Environmental variability is especially important in modeling zoonotic infectious diseases, vector-borne diseases, and waterborne diseases e. In this primer, the emphasis is on demographic variability. In the following sections, these two stochastic processes are formulated for the well-known SIR Susceptible-Infectious-Recovered epidemic model and the Ross malaria host-vector model.
The Gillespie algorithm and the Euler-Maruyama numerical method are described for the two types of stochastic processes. In addition, some analytical methods from branching processes that are related to the CTMC models are used to approximate the probability of an outbreak. In the last section, some stochastic methods for modeling environmental variability are presented.
In the SIR deterministic model, S t , I t , and R t are the number of susceptible, infectious, and recovered individuals, respectively. In the simplest model, there are no births and deaths, only infection and recovery:. The stochastic formulation of the CTMC and SDE models requires defining two random variables for S and I whose dynamics depend on the probabilities of the two events: infection and recovery.
For simplicity, the same notation is used in the stochastic and the deterministic formulations. Differential equations for the transition probabilities can be derived from 2. These differential equations are often referred to as the forward or the backward Kolmogorov differential equations. The general form for the forward Kolmogorov differential equations are.
That is,. A similar derivation applies to the backward equations. The backward equations 4 are. For a more thorough derivation of these equations, consult the references, e. Matrix Q is straightforward to define for a single random variable whose states are already linearly ordered from 0 to N e. But for a bivariate process with states s , i the form of matrix Q depends on how the set of ordered pairs are linearly ordered. In general, matrix Q has negative diagonal entries and nonnegative off-diagonal entries.
In addition, matrix Q has the property that the row sums are zero. It follows from Equations 3 , B. Matrix P t has diagonal entries p a , a t , where a is one of the linearly ordered states. In this case, the transpose of these equations is applied with matrix Q T instead of Q Allen, , Allen, In this brief introduction, we study the stochastic behavior near the disease-free equilibrium to determine whether an epidemic major outbreak occurs when a few infectious individuals are introduced into the population.
The probability of no major outbreak a minor outbreak for the CTMC model near the disease-free equilibrium is approximated by applying branching process theory and techniques from probability generating functions pgfs. Most important is the fact that when I hits zero, it stays in state zero.
The state I has reached an absorbing state and disease transmission stops. But finite time extinction of I occurs in the stochastic model. We are interested in the stochastic dynamics at the initiation of an epidemic, when almost everyone in the population is susceptible. The duration of the epidemic, once initiated, is another interesting stochastic problem, not considered here, see e.
The branching process is the linear approximation of the SIR stochastic process near the disease-free equilibrium. For a few initial infectious individuals, the branching process either grows exponentially or hits zero. These two phenomena are captured in the branching process approximation of the CTMC model near the disease-free equilibrium. If the number of infectious individuals increases substantially to a large number of cases, then there is a major outbreak.
However, if there are only a few additional cases, above the initial number of cases, then there is a minor outbreak. The branching process is a good approximation of the CTMC model, if the susceptible population size is sufficiently large. Then the two outcomes, either a major or minor outbreak, are clearly distinguishable. The branching process is a birth and death process for I ; the variables S and R are not considered in this approximation.
The process begins with just a few infectious individuals. The branching process approximation is a CTMC, but near the disease-free equilibrium, the rates are linear Table 2. Each infectious individual has the same probability of recovery and the same probability of transmitting an infection.
Assumption 1 is reasonable if a small number of infectious individuals is introduced into a large homogeneously-mixed population assumption 3.
Two probability generating functions pgfs are used in the study of the probability of extinction. The first one applies to each infectious individual, known as the offspring pgf, and the second one applies to the entire infectious class I t at time t. For our purposes, the offspring pgf is the most important one.
In general, an offspring pgf has the form:. The pgf has some properties that are useful in the analysis. The first term in f u is the probability that an infectious individual recovers and the coefficient of the second term is the probability that an infectious individual infects another individual. The power to which u is raised is the number of infectious individuals generated from one infectious individual. If an individual recovers, then no new infections are generated u 0 and if the infection is transmitted to another individual, there are now two individuals infectious u 2.
