Dengue fever, a vector-borne disease, has affected the whole world in general and the Indian subcontinent in particular for the last three decades. Dengue fever has a significant economic and health impact worldwide; it is essential to develop new mathematical models to study not only the dynamics of the disease but also to suggest cost-effective mechanisms to control disease. In this paper, we design modified facts about the dynamics of this disease more realistically by formulating a new basic $S_h{E}_h{I}_h{R}_h$ host population and $S_v{I}_v$ vector population integer order model, later converting it into a fractional-order model with the help of the well-known Atangana-Baleanu derivative. In this design, we introduce two more compartments, such as the treatment compartment $T_h$, and the protected traveler compartment $P_h$ in the host population to produce $S_h{E}_h{I}_h{T}_h{R}_h{P}_h$. We present some observational results by investigating the model for the existence of a unique solution as well as by proving the positivity and boundedness of the solution. We compute reproduction number ${\mathcal{R}}_0$ by using a next-generation matrix method to estimate the contagious behavior of the infected humans by the disease. In addition, we prove that disease free and endemic equilibrium points are locally and globally stable with restriction to reproduction number ${\mathcal{R}}_0$. The second goal of this article is to formulate an optimal control problem to study the effect of the control strategy. We implement the Toufik-Atangana scheme for the first time to solve both of the state and adjoint fractional differential equations with the ABC derivative operator. The numerical results show that the fractional order and the different constant treatment rates affect the dynamics of the disease. With an increase in the fractional order and the treatment rate, exposed and infected humans, as well as the infected mosquitoes, decrease. However, the optimal control analysis reveals that the implemented optimal control strategy is very effective for disease control.

In the present paper, we focus on the reducibility of an almost-periodic linear Hamiltonian system

$$\frac{dX}{dt}=J[A+\epsilon Q(t)]X,X\in {\mathbb{R}}^{2d},$$

where $J$ is an anti-symmetric symplectic matrix, $A$ is a symmetric matrix, $Q(t)$ is an analytic almost-periodic matrix with respect to $t$, and $\epsilon$ is a parameter which is sufficiently small. Using some non-resonant and non-degeneracy conditions, rapidly convergent methods prove that, for most sufficiently small $\epsilon$, the Hamiltonian system is reducible to a constant coefficients Hamiltonian system through an almost-periodic symplectic transformation with similar frequencies as $Q(t)$. At the end, an application to Schrödinger equation is given.

A co-infection with Covid-19 and dengue fever has had worse outcomes due to high mortality rates and longer stays either in isolation or at hospitals. This poses a great threat to a country's economy. To effectively deal with these threats, comprehensive approaches to prevent and control Covid-19/dengue fever co-infections are desperately needed. Thus, our focus is to formulate a new co-infection fractional model with the Atangana-Baleanu derivative to suggest effective and feasible approaches to restrict the spread of co-infection. In the first part of this paper, we present Covid-19 and dengue fever sub-models, as well as the co-infection model that is locally asymptotically stable when the respective reproduction numbers are less than unity. We establish the existence and uniqueness results for the solutions of the co-infection model. We extend the model to include a vaccination compartment for the Covid-19 vaccine to susceptible individuals and a treatment compartment to treat dengue-infected individuals as optimal control strategies for disease control. We outline the fundamental requirements for the fractional optimal control problem and illustrate the optimality system for the co-infection model using Pontraygin's principle. We implement the Toufik-Atangana approximating scheme to simulate the optimality system. The simulations show the effectiveness of the implemented strategy in determining optimal vaccination and treatment rates that decrease the cost functional to a minimum, thus significantly decreasing the number of infected humans and vectors. Additionally, we visualize a meaningful decrease in infection cases with an increase in the memory index. The findings of this study will provide reasonable disease control suggestions to regions facing Covid-19 and dengue fever co-infection.

Malaria is a significant global health challenge. This devastating disease continues to affect millions, especially in tropical regions. It is caused by Plasmodium parasites transmitted by female Anopheles mosquitoes. This study introduces a nonlinear mathematical model for examining the transmission dynamics of malaria, incorporating both human and mosquito populations. We aim to identify the key factors driving the endemic spread of malaria, determine feasible solutions, and provide insights that lead to the development of effective prevention and management strategies. We derive the basic reproductive number employing the next-generation matrix approach and identify the disease-free and endemic equilibrium points. Stability analyses indicate that the disease-free equilibrium is locally and globally stable when the reproductive number is below one, whereas an endemic equilibrium persists when this threshold is exceeded. Sensitivity analysis identifies the most influential mosquito-related parameters, particularly the bite rate and mosquito mortality, in controlling the spread of malaria. Furthermore, we extend our model to include a treatment compartment and three disease-preventive control variables such as antimalaria drug treatments, use of larvicides, and the use of insecticide-treated mosquito nets for optimal control analysis. The results show that optimal use of mosquito nets, use of larvicides for mosquito population control, and treatment can lower the basic reproduction number and control malaria transmission with minimal intervention costs. The analysis of disease control strategies and findings offers valuable information for policymakers in designing cost-effective strategies to combat malaria.