We formulate a mathematical model to explore the impact of vaccination and treatment on the transmission dynamics of tuberculosis (TB). We develop a technique to prove that the basic reproduction number is the threshold of global stability of the disease-free and endemic equilibria. We then incorporate a control term and evaluate the cost of control strategies, and then perform an optimal control analysis by Pontryagin's maximum principle. Our numerical simulations suggest that the maximum vaccination strategy should be enforced regardless of its efficacy.

In a model of tuberculosis with relapse, the basic reproduction number R₀ includes new and relapse infections. Lyapunov functions help to prove that the global dynamic is completely determined by R₀. Replicated Latin hypercube sampling shows that early diagnosis and treatment are more efficient when relapse cases are considered.

The statistical data of monthly pulmonary tuberculosis (TB) incidence cases from January 2004 to December 2012 show the seasonality fluctuations in Shaanxi of China. A seasonality TB epidemic model with periodic varying contact rate, reactivation rate, and disease-induced death rate is proposed to explore the impact of seasonality on the transmission dynamics of TB. Simulations show that the basic reproduction number of time-averaged autonomous systems may underestimate or overestimate infection risks in some cases, which may be up to the value of period. The basic reproduction number of the seasonality model is appropriately given, which determines the extinction and uniform persistence of TB disease. If it is less than one, then the disease-free equilibrium is globally asymptotically stable; if it is greater than one, the system at least has a positive periodic solution and the disease will persist. Moreover, numerical simulations demonstrate these theorem results.

In this paper, stochastic finite-time stability analysis and stabilization problems for stochastic delayed recurrent neural networks(SDRNNs) are investigated, and the stochastic settling time function is also estimated. The sufficient condition of stochastic finite-time stable(SFTS) for SDRNNs is proposed at first. Then the nonlinear delay-feedback controller to stochastic finite-time stabilize the SDRNNs is designed. Finally, an numerical example is given to show the corrections of the proposed results and the effectiveness of the provided controller.

This paper investigates an optimal cooperative guidance strategy for the active defense of an early-warning aircraft (EWA) escorted by two fighters against an incoming missile. The proposed framework extends classical three-body defense models (Target–Missile–Interceptor) into a more realistic four-body engagement (Target–Missile–Interceptor 1–Interceptor 2), allowing explicit coordination among multiple defenders. By projecting the 3D engagement kinematics onto two orthogonal 2D planes—a validated simplification for typical aerial combat geometries—a tractable dynamic model is obtained. Within this model, an analytical cooperative guidance law is derived using optimal control theory and the calculus of variations, minimizing a multi-objective cost function that combines miss distance, control effort, intercept geometry, and coordination terms. Extensive Monte Carlo simulations across 23 attack directions and multiple initial ranges demonstrate that the proposed method achieves an interception success rate of 99%, with an average miss distance of below 5 m. Robustness tests further confirm stable performance under target maneuver uncertainty, sensor noise, and modeling deviations. The algorithm features closed-form control commands with low computational complexity, enabling real-time onboard implementation.