled Stochastic Reaction Systems

The relative contributions of infectious and mitotic spread to HTLV-1 persistence

“…Thus the total number of possible reactions C ∈ ℕ at time t is C = 3S(t) . Following the formulation given in [26, 27], let X ( t ) = (( X i ( t )) i ∈ S ( t ) ) T be the state vector at time t of all clones. X(t) is a random variable in that consists of the random variables X i ( t ) ∈ ℕ 0 = ℕ ⋃{0} of the frequencies x i (t) of clones i = 1,…, S max , where S max is chosen to always be larger than S(t) for all t .…”

“…X(t) is a random variable in that consists of the random variables X i ( t ) ∈ ℕ 0 = ℕ ⋃{0} of the frequencies x i (t) of clones i = 1,…, S max , where S max is chosen to always be larger than S(t) for all t . The state vector X(t) evolves through a Markov jump process that depends only on the current state , and its evolution is given by where V c and α c respectively denote the stoichiometric vector and propensity function of reaction c [26, 27]. Equation (2) states that the population X(t) at time t is equal to the initial population y 0 plus the sum of the changes induced by all reactions.…”

“…There exists a probability distribution associated with the random variable in (2), given by ℙ ( X ; t ) = ℙ ( X ( t ) = y | X (0) = y 0 ), where is a column vector where each entry is a probability associated with a potential state of the random variable at time t . It can be shown [27–30] that ℙ( X ; t ) is a solution of the Chemical Master Equation (CME) which describes the rate of change in the probability distribution associated with X(t) . The first term is the sum over all reactions of the probability of arriving at state X(t) = y from state X(t) = y - V c via reaction c , and the second term is the sum over all reactions of the probability of leaving state X(t) = y via reaction c .…”

“…If we impose a maximum frequency for a particular clone i (supplementary information) [Figure 3B], we can summarise Equation (3) using multiple, simpler differential equations below where A is the transition matrix or “matrix of connections” [supplementary information] [27, 31, 32]. Further, because the proliferation rate is constant at equilibrium, rates are independent of time, and so Equation (8) has solution where ℙ 0, i = ℙ( X i ; t = 0) is the initial probability distribution and e At is the matrix exponential [33].…”

The relative contributions of infectious and mitotic spread to HTLV-1 persistence

“…PLOS COMPUTATIONAL BIOLOGY C = 3S(t). Following the formulation given in [26,27], let X(t) = (X i (t) i2S(t) ) T be the state vector at time t of all clones. X(t) is a random variable in N S max that consists of the random variables X i ðtÞ 2 N 0 ¼ N [ f0g of the frequencies x i (t) of clones i = 1,. .…”