Exploring the effective strategies for allocating limited resources to minimize cholera outbreaks.

In this paper, a stochastic continuous-time Markov chain (CTMC) model is developed and analyzed to explore the dynamics of cholera. The multitype branching process is used to compute a stochastic threshold for the CTMC model. Latin hypercube sampling/partial rank correlation coefficient (LHS/PRCC) sensitivity analysis methods are implemented to derive sensitivity indices of model parameters. The results show that the natural death rate μv of a vector is the most sensitive parameter for controlling disease outbreaks. Numerical simulations indicate that the solutions of the CTMC stochastic model are relatively close to the solutions of the deterministic model. Numerical simulations estimate the probability of both disease extinction and outbreak. The probability of cholera extinction is high when it emerges from bacterial concentrations in non-contaminated/safe water in comparison to when it emerges from all infected groups. Thus, any intervention that focuses on reducing the number of infections at the beginning of a cholera outbreak is essential for reducing its transmission.

Single-type and multitype branching processes have been used to study the dynamics of a variety of stochastic birth-death type phenomena in biology and physics. Their use in epidemiology goes back to Whittle's study of a susceptible-infected-recovered (SIR) model in the 1950s. In the case of an SIR model, the presence of only one infectious class allows for the use of single-type branching processes. Multitype branching processes allow for multiple infectious classes and have latterly been used to study metapopulation models of disease. In this article, we develop a continuous time Markov chain (CTMC) model of infectious salmon anemia virus in two patches, two CTMC models in one patch and companion multitype branching process (MTBP) models. The CTMC models are related to deterministic models which inform the choice of parameters. The probability of extinction is computed for the CTMC via numerical methods and approximated by the MTBP in the supercritical regime. The stochastic models are treated as toy models, and the parameter choices are made to highlight regions of the parameter space where CTMC and MTBP agree or disagree, without regard to biological significance. Partial extinction events are defined and their relevance discussed. A case is made for calculating the probability of such events, noting that MTBPs are not suitable for making these calculations.

In this thesis, a single serotype (j = 1) and patch (i = 1) ordinary difference equation (ODE) model is formulated and analysed for the effects of direct and transplacental transmission on the probability of bluetongue virus (BTV) persistence. Using the next generation approach, the basic reproduction number (R0) is determined. When R0 < 1, the model exhibits a backward bifurcation indicating that the virus persists. When R0 > 1, a continuous-time Markov chain (CTMC) model derived from the ODE model is used to estimate the probability of BTV persistence. By approximating the CTMC model with a multitype branching process, it is shown that both direct and transplacental transmission can have a large effect on the probability of persistence in regions with temperature T < 12◦C and a small effect for those with T > 12◦C. The ODE and CTMC models are extended to include r serotypes and n patches with the aim of determining the effects of midge movement on the outbreak and coexistence of multiple BTV serotypes in an environment divided into patches depending on the risk of infection. An estimate for the probability of a major outbreak of two BTV serotypes in two patches is obtained by approximating the CTMC model with a multitype branching process. It is shown that without movement a major outbreak occurs in the high-risk patch, but with cattle or midge movement it occurs in both patches. When a major outbreak occurs, numerical simulations of the ODE model illustrate possible coexistence in both patches if the patches are connected by midge or cattle movement. The multi-patch single-serotype ODE model is then modified as an optimal control problem to evaluate the effectiveness of vaccination, quarantine, insecticide spraying and the use of repellent control strategies in reducing the within- and between-patches transmission. By using optimal control theory, the effectiveness of these strategies is established. In a single patch, vaccination, insecticide spraying and the use of a repellent are all highly effective in minimising transmission, but the most costeffective is vaccination. In patches connected by host and midge movements, if any of these controls is applied in the high-risk patch, a disease-free status is achieved in both patches, but if implemented in the low-risk patch, it is not attained in any patch. If hosts and midges move, quarantine has no effect, but for no midge movement, the effect can be large in the low-risk patch if it’s internally imposed.

