Limits of a multi-patch SIS epidemic model.

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

A brief introduction to the formulation of various types of stochastic epidemic models is presented based on the well-known deterministic SIS and SIR epidemic models. Three different types of stochastic model formulations are discussed: discrete time Markov chain, continuous time Markov chain and stochastic differential equations. Properties unique to the stochastic models are presented: probability of disease extinction, probability of disease outbreak, quasistationary probability distribution, final size distribution, and expected duration of an epidemic. The chapter ends with a discussion of two stochastic formulations that cannot be directly related to the SIS and SIR epidemic models. They are discrete time Markov chain formulations applied in the study of epidemics within households (chain binomial models) and in the prediction of the initial spread of an epidemic (branching processes).

We study the crossover phenomena from the dynamical percolation class (DyP) to the directed percolation class (DP) in the model of disease spreading, susceptible-infected-refractory-susceptible (SIRS) on a two-dimensional lattice. In this model, agents of three species S, I, and R on a lattice react as follows: S + I → I + I with probability λ, I → R after infection time τ I and R → I after recovery time τ R . Depending on the value of the parameter τ R , the SIRS model can be reduced to the following two well-known special cases. On the one hand, when τ R → 0 , the SIRS model reduces to the SIS model. On the other hand, when τ R → ∞ the model reduces to the SIR model. It is known that whereas the SIS model belongs to the DP universality class, the SIR model belongs to the DyP universality class. We can deduce from the model dynamics that SIRS will behave as the SIS model for any finite values of τ R . The model will behave as SIR only when τ R = ∞ . Using Monte Carlo simulations, we show that as long as the τ R is finite the SIRS belong to the DP university class. We also study the phase diagram and analyze the scaling behavior of this model along the critical line. By numerical simulations and analytical arguments, we find that the crossover from DyP to DP is described by the crossover exponent 1 / ϕ = 0.67 ( 2 ) .

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.

We introduce stochasticity into the SIS model with saturated incidence. The existence and uniqueness of the positive solution are proved by employing the Lyapunov analysis method. Then, we carry out a detailed analysis on both its almost sure exponential stability and itspth moment exponential stability, which indicates that thepth moment exponential stability implies the almost sure exponential stability. Additionally, the results show that the conditions for the disease to become extinct are much weaker than those in the corresponding deterministic model. The conditions for the persistence in the mean and the existence of a stationary distribution are also established. Finally, we derive the expressions for the mean and variance of the stationary distribution. Compared with the corresponding deterministic model, the threshold value for the disease to die out is affected by the half saturation constant. That is, increasing the saturation effect can reduce the disease transmission. Computer simulations are presented to illustrate our theoretical results.

Forest fire spreading is a complex phenomenon characterized by a stochastic behavior. Nowadays, the enormous quantity of georeferenced data and the availability of powerful techniques for their analysis can provide a very careful picture of forest fires opening the way to more realistic models. We propose a stochastic spreading model continuous in space and time that has the potentiality to use such data in their full power. The state of the forest fire is described by the subprobability densities of the green trees and of the trees on fire that can be estimated thanks to data coming from satellites and earth detectors. The fire dynamics is encoded into a kernel function that can take into account wind conditions, land slope, spotting phenomena, and so on, bringing to a system of integrodifferential equations whose solutions provide the evolution in time of the subprobability densities. That makes the model complementary to models based on cellular automata that furnish single instantiations of the stochastic phenomenon. Moreover, stochastic models based on cellular automata can be derived from the present model by space and time discretization. Existence and uniqueness of the solutions is proved by using Banach's fixed‐point theorem. The asymptotic behavior of the model is analyzed as well. By specifying a particular structure for the kernel, we obtain numerical simulations of the fire spreading under different conditions. For example, in the case of a forest fire evolving toward a river, the simulations show that the probability density of the trees on fire is different from zero beyond the river due to the spotting phenomenon. The kernel could be slightly modified to include firefighters interventions and weather changes.

A stochastic modelling approach for the characterisation of collision exchange processes

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.

