Here, we refer to them as transition probabilities. A transition probability for a birth is the probability associated with the change.
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A transition probability for a death is the probability associated with the change. Denote the transition probability for a change in state from i to state j in a short time as the following conditional probability: 1. When the transition probabilities depend only on the length of time between the transitions and not on the particular time t at which these transitions occur, as in 1 , then the stochastic process is referred to as a time-homogeneous process.
Let b be the per capita birth rate and d be the per capita death rate of the process. Then the transition probabilities for a small interval of time , 2. Definition 2 states that when is small and , the probability of a birth during the time interval is approximately , the probability of a death is approximately , the probability of no change is approximately and all other changes are negligible. The sum of all the probabilities must equal one.
Definition 2 also states that jumps occur in the population size when there is a birth or a death. The time that elapses between jumps a birth or a death is referred to as the interevent time. This simple birth and death process is an example of a continuous-time Markov chain. The term chain implies the random variable X t is discrete-valued and the term Markov named after Andrey Andreyevich Markov implies the future state at , given the present state at time t , is independent of the past.
That is, for time ,.
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This is referred to as the memoryless property. The only continuous probability distribution with the memoryless property is the exponential distribution. The interevent time in a Markov chain is a continuous random variable with an exponential distribution. This property is demonstrated in the birth and death process in Equation 4. It is important to note that if the initial state is known, then the transition probabilities are actually the probabilities associated with the random variables X t : 3. Because the values of the random variables are discrete, any sample path X t of a continuous-time Markov chain is not a continuous function of time.
For example, if the process is in state i and there is a birth, then the process jumps to state or if there is a death, the process jumps to state. From Definition 2 , if and if the interevent time is , then a change of state occurs at. It follows that the left-hand limit of the process at is i and the right-hand limit is either or. The stochastic process is continuous from the right but not from the left see Figure 1. That is,. The probability of a birth, , is.
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Figure 1. A sample path is continuous from the right but not from the left. In this example, the jump times are 1, 1. To numerically simulate a birth or a death the uniform distribution U on [0, 1] is applied. In general, for k different events, each representing a jump in the process and each with a given positive probability , , the unit interval [0, 1] is divided into k subintervals of length ,.
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Then, if a uniform random number lies in the subinterval , event j occurs. In the birth and death process there are only two events, and.
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If , there is a birth but if not, there is a death. In addition to the birth and death probabilities, numerical simulation of a sample path requires computation of the values for the interevent times. The interevent time is a continuous random variable with values. As noted earlier, for Markov chains, the interevent time is exponentially distributed. Given that the process is in state i at time t , , the interevent time is exponentially distributed with parameter the sum of the rates corresponding to all possible events.
In particular, the probability density function of is and the cumulative distribution is. The mean and standard deviation of the exponential distribution are equal to. Therefore, as the population size i increases, the mean interevent time decreases. The Markov property of the interevent time can be easily demonstrated.
Predicting population extinction or disease outbreaks with stochastic models
The interevent time does not depend on the length of time t to reach state i but only on the current state i of the process independent of past history or memoryless property : 4. The uniform distribution U and the cumulative distribution are used to compute the interevent time of a sample path. We derive an identity for in terms of U that depends on the following properties of U :. The preceding calculation is part of an algorithm for the interevent time which is often referred to as the Gillespie algorithm Gillespie, Gillespie, D. Exact stochastic simulation of coupled chemical reactions.
Journal of Physical Chemistry , 81 , — A uniform random number generates a value for the interevent time. The deterministic analogue of the simple birth and death process is the well-known Malthusian exponential growth model. Expressed as a differential equation, the model is 5. In fact, the exponential growth model 5 is the mean of the simple birth and death process. To compare the dynamics of the exponential growth model to the simple birth and death process, we assume both models have the same initial size and the same birth and death rates.
In the example in Figure 2 , , and and. Figure 2. The exponential solution black dashed curve and five sample paths of the simple birth and death process black, red, blue, green and magenta curves are plotted for parameter values , and initial value. In the simple birth and death process, the zero state is an absorbing state, corresponding to population extinction. That is, if , then for.
Absorption into the zero state is illustrated in two of the five sample paths in Figure 2. Although extinction cannot occur in the deterministic model when , the probability of extinction in the stochastic model is always positive. In the next section, we give an analytical expression for the probability of extinction. The probability of extinction can be derived from differential equations that follow from the transition probabilities, Definition 2 and techniques from probability generating functions e. Allen, Allen, L. An introduction to stochastic processes with applications to biology 2nd ed.
The elements of stochastic processes with applications to the natural sciences. New York, NY : Wiley. The differential equations are known as the Kolmogorov differential equations to honour the contributions of the mathematician Andrey Nikolaevich Kolmogorov. The derivation is relegated to Appendix 1. The derivation for the probability of extinction can also be found in the classic textbook by Feller Feller, W. An introduction to probability theory and its applications 3rd ed. Wiley Series in Probability and Mathematical Statistics.
The derivation in Appendix 1 yields an expression for the probability generating function,. The probability generating function is useful for generating the probabilities and the moments. Evaluating at gives and differentiating with respect to s and evaluating at gives.