An Introduction to Stochastic Epidemic Models

We study SIR and SIS epidemic models with multiple pathogen strains. In our models we assume total cross immunity, standard incidence, and density-dependent host mortality. We derive conditions on the models parameters which guarantee competitive exclusion between the n strains. An example is given to show that if these conditions are not satisfied then coexistence between the strains is possible. Furthermore, numerical results are presented to indicate that our conditions on the parameters are sufficient but not necessary for competitive exclusion.

Thresholds for disease extinction provide essential information for the prevention and control of diseases. In this paper, a stochastic epidemic model, a continuous-time Markov chain, for the transmission dynamics of West Nile virus in birds is developed based on the assumptions of its analogous deterministic model. The branching process is applied to derive the extinction threshold for the stochastic model and conditions for disease extinction or persistence. The probability of disease extinction computed from the branching process is shown to be in good agreement with the probability approximated from numerical simulations. The disease dynamics of both models are compared to ascertain the effect of demographic stochasticity on West Nile virus dynamics. Analytical and numerical results show differences in model predictions and asymptotic dynamics between stochastic and deterministic models that are crucial for the prevention of disease outbreaks. It is found that there is a high probability of disease extinction if the disease emerges from exposed mosquitoes unlike if it emerges from infectious mosquitoes and birds. Finite-time to disease extinction is estimated using sample paths and it is shown that the epidemic duration is shortest if the disease is introduced by exposed mosquitoes.

ABSTRACTIn epidemic modelling, the emergence of a disease is characterized by the low numbers of infectious individuals. Environmental randomness impacts outcomes such as the outbreak or extinction of the disease in this case. This randomness can be accounted for by modelling the system as a continuous time Markov chain, . The probability of extinction given some initial state is the probability of hitting a subset of the state space associated with extinction for the initial state. This hitting probability can be studied by passing to the discrete time Markov chain (DTMC), . An approach is presented to approximate a DTMC on a countably infinite state space by a DTMC on a finite state space for the purpose of solving general hitting problems. This approach is applied to approximate the probability of disease extinction in an epidemic model. It is also applied to evaluate a heterogeneous disease control strategy in a metapopulation.

This article presents an analysis of deterministic models of computer virus epidemic propagation (SIR models) and their classification. The main areas of research into these models are highlighted. An analysis of existing stochastic models based on the SIR model and their diversity are presented. A method for constructing a stochastic SIR model based on the classical SIR model, represented by a system of Ito stochastic differential equations with a Wiener process, is proposed. The possibility of constructing a stochastic and deterministic model of an information system protected against computer virus infection with probability 1 is demonstrated: a stochastic model, in which infection by viruses occurs continuously, and a deterministic model, in which the virus is present in the information system. A mathematical stochastic model of an information system protected from computer virus outbreaks is constructed as a system of stochastic differential equations whose first integrals are invariants preserved with probability 1. A certain functional relationship between model variables, maintaining a constant value, is considered as the system's security indicator. Introducing a compensator (program control with probability 1 (PCP1)) into the model allows the specified security indicator, described by the model variables, to be maintained with probability 1. Introducing a compensator (program control with probability 1 (PCP1)) into the model allows the specified security indicator, described by the model variables, to be maintained with probability 1. Similarly, based on the proposed algorithm, a deterministic model of an information system protected from computer virus infection is constructed. A control similar to programmed control with probability 1 (PCP1) is introduced into the constructed model, which allows the invariants to be maintained. A distinctive feature of the proposed models is that they preserve invariants associated with the properties that ensure the security of the information system. The behavior of the constructed models is studied using numerical simulation in the MathCad environment. Based on the research results, conclusions were drawn on the possibility of using the proposed method in constructing stochastic models based on other models of epidemic spread, as well as for models of protecting an information system from the spread of a computer virus epidemic.

Comparison of deterministic and stochastic SIS and SIR models in discrete time

The presence of a pathogen among multiple competing species has important ecological implications. For example, a pathogen may change the competitive outcome, resulting in replacement of a native species by a non-native species. Alternately, if a pathogen becomes established, there may be a drastic reduction in species numbers. Stochastic variability in the birth, death and pathogen transmission processes plays an important role in determining the success of species or pathogen invasion. We investigate these phenomena while studying the dynamics of deterministic and stochastic models for n competing species with a shared pathogen. The deterministic model is a system of ordinary differential equations for n competing species in which a single shared pathogen is transmitted among the n species. There is no immunity from infection, individuals either die or recover and become immediately susceptible, an SIS disease model. Analytical results about pathogen persistence or extinction are summarized for the deterministic model for two and three species and new results about stability of the infection-free state and invasion by one species of a system of n-1 species are obtained. New stochastic models are derived in the form of continuous-time Markov chains and stochastic differential equations. Branching process theory is applied to the continuous-time Markov chain model to estimate probabilities for pathogen extinction or species invasion. Finally, numerical simulations are conducted to explore the effect of disease on two-species competition, to illustrate some of the analytical results and to highlight some of the differences in the stochastic and deterministic models.