Stochastic dynamics of an SIS epidemiological model with media coverage

A tropical cyclone (TC) vortex is an immense, coherent, organized-convective system. Beneath this large-scale organization, is a litany of azimuthally asymmetric convective motions that exist on a spectrum of scales. These asymmetries are especially dominant during periods when the vortex undergoes critical transitions in its intensity and structure. However, the precise nature of influence of the organization of asymmetries on TC intensity change remains an enigma. The inherent difficulty in predicting their behavior is because asymmetries may arise due to different external or intrinsic sources and occur at different spatial and temporal scales while several complex mechanisms act near-simultaneously to dictate their evolution in time. As a result, multiple pathways are possible for a TC vortex that is influenced by these asymmetries. Our preliminary investigations using numerical models made it apparent that there wasn't a single, unifying way to address this problem. In this thesis, I outline multiple novel techniques of diagnosing and predicting which of the many pathways are likely for a TC vortex that is influenced by azimuthal asymmetries. First, using three-dimensional numerical simulations of a pair of sheared and non-sheared vortices, I demonstrate the diagnostic potential of the juxtaposition in the azimuthal phasing of: (i) the asymmetrically distributed vertical eddy flux of moist-entropy across the top of the boundary layer, and the radial eddy flux of moist-entropy within the boundary layer; and (ii) eddy relative vorticity, eddy moist-entropy, and vertical velocity throughout the depth of the vortex. Second, I introduce an energetics-based diagnostic framework that computes the energy transactions occurring at asymmetries across various length-scales in the wavenumber domain. By applying it to select cases, this thesis uncovers the relative importance of all the energy pathways that support or disrupt the growth of asymmetries within the vortex. Contrary to the traditional explanations of convective aggregation/disaggregation and axi/asymmetrization through barotropic mean-eddy transactions, my thesis reveals that the growth or disruption of asymmetries are predominantly due to (i) the baroclinic conversion from available potential to kinetic energy at individual scales of asymmetries and (ii) the transactions of kinetic energy across asymmetries of different length scales. Finally, this thesis introduces two further diagnostic frameworks targeted at tackling the problem of real-time forecasting of TC rapid intensity changes. The first is an empirical framework which examines symmetric and asymmetric convection and other state variables within the vortex, and in the environment across a suite of TCs and identifies a set of `important' variables that are significantly different during time periods that precede a rapid intensification as opposed to a rapid weakening. My framework then ranks the variables identified based on how significantly they influence a rapid intensity change in a TC and the amplification factor of any associated variability. We recommend that future observational, and consequent TC modeling and data assimilation efforts prioritize the highest ranked variables identified here. The second is a stochastic model wherein a scale-specific stochastic term is added to the equations describing the energy transactions within the TC vortex. By simulating a stochastic forcing that may arise from any scale, I compute the probability of the vortex transitioning into a rapidly intensifying or a rapidly weakening configuration across an ensemble of scenarios. In summary, this thesis introduces and applies a variety of diagnostic techniques that help determine the impact of azimuthal asymmetries on TC intensity evolution.

The study of spreading dynamics is an important foundation to analyze the spread of viruses or rumors. The general method of analyzing spreading dynamics is based on model, such as SI model, SIS model, SIR model,ect. However, the spreading dynamics in the network is affected by many factors and represents some randomicity. So, in this paper, we propose Random Walks(RW) spreading model for opinions to illustrate the spreading behavior in networks, for instance, the random networks and BA scale-free networks. With MonteCarlo simulation method, the results show that the spreading speed of BA scale-free network is more quickly than that of ER random network, when the spreading probability β is small. But, with the increase of β, the spreading speed of ER random network increases gradually and exceeds that of BA scale-free network.

This chapter examines mathematical models that describe disease spread, beginning with the classical Kermack–McKendrick SIR model, which divides populations into Susceptible, Infected, and Removed compartments. It explains how the basic reproduction number (R0) determines whether an epidemic grows or dies out, and explores strategies to reduce R0, such as vaccination and social distancing. The chapter extends to SIR model variations, including the SEIR and SIS models, which account for incubation periods and diseases without lasting immunity, respectively. It also analyses real-world applications, such as Covid-19 lockdown measures and foot-and-mouth disease control, where mathematical models were used to guide policy decisions. It introduces structured models, where parameters may vary depending on the age of an individual or on their behaviour. Understanding how drugs are absorbed in the body and how they tackle disease is the subject of the area of Pharmacokinetic/Pharmacodynamic (PK/PD) modelling and the key building blocks of this approach are presented. The chapter also introduces within-host models, which aim to capture how a virus grows in the host. Coupling such models with those at the population level can then capture key aspects of the multiscale complexity of disease spread.

This volume presents explicit approximations of the quasi-stationary distribution and of the expected time to extinction from the state one and from quasi-stationarity for the stochastic logistic SIS

A stochastic SIS epidemic model with saturating contact rate

Bifurcation analysis of an SIS epidemic model with saturated incidence rate and saturated treatment function