The effect of demographic stochasticity on Zika virus transmission dynamics: Probability of disease extinction, sensitivity analysis, and mean first passage time.

Viral infections spread by mosquitoes are a growing threat to human health and welfare. Zika virus (ZIKV) is one of them and has become a global worry, particularly for women who are pregnant. To study ZIKV dynamics in the presence of demographic stochasticity, we consider an established ZIKV transmission model that takes into consideration the disease transmission from human to mosquito, mosquito to human, and human to human. In this study, we look at the local stability of the disease-free and endemic equilibriums. By conducting the sensitivity analysis both locally and globally, we assess the effect of the model parameters on the model outcomes. In this work, we use the continuous-time Markov chain (CTMC) process to develop and analyze a stochastic model. The main distinction between deterministic and stochastic models is that, in the absence of any preventive measures such as avoiding travel to infected areas, being careful from mosquito bites, taking precautions to reduce the risk of sexual transmission, and seeking medical care for any acute illness with a rash or fever, the stochastic model shows the possibility of disease extinction in a finite amount of time, unlike the deterministic model shows disease persistence. We found that the numerically estimated disease extinction probability agrees well with the analytical probability obtained from the Galton-Watson branching process approximation. We have discovered that the disease extinction probability is high if the disease emerges from infected mosquitoes rather than infected humans. In the context of the stochastic model, we derive the implicit equation of the mean first passage time, which computes the average amount of time needed for a system to undergo its first state transition.

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 sensitivity analysis and early warning signals of critical transitions in a tri-stable prey-predator system with noise.

Near a tipping point, small changes in a certain parameter cause an irreversible shift in the behavior of a system, called critical transitions. Critical transitions can be observed in a variety of complex dynamical systems, ranging from ecology to financial markets, climate change, molecular bio-systems, health, and disease. As critical transitions can occur suddenly and are hard to manage, it is important to predict their occurrence. Although it is very tough to predict such critical transitions, various recent works suggest that generic early warning signals can detect the situation when systems approach a critical point. The most important indicator that predicts the risk of an upcoming critical transition is critical slowing down (CSD). CSD indicates a slow recovery rate from external perturbations of the stable state close to a bifurcation point. In this contribution, we study a two dimensional prey-predator model. Without any noise, the prey-predator model shows bistability and tri-stability due to the Allee effect in predators. We explore the critical transitions when external noise is added to the prey-predator system. We investigate early warning indicators, e.g., recovery rate, lag-1 autocorrelation, variance, and skewness to predict the critical transition. We explore the confidence domain method using the stochastic sensitivity function (SSF) technique near a stable equilibrium point to find a threshold value of noise intensity for a transition. The SSF technique in a two stage transition through confidence ellipse is described. We also show that the possibility of a transition to the predator-free state is independent of initial conditions. Our result may serve as a paradigm to understand and predict the critical transition in a two dimensional system.