A stochastic epidemic model for the dynamics of two pathogens in a single tick population

Analysis of deterministic and stochastic mathematical models with resistant bacteria and bacteria debris for bacteriophage dynamics

A stochastic SIRS epidemic model with nonlinear incidence rate

ABSTRACTTwo Itô stochastic differential equation (SDE) systems are constructed for a Susceptible-Infected-Susceptible epidemic model with temporary vaccination. A constant number of new members enter the population and total size of the population is variable. Some conditions for disease extinction in the stochastic models are established and compared with conditions in deterministic one. It is shown that the two stochastic models are equivalent in the sense that their solutions come from same distribution. In addition, the SDE models are simulated and the equivalence of the two stochastic models is confirmed by numerical examples. The probability distribution for extinction is also obtained numerically, provided there exists a probability for disease persistence whereas the expected duration of epidemic is acquired when extinction occurs with probability 1.

A stochastic SIS epidemic model with saturating contact rate

We study a quantum entanglement switch that serves $k$ users in a star topology. We model variants of the system using Markov chains and standard queueing theory and obtain expressions for switch capacity and the expected number of qubits stored in memory at the switch. While it is more accurate to use a discrete-time Markov chain (DTMC) to model such systems, we quickly encounter practical constraints of using this technique and switch to using continuous-time Markov chains (CTMCs). Using CTMCs allows us to obtain a number of analytic results for systems in which the links are homogeneous or heterogeneous and for switches that have infinite or finite buffer sizes. In addition, we can model the effects of decoherence of quantum states fairly easily using CTMCs. We also compare the results we obtain from the DTMC against the CTMC in the case of homogeneous links and infinite buffer, and learn that the CTMC is a reasonable approximation of the DTMC. From numerical observations, we discover that decoherence has little effect on capacity and expected number of stored qubits for homogeneous systems. For heterogeneous systems, especially those operating close to stability constraints, buffer size and decoherence can have significant effects on performance metrics. We also learn that in general, increasing the buffer size from one to two qubits per link is advantageous to most systems, while increasing the buffer size further yields diminishing returns.

We study SIS epidemic spreading models under population dispersal on multi-layer networks. We consider a patchy environment in which each patch comprises individuals belonging to different classes. Individuals disperse to other patches on a multi-layer network in which each layer corresponds to a class. The dispersal on each layer is modeled by a Continuous Time Markov Chain (CTMC). At each time, individuals disperse according to their CTMC and subsequently interact with the local individuals in the patch according to an SIS model. We establish the existence of various equilibria under different parameter regimes and establish their (almost) global asymptotic stability using Lyapunov techniques. We also derive simple conditions that highlight the influence of the multi-layer network on the stability of these equilibria. For this model, we study optimal intervention strategies using a convex optimization framework. Finally, we numerically illustrate the influence of the multi-layer network structure and the effectiveness of the optimal intervention strategies.

In previous work a deterministic model for the compartment level was built, taking into account the two different syndromes with which Salmonella Typhimurium appears at pig farms. Based on this model, a stochastic one was built in this work that simulated different compartmental sizes, taking into account compartments of 200 to 400 pigs. Multiple scenarios of starting conditions of infection (SCI) ranging from 0.25 to 100% were tested for each population size. The effect of each of these two factors on the probability of disease extinctions and the prevalence of each of the classes of the model and the risk groups of pigs were estimated. The results showed that the compartment population had an inverse effect on the probability of disease extinction. On the other hand, low SCI resulted in high levels of early extinctions reaching 45%, while higher SCI led to high levels of late extinctions. Early extinctions resulted in the absence of the pathogen from the compartment, while late extinctions did not assure it. This effect shows that reducing the population of the compartment combined with appropriate cleaning and good farming practices could have a positive effect in the reduction of the risk of introducing S. Typhimurium into the slaughtering procedure. On the other hand, the profile of seroprevalence at slaughter age allows for risk characterization of the farm, given the relative stability and the small variation for higher SCI.

An approach to analysis of impacts of preventive maintenance on safety integrity level of a control system is presented in the paper. Suggested approach incorporates combination of Continuous Time and Discrete Time Markov Chain which allows full employment of advantages of Continuous Time Markov Chains as a stochastic modelling method. Deterministic behaviour of either periodic or non-periodic preventive maintenance is consecutively implemented through Discrete Time Markov Chain.