This paper is concern with modeling cholera epidemic. Despite the advances made in understanding this disease and its treatment, cholera continues to be a major public health problem in many countries. Deterministic and stochastic models emerged in modeling of cholera epidemic, in order to understand the mechanism by which cholera disease spread, conditions for cholera disease to have minor and large outbreaks. We formulate a continuous time Markov chain model for cholera epidemic transmission from the deterministic model. The basic reproduction number (R0) and the extinction thresholds of corresponding cholera continuous time Markov chain model are derived under certain assumptions. We find that, the probability of extinction (no outbreak) is 1 if R0 < 1, but less than 1 if R0 > 1. We also carry out numerical simulations using Gillespie algorithm and Runge–Kutta method to generate the sample path of cholera continuous time Markov chain model and the solution of ordinary differential equation respectively. The results show that the sample path of continuous time Markov chain model fluctuates within the solution of the ordinary differential equation.

Stochastic analysis of in-host HCV dynamics through budding and bursting process

Determining the effects of wind-aided midge movement on the outbreak and coexistence of multiple bluetongue virus serotypes in patchy environments

The major goal of this study was to create a continuous time Markov chain (CTMC) models of voltage gating of gap junction (GJ) channels formed of connexin protein. This goal was achieved by using the Piece Linear Aggregate (PLA) formalism to describe the function of GJs and transforming PLA into Markov process. Infinitesimal generator of CTMC was used to automate construction of Markov chain model from description of the system using PLA formalism. Developed Markov chain models were used to simulate gap junctional conductance dependence on transjunctional voltage. The proposed method was implemented to create models of voltage gating of GJ channels containing 4 and 12 gates. CTMC modeling results were compared with the results obtained using a discrete time Markov chain (DTMC) model. It was shown that CTMC modeling requires less CPU time than an analogous DTMC model. DOI: http://dx.doi.org/10.5755/j01.itc.43.2.3198

Outbreak or extinction of bovine cysticercosis and human taeniasis: A Stochastic modelling approach

Indirect transmission through the environment, pathogen shedding by infectious hosts, replication of free-living pathogens within the environment, and environmental decontamination are suspected to play important roles in the spread and control of environmentally transmitted infectious diseases. To account for these factors, the classic Susceptible–Infectious–Recovered–Susceptible epidemic model is modified to include a compartment representing the amount of free-living pathogen within the environment. The model accounts for host demography, direct and indirect transmission, replication of free-living pathogens in the environment, and removal of free-living pathogens by natural death or environmental decontamination. Based on the assumptions of the deterministic model, a continuous-time Markov chain model is developed. An estimate for the probability of disease extinction or a major outbreak is obtained by approximating the Markov chain with a multitype branching process. Numerical simulations illustrate important differences between the deterministic and stochastic counterparts, relevant for outbreak prevention, that depend on indirect transmission, pathogen shedding by infectious hosts, replication of free-living pathogens, and environmental decontamination. The probability of a major outbreak is computed for salmonellosis in a herd of dairy cattle as well as cholera in a human population. An explicit expression for the probability of disease extinction or a major outbreak in terms of the model parameters is obtained for systems with no direct transmission or replication of free-living pathogens.

This paper analyzes the throughput performance of dense Wireless Local Area Networks (WLANs) with random topologies using the continuous time Markov chain (CTMC) model. Because the main factor that affects the accuracy of throughput analysis of dense WLANs is the interference caused by the simultaneous transmitting access points (APs). The accuracy decreases as the interference becomes increasingly significant. Therefore, we introduce coverage probability model into the CTMC model, thus capturing the effects of interference on the transmission process. In order to capture all the feasible CTMC states of different network topologies, we present a Feasible State Searching Algorithm (FSSA) which can calculate the number of states that each basic service set (BSS) belongs to and the total number of states in a CTMC model. Simulation results indicate that the proposed CTMC based throughput analysis method can capture the throughput properties of dense WLANs with random topologies.

