A stochastic continuous-time Markov chain (CTMC) model, rooted in a deterministic approach, investigates rabies dynamics within human and stray dog populations. Stability analysis confirms the disease-free equilibrium is globally asymptotically stable for [Formula: see text], while an endemic equilibrium persists for [Formula: see text]. Through a Galton-Watson branching process, the stochastic threshold is derived, providing extinction probabilities that align closely with 10,000 numerical sample paths. The CTMC model predicts two distinct outcomes, disease extinction or persistence, and demonstrates a positive probability of rabies extinction in finite time even when [Formula: see text]. This contrast is significant because the deterministic model predicts a certain outbreak under identical conditions, whereas the stochastic framework captures the possibility of early extinction due to demographic fluctuations. Sensitivity analysis identifies the natural death rate and culling intensity as the most influential parameters. Results show a direct correlation between culling intensity and the probability of rabies elimination.
From an experimental point of view, it is difficult to investigate the propagation mechanism of infectious disease and predict its trend. However, applying a nonlinear model offers a more straightforward and effective approach for analyzing transmission characteristics of infectious diseases. This paper proposes an SLVEIMQR dynamic model by dividing the population into susceptible (S), low-risk (L), vaccinated (V), exposed (E), infected (I), mutant (M), quarantined (Q) and recovered (R). The basic reproduction number and the equilibrium of the proposed model are computed, and the local asymptotic stability of the disease-free equilibrium is proved. Furthermore, the transcritical bifurcation at equilibrium and the sensitivity of key parameters are analyzed. To estimate the model parameters and predict the trend of infectious disease, the least squares method is applied by using epidemic data from different phases in Italy. The results show that the average relative errors of the model in the two prediction stages are 1.72% and 3.66%, respectively.
This study investigates a discrete nonlinear resource-effort system with a state-dependent carrying capacity, where the maximum exploitation capacity adapts to resource availability. Unlike classical fixed-capacity models, the proposed framework incorporates adaptive feedback between resource and exploitation variables, providing a generalized approach for analyzing complex dynamics in nonlinear discrete systems. Using the method of piecewise constant argument, the continuous model is transformed into a nonlinear discrete map. Fundamental properties, including positivity, boundedness, equilibrium points and local stability, are analyzed. The loss of stability at the interior equilibrium is explored through flip and Neimark–Sacker bifurcations, leading to period-doubling cascades, quasiperiodic oscillations and chaos. Normal form theory is applied to determine bifurcation direction and stability. Numerical simulations using bifurcation diagrams, phase portraits, time series and maximum Lyapunov exponents confirm complex transitions. A feedback control strategy is further proposed to stabilize unstable dynamics and enlarge the stability region.
We formulate and analyse a mathematical model for the within-host co-infection dynamics of Zika virus (ZIKV) and chikungunya virus (CHIKV), incorporating latently infected cells and cytotoxic T lymphocyte (CTL) immune responses. We establish the nonnegativity and boundedness of solutions. Using the next-generation matrix method, we derive the basic reproduction numbers ([Formula: see text]) and ([Formula: see text]) for the ZIKV and CHIKV submodels, respectively, and the overall reproduction number ([Formula: see text]). Invasion reproduction numbers, ([Formula: see text]) and ([Formula: see text]), are derived to characterise the ability of either virus to invade an established infection. Global stability of the equilibria is established using Lyapunov functions. We investigate three therapeutic protocols: antiviral treatment blocking viral entry, treatment reducing viral production, and IL-2 therapy enhancing CTL responses. Sensitivity analysis of ([Formula: see text]) and ([Formula: see text]) is performed, and numerical simulations support the stability results. Simulations indicate that combined antiviral and immune-based interventions suppress co-infection by reducing viral replication and enhancing immune-mediated clearance.
Pine wilt disease is a destructive forest disease involving pine trees, pinewood nematodes, and insect vectors. This study develops deterministic and stochastic delayed host-vector models to examine waiting periods and variability in transmission. For the deterministic system, positivity, boundedness, equilibria, and the basic reproduction number R-0 are established. The disease-free equilibrium is locally and globally asymptotically stable when R-0<1 , while the endemic equilibrium is assessed through delay-dependent local stability. The stochastic model includes multiplicative environmental noise and is shown to possess positive solutions with bounded first moments. Exact It & ocirc; logarithmic identities demonstrate that long-term infection behaviour depends on current and delayed host-vector states. A delay-aware stochastic nonstandard finite difference scheme is constructed using delayed indices, nonstandard denominator functions, and positive stochastic exponential factors. Numerical experiments illustrate the effects of delay, noise, and step size, with the proposed scheme showing controlled behaviour at larger steps.
