In this study, we investigate two mathematical models, formulated using delay differential equations, to capture HIV-1 transmission dynamics. Both models incorporate CD4 ${}^{+}$T cells and macrophages as target cells and saturated non-linear infection transmission terms. The first model introduces a discrete time delay, while the second employs a distributed delay to reflect more realistic biological insights. Our theoretical analysis explores key properties such as the basic reproduction number, non-negativity, boundedness, and the existence of equilibria. The global stability analysis of the disease-free and endemic equilibrium are analysed using the LaSalle invariance principle by proposing Lyapunov functions. We conducted sensitivity analysis to find the significant parameters related to the infection dynamics. Numerical simulations are performed to validate our theoretical results and visualize the behavior of the system under different parametric conditions. This study shows the critical role of time delays in shaping HIV-1 infection dynamics. Incorporation of such delays into mathematical models are essential for accurately capturing the progression of the disease and proposing effective intervention strategies for HIV-1.

A mathematical model is developed for analysis of the spread of mosaic disease in plants, which account for incubation period and latency that are represented by time delays. Feasibility and stability of different equilibria are studied analytically and numerically. Conditions that determine the type of behavior exhibited by the system are found in terms of various parameters. We have derived the basic reproduction number and identify the conditions resulting in eradication of the disease, as well as those that lead to the emergence of stable oscillations in the population of infected plants, as a result of Hopf bifurcation of the endemic equilibrium. Numerical simulations are performed to verify the analytical results and also to illustrate different dynamical regimes that can be observed in the system. In this research, the stabilizing role of both the time delay has been established i.e. when delay time is large, disease will persist if the infection rate is higher. The results obtained here are useful for plant disease management.

In this work, we have studied an eco-epidemic model using the Crowley-Martin functional response that includes disease in prey and gestation delay in the predator population. The model possesses three equilibria, namely the disease-free, Predator-free, and the interior equilibrium point. In addition, we examined the stability of the equilibrium points varying the infection rate and time delay parameter. Detailed analysis of Hopf bifurcation of the interior equilibrium point contains two situations: with delay and without delay. Moreover, we have studied the direction of the Hopf bifurcation and the stability of periodic solutions utilizing normal form theory and the center manifold theorem. It is emphasized that Hopf bifurcation occurs when the time delay exceeds the critical value and that the critical value of the delay is strongly impacted by the infection rate in prey. A detailed numerical simulation is provided to verify the analytical results.

Mathematical modeling and analysis of a crop-pest interacting system helps us to understand the dynamical properties of the system such as stability, bifurcations and chaos. In this article, a predator-prey type mathematical model for pest control using bio-pesticides has been analysed to study the global stability property of the interior equilibrium point. Moreover, the occurrence and orbital stability of Hopf bifurcating limit cycle solutions have been studied using ref30's conditions. Analytical and numerical results show that the interior equilibrium of the pest control model is globally asymptotically stable. Also, Hopf bifurcating occurs when the bifurcation parameter crosses the critical value, and the bifurcating periodic solution is found to be stable.

Plant viral infections are primarily propagated by adult insect vectors, which generally require a maturation period of approximately ten to twelve days. Following ingestion of the virus from an infected host plant, these vectors become capable of transmitting the pathogen to susceptible plants. In this study, a stage-structured mathematical model was formulated and analyzed to characterize the transmission dynamics of plant viral diseases mediated by adult insect vectors. Particular emphasis was placed on assessing how the maturation period of vectors influences the progression of infection transmission. To establish the mathematical validity of the model, it was shown to possess non-negative and bounded solutions, which confirms its well-posedness. We identified all steady states and studied their stability. The results show how infection rate, maturation rate, and maturation time can cause stability changes in steady states. Numerical stability and simulations were presented to analyze the behaviors of the system in different dynamical regimes. The stabilizing effect of the maturation period can help develop control methods for the management of plant viral diseases.