The difference is due to the fact that in a small period of time, in a continuous-time process, the infectious individual that infects another person is counted as still being infectious two infectious individuals.
This latter expression is not the same as the basic reproduction number, the average number of infectious individuals generated by one infectious individual during the period of infectivity. It is well-known from the theory of branching processes that a fixed point of the offspring pgf yields the asymptotic probability of extinction Athreya and Ney, , Dorman et al. It is shown in Appendix B that the fixed points of f are the stationary solutions time-independent solutions of the branching process approximation for the probability of extinction of the infectious class I t.
The preceding results were first applied to the stochastic SIR epidemic model by Whittle in Whittle, As these estimates for extinction no outbreak are asymptotic approximations from the branching process, they are more accurate for a large susceptible population size N and a few infectious individuals.
The results are summarized below:. In general, for multivariate processes, it is difficult to find analytical solutions for the transition probabilities from the forward and backward Kolmogorov differential equations. For multivariate processes, it is often simpler to numerically simulate stochastic realizations sample paths of the process.
This method is known as the Gillespie algorithm or the Stochastic Simulation algorithm. See Appendix A for the derivation of the interevent time. The second random number u 2 tells which particular event occurs. If u 2 lies in the k th subinterval, then the k th event occurs. The close-up of the dynamics on the right side illustrates exponential growth at the initiation of an outbreak, the region where the branching process approximation is applicable.
The graphs on the right are a close-up view on the time interval [ 0,15 ] of the graphs on the left. Stochastic differential equations for the SIR epidemic model follow from a diffusion process. The random variables are continuous,.
Forward and backward Kolmogorov partial differential equations for the transition probability density functions can be derived and they in turn lead directly to the SDEs, e. The SDEs are useful in simulating sample paths of the continuous-state process.
In addition, the SDEs are much easier to solve numerically than the Kolmogorov differential equations and faster than simulating sample paths of the CTMC model. The following matrix G has this latter property G is not unique Allen, , Allen et al. Matrix G is straightforward to compute as each column represents the square root of the rates as given in Table 1 ,. The Euler-Maruyama method is a simple numerical method that can be used to simulate sample paths of SDEs. In general, for a system of SDEs of the form,.
The sample paths are continuous, but not differentiable a property of the Wiener process. Numerical simulation of sample paths for SDE models is faster and simpler than computing sample paths for CTMC models when the population size is large. Numerical methods with greater accuracy than the Euler-Maruyama method are discussed in the references, e. In addition, methods have been developed to speed up the stochastic simulation see, e. Malaria infection is caused by a Plasmodium parasite.
An infectious mosquito transmits the parasite to a susceptible host through a mosquito bite. It is the female mosquito that bites the host to acquire blood for reproduction. According to the website of the World Health Organization World Health Organization, there were approximately million new cases of malaria and , deaths worldwide in Most cases were reported in the African region.
Sir Ronald Ross was one of the first scientists to formulate mathematical models for the spread of malaria between insect vectors and human hosts Ross, , Ross, , Smith et al. In the Ross malaria model, the total number of vectors M and hosts H are constant. The variables S and I are the number of susceptible and infectious hosts, respectively, and U and V are the number of healthy and infectious vectors, respectively.
A female mosquito requires a certain number of blood meals for reproduction and it is assumed that a single mosquito takes k bites per unit time to fulfill this blood requirement. Another important assumption is that the total number of bites by the mosquito population is dependent on the total number of mosquitoes but it is not dependent on the number of human hosts only the proportion of human hosts. The probability per bite that an infectious mosquito transmits malaria is p and the probability per bite that a healthy mosquito acquires infection is q.
The birth rate and death rate of the mosquito population are equal. The natural birth and death rates of humans are negligible with respect to the modeling time frame and are assumed to be zero. We cannot process tax exempt orders online.
If you wish to place a tax exempt order please contact us. Add to cart. Dirk P. He has published over articles and four books in a wide range of areas in applied probability and statistics, including Monte Carlo methods, cross-entropy, randomized algorithms, tele-traffic c theory, reliability, computational statistics, applied probability, and stochastic modeling.
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