The chikungunya virus (CHIKV) infects macrophages and adherent cells and it can be transmitted via a direct contact with the virus or with an already infected cell. Thus, the CHIKV infection can have two routes. Furthermore, it can exhibit seasonal peak periods. Thus, in this paper, we consider a dynamical system model of the CHIKV dynamics under the conditions of a seasonal environment with a general incidence rate and two routes of infection. In the first step, we studied the autonomous system by investigating the global stability of the steady states with respect to the basic reproduction number. In the second step, we establish the existence, uniqueness, positivity and boundedness of a periodic orbit for the non-autonomous system. We show that the global dynamics are determined by using the basic reproduction number denoted by ${\mathcal{R}}_0$ and they are calculated using the spectral radius of an integral operator. We show the global stability of the disease-free periodic solution if ${\mathcal{R}}_0<1$ and we also show the persistence of the disease if ${\mathcal{R}}_0>1$ where the trajectories converge to a limit cycle. Finally, we display some numerical investigations supporting the theoretical findings.

In order to study the impact of seasonality on Zika virus dynamics, we analyzed a non-autonomous mathematical model for the Zika virus (ZIKV) transmission where we considered time-dependent parameters. We proved that the system admitted a unique bounded positive solution and a global attractor set. The basic reproduction number, ${\mathcal{R}}_0$, was defined using the next generation matrix method for the case of fixed environment and as the spectral radius of a linear integral operator for the case of seasonal environment. We proved that if ${\mathcal{R}}_0$ was smaller than the unity, then a disease-free periodic solution was globally asymptotically stable, while if ${\mathcal{R}}_0$ was greater than the unity, then the disease persisted. We validated the theoretical findings using several numerical examples.

In this paper, we develop a comprehensive mathematical model to investigate the transmission dynamics of dual Nosema infections (Nosema apis and Nosema ceranae) in two interacting honeybee colonies. The model incorporates distributed time delays to capture biological realism in latency, incubation, and parasite maturation periods, and includes an environmental pathogen compartment to account for indirect, environment-mediated transmission. First, we analyze a simplified ordinary differential equation (ODE) version of the model, thereby deriving the basic reproduction number ${\mathcal{R}}_0$ and establishing the global asymptotic stability of both disease-free and endemic equilibria using Lyapunov functions. Then, the analysis is extended to the full distributed-delay system, where we derive the delayed basic reproduction number ${\mathcal{R}}_0^d$ and prove the global stability of its equilibria via carefully constructed Lyapunov functionals. A sensitivity analysis identifies key parameters—most notably transmission rates, spore shedding rates, and natural mortality—that dominate the infection dynamics. Furthermore, we introduce an antiviral treatment term to quantify the efficacy required to drive ${\mathcal{R}}_0^d$ below unity and achieve disease eradication. Numerical simulations validate the analytical results and illustrate how distributed delays and treatment interventions critically influence the long-term disease outcomes. The study provides a robust theoretical framework to understand Nosema spread in multi-colony settings. Its key contributions are as follows: (1) The derivation of an additive basic reproduction number reveals the necessity of apiary-wide management; (2) provides rigorous global stability proofs for the delayed system; and (3) provides actionable quantitative insights to design effective apiary management, identify critical intervention targets, and establish treatment efficacy thresholds for disease eradication.