The effect of demographic and environmental variability on disease outbreak for a dengue model with a seasonally varying vector population

A stochastic multi-host model for West Nile virus transmission

When initially introduced into a susceptible population, a disease may die out or result in a major outbreak. We present a Continuous-Time Markov Chain model for enzootic WNV transmission between two avian host species and a single vector, and use multitype branching process theory to determine the probability of disease extinction based upon the type of infected individual initially introducing the disease into the population - an exposed vector, infectious vector, or infectious host of either species. We explore how the likelihood of disease extinction depends on the ability of each host species to transmit WNV, vector biting rates on host species, and the relative abundance of host species, as well as vector abundance. Theoretical predictions are compared to the outcome of stochastic simulations. We find the community composition of hosts and vectors, as well as the means of disease introduction, can greatly affect the probability of disease extinction.

Lyme Disease Models of Tick-Mouse Dynamics with Seasonal Variation in Births, Deaths, and Tick Feeding

Lyme disease is the most common vector-borne disease in the United States impacting the Northeast and Midwest at the highest rates. Recently, it has become established in southeastern and south-central regions of Canada. In these regions, Lyme disease is caused by Borrelia burgdorferi, which is transmitted to humans by an infected Ixodes scapularis tick. Understanding the parasite-host interaction is critical as the white-footed mouse is one of the most competent reservoir for B. burgdorferi. The cycle of infection is driven by tick larvae feeding on infected mice that molt into infected nymphs and then transmit the disease to another susceptible host such as mice or humans. Lyme disease in humans is generally caused by the bite of an infected nymph. The main aim of this investigation is to study how diapause delays and demographic and seasonal variability in tick births, deaths, and feedings impact the infection dynamics of the tick-mouse cycle. We model tick-mouse dynamics with fixed diapause delays and more realistic Erlang distributed delays through delay and ordinary differential equations (ODEs). To account for demographic and seasonal variability, the ODEs are generalized to a continuous-time Markov chain (CTMC). The basic reproduction number and parameter sensitivity analysis are computed for the ODEs. The CTMC is used to investigate the probability of Lyme disease emergence when ticks and mice are introduced, a few of which are infected. The probability of disease emergence is highly dependent on the time and the infected species introduced. Infected mice introduced during the summer season result in the highest probability of disease emergence.

A practical guide to mathematical methods for estimating infectious disease outbreak risks

Mathematical models are increasingly used throughout infectious disease outbreaks to guide control measures. In this review article, we focus on the initial stages of an outbreak, when a pathogen has just been observed in a new location (e.g., a town, region or country). We provide a beginner's guide to two methods for estimating the risk that introduced cases lead to sustained local transmission (i.e., the probability of a major outbreak), as opposed to the outbreak fading out with only a small number of cases. We discuss how these simple methods can be extended for epidemiological models with any level of complexity, facilitating their wider use, and describe how estimates of the probability of a major outbreak can be used to guide pathogen surveillance and control strategies. We also give an overview of previous applications of these approaches. This guide is intended to help quantitative researchers develop their own epidemiological models and use them to estimate the risks associated with pathogens arriving in new host populations. The development of these models is crucial for future outbreak preparedness. This manuscript was submitted as part of a theme issue on "Modelling COVID-19 and Preparedness for Future Pandemics".

Impact of demographic variability on the disease dynamics for honeybee model

For the last few years, annual honeybee colony losses have been center of key interest for many researchers throughout the world. The spread of the parasitic mite and its interaction with specific honeybee viruses carried by Varroa mites has been linked to the decline of honeybee colonies. In this investigation, we consider honeybee-virus and honeybee-infected mite-virus models. We perform sensitivity analysis locally and globally to see the effect of the parameters on the basic reproduction number for both models and to understand the disease dynamics in detail. We use the continuous-time Markov chain model to develop and analyze stochastic epidemic models corresponding to both deterministic models. By using the disease extinction process, we compare both deterministic and stochastic models. We have observed that the numerically approximated probability of disease extinction based on 30 000 sample paths agrees well with the calculated probability using multitype branching process approximation. In particular, it is observed that the disease extinction probability is higher when infected honeybees spread the disease instead of infected mites. We conduct a sensitivity analysis for the stochastic model also to examine how the system parameters affect the probability of disease extinction. We have also derived the equation for the expected time required to reach disease-free equilibrium for stochastic models. Finally, the effect of the parameters on the expected time is represented graphically.

Stochastic dynamics of an SIS epidemic on networks

We derive a stochastic SIS pairwise model by considering the change of the variables of this system caused by an event. Based on approximations, we construct a low-dimensional deterministic system that can be used to describe the epidemic spread on a regular network. The mathematical treatment of the model yields explicit expressions for the variances of each variable at equilibrium. Then a comparison between the stochastic pairwise model and the stochastic mean-field SIS model is performed to indicate the effect of network structure. We find that the variances of the prevalence of infection for these two models are almost equal when the number of neighbors of every individual is large. Furthermore, approximations for the quasi-stationary distribution of the number of infected individuals and the expected time to extinction starting in quasi-stationary are derived. We analyze the approximations for the critical number of neighbors and the persistence threshold based on the stochastic model. The approximate performance is then examined by numerical and stochastic simulations. Moreover, during the early development phase, the temporal variance of the infection is also obtained. The simulations show that our analytical results are asymptotically accurate and reasonable.

A Comparison of Deterministic and Stochastic Plant-Vector-Virus Models Based on Probability of Disease Extinction and Outbreak

In this investigation, we formulate and analyse a stochastic epidemic model using the continuous-time Markov chain model for the propagation of a vector-borne cassava mosaic disease in a single population. The stochastic model is based upon a pre-existing deterministic plant-vector-virus model. To see how demographic stochasticity affects the vector-borne cassava mosaic disease dynamics, we compare the disease dynamics of both deterministic and stochastic models through disease extinction process. The probability of disease extinction and therefore the major outbreak are estimated analytically using the multitype Galton-Watson branching process (GWbp) approximation. Also, we have found the approximate probabilities of disease extinction numerically based on 30000 sample paths, and it is shown to be good estimate with the calculated probabilities from GWbp approximation. In particular, it is observed that there is a very high probability of disease extinction when the disease is introduced via the infected vectors rather than through infected plants.

Three-dimensional spread analysis of a Dengue disease model with numerical season control

Dengue is among the most important infectious diseases in the world. The main contribution of our paper is to present a mixed system of partial and ordinary differential equations. This combined model is a generalization of the two presented mathematical models (A. L. de Araujo, J. L. Boldrini and B. M. Calsavara, An analysis of a mathematical model describing the geographic spread of dengue disease, J. Math. Anal. Appl. 444 (2016) 298–325) and (L. Cai, X. Li, N. Tuncer, M. Martcheva and A. A. Lashari, Optimal control of a malaria model with asymptomatic class and superinfection, Math. Biosci. 288 (2017) 94–108), describing the geographic spread of dengue disease. Our model has the ability to consider the possibility of asymptomatic infection, which leads to investigate the effect of dengue asymptomatic individuals on disease dynamics and to go into the possibility of superinfection of asymptomatic individuals. In the light of considering these factors, as well as the movements of human and mature female mosquitoes, more realistic modeling of dengue disease can be achieved. We present a mathematical analysis and show the global existence of a unique non-negative solution to this model and then establish ways to control dengue disease using numerical simulations and sensitivity analysis of model parameters (which are related to the contact rates and death rate of winged mosquitoes). To show different biological behaviors, we provide several numerical results, showing the role of parameters in controlling dengue disease transmission. From our numerical simulations, it can also be concluded that local control of dengue transmission can be done at a lower cost.