Dynamic Spectrum Access (DSA) networks are vulnerable to hackers who normally pretend themselves to be the primary users and called the Primary User Emulation Attack (PUEA). Research communities have already reported a vast use of PUEA in the existing research. Other potential attackers such as greedy users should not be ignored when investigating the dynamic spectrum access networks. In this paper, we propose a multi-states Continuous Time Markov Chain (CTMC) model to describe the behaviour of DSA, analysis of the channel states and discussion on the impacts of normal, normal greedy and greedy malicious users in DSA network. The CTMC model is simulated and the simulation results have been discussed and validated by comparing with the existing models. Finally, it is proved that CTMC model is an improved method to analyse the performance of the DSA networks when PUEA occurs.

We present an experimental concurrent programming language LPEP which supports performance evaluation programming. An LPEP program is a collection of modules which are executed concurrently. The structure of an LPEP program is similar to the structure of a Continuous Time Markov Chain (CTMC) model expressed in the PRISM language (of the PRISM probabilistic model checker). However, LPEP is a programming language, not a model checker. Activities are abstracted in a PRISM CTMC model by their rates. In the language LPEP an activity is the evaluation of a function (expressed in a functional sub-language of LPEP ) in a certain state of a module. Each LPEP module contains a number of variables (which describe the possible states of the module) and a list of commands that can specify activities. LPEP supports variables of both primitive types (booleans and integers) and non-primitive types, e.g., lists. For the purpose of performance evaluation the programmer must bound the ranges of variables. Any value of a non-primitive type is handled according to its complexity (rather than its value). For example, the complexity of a list could be the length of the list, which must also be bounded. It is also the responsibility of the programmer to design each LPEP function by induction on an appropriate complexity measure which always decreases upon a recursive call. Performance evaluation is supported in LPEP by constructing a CTMC model from an LPEP program, and by analyzing the CTMC model using the PRISM tool.

Pasteurellosis transmission dynamics in free range chicken and wild birds: A deterministic and stochastic modeling approach

Considering common mode failure (CMF) in the rendering cluster systems, the availability of rendering cluster systems with the increase of cluster’s number was studied. Firstly, based on availability of system with one cluster node, system with two cluster nodes was modeled with continuous time markov chain (CTMC) model. Then, the CTMC model was extended to the case of system with three cluster nodes. Furthermore, by solving these three CTMC models, availability for different cases were numerically deduced. Additionally, during one year the unavailable time for different cases was calculated and analysis by comparison was conducted. Finally, conclusions on different cases’ advantages and disadvantages are derived thereby, which offers theoretical foundations for establishing rendering cluster systems.

Mycoplasma pneumoniae (Mp) is one of the most common causes of community-acquired pneumonia in children. To uncover the effective interventions during an epidemic in crowded settings, we first develop a novel staged progression ordinary differential equation model for the transmission of Mp, incorporating the effects of isolation measures and correct diagnosis rate. The basic reproduction number is obtained by the next generation matrix approach. Based on the deterministic model, a continuous-time Markov chain (CTMC) model is formulated to account for demographic variability. An analytic estimate for the probability of a disease outbreak, as well as an explicit expression for the mean (variance) of the disease extinction time in the absence of an outbreak, is derived by a multi-type branching process approximation of the CTMC model. By fitting the model to real data from a primary school, we estimate some key parameters of our model. Numerical simulations indicate that: (i) if the effects of demographic variability are ignored, the time to extinction after an outbreak is likely to be significantly underestimated or overestimated, depending on the isolation proportion; (ii) the impact of disease transmission rate, isolation proportion, and correct diagnosis rate on the probability of a disease outbreak depends on the stage of infection in which an infected individual is first introduced; (iii) decreasing the transmission rate, increasing the isolation proportion, or improving the correct diagnosis rate can significantly reduce the mean final size after an outbreak; and (iv) improving the correct diagnosis rate can help reduce the number of severe pneumonia cases.