A mathematical model for forecasting the transmission of the COVID-19 outbreak is proposed to investigate the effects of quarantined and hospitalized individuals. We analyze the proposed model by considering the existence and the positivity of the solution. Then, the basic reproduction number $\left(R_0\right)$—the expected number of secondary cases produced by a single infection in a completely susceptible population—is computed by using the next-generation matrix to carry out the stability of disease-free equilibrium and endemic equilibrium. The results show that the disease-free equilibrium is locally asymptotically stable if $R_0<1$, and the endemic equilibrium is locally asymptotically stable if $R_0>1$. Numerical simulations of the proposed model are illustrated. The sensitivity of the model parameters is considered in order to control the spread by intervention strategies. Numerical results confirm that the model is suitable for the outbreak that occurred in Thailand.

In this paper, we propose an impulsive mathematical model of bone formation and resorption accounting for the number of active osteoclastic cells, bone resorbing cells, and the number of active osteoblastic cells, bone forming cells, based on the effects of parathyroid hormone and calcitonin with impulsive estrogen supplement. The model is then analyzed theoretically in terms of its stability, permanence and oscillatory behavior. The conditions on the model parameters, for which the desirable behaviors of the solution of the system can be expected, are derived. Numerical simulations are also carried out in order to support our theoretical predictions. The results indicate that the frequency and dosage of the estrogen supplements are important since the behavior of the solution of the system depends on the frequency and dosage of the estrogen supplements.

This study examines an epidemiological model known as the susceptible-exposed-infected-hospitalized-recovered (SEIHR) model, with and without impulsive vaccination strategies. First, the model was analyzed without impulsive vaccination in the presence of a reinfection effect. Subsequently, it was studied as part of a periodic impulsive vaccination strategy targeting the susceptible population. These vaccination impulses were administered in very brief intervals at specific time instants, with a fixed time gap between each impulse. The two approaches can be modified to respond to different amounts of susceptibility, with control efforts intensifying as susceptibility levels rise. The model's analysis includes crucial aspects such as the non-negativity of solutions, the existence of steady states, and the stability corresponding to the basic reproduction number. We demonstrate that when vaccination measures are taken into account, the basic reproduction number remains as less than one. Therefore, the disease-free equilibrium in the case of vaccination could still be asymptotically stable at the higher disease transmission rate, as compared to the case of no vaccination in which the disease-free equilibrium may no longer be asymptotically stable. Furthermore, we show that when the disease-free equilibrium is stable, the endemic equilibrium cannot be attained, and that when the reproduction number rises above unity, the disease-free equilibrium becomes unstable while the endemic equilibrium becomes stable. We have also derived conditions for the global stability of both equilibriums. To support our theoretical results, we have constructed a time series of numerical simulations and compared them with real-world data from the ongoing SARS-CoV-2 (COVID-19) pandemic.

In this paper, a mathematical model for African swine fever is modified by considering the swine farm with the contaminated human vector that is able to infect and spread the disease among swine farms. In the developed model, we have divided the swine farm density into three related groups, namely the susceptible swine farm compartment, latent swine farm compartment, and infectious swine farm compartment. On the other hand, the human vector population density has been separated into two classes, namely the susceptible human vector compartment and the infectious human vector compartment. After that, we use this model and a quarantine strategy to analyze the spread of the infection. In addition, the basic reproduction number $R_0$ is determined by using the next-generation matrix, which can analyze the stability of the model. Finally, the numerical simulations of the proposed model are illustrated to confirm the results from theorems. The results showed that the transmission coefficient values per unit of time per individual between the human vector and the swine farm resulted in the spread of African swine fever.

In this paper, a mathematical model for Streptococcus suis infection is improved by using the fractional order derivative. The modified model also investigates the transmission between pigs and humans. The proposed model can classify the pig population density into four classes, which are pig susceptible class, pig infectious class, pig quarantine class, and pig recovery class. Moreover, the human population density has been separated into three classes, these are human susceptible class, human infectious class, and human recovery class. The spread of the infection is analyzed by considering the contact between humans and pigs. The basic reproduction number $\left(R_0\right)$, the infectious indicator, is carried out using the next generation matrix. The disease-free equilibrium is locally asymptotically stable if $R_0<1$, and the endemic equilibrium is locally asymptotically stable if $R_0>1$. The theoretical analyses of the fractional order derivative model, existence and uniqueness, have been proposed. The numerical examples were illustrated to support the proposed stability theorems. The results show that the fractional order derivative model provides the various possible solution trajectories with different fractional orders for the same parameters. In addition, transmission between pigs and humans resulted in the spread of Streptococcus suis infection.

Bone remodeling is a dynamic biological process that preserves bone strength and structure through the coordinated actions of osteoblasts, osteoclasts, osteocytes, and bone mass density. Traditional models based on ordinary differential equations often fail to capture the memory-dependent nature of these interactions. In this study, we propose a novel mathematical model of bone remodeling using the Atangana–Baleanu–Caputo fractional derivative, which accounts for the non-local and hereditary characteristics of biological systems. The model introduces fractional-order dynamics into a previously established ODE framework while maintaining the intrinsic symmetry between bone-forming and bone-resorbing mechanisms, as well as the balance mediated by porosity-related feedback. We establish the existence, uniqueness, and positivity of solutions, and analyze the equilibrium points and their global stability using a Lyapunov function. Numerical simulations under various fractional orders demonstrate symmetric convergence toward equilibrium across all biological variables. The results confirm that fractional-order modeling provides a more accurate and balanced representation of bone remodeling and reveal the underlying symmetry in the regulation of bone tissue. This work contributes to the growing use of fractional calculus in modeling physiological processes and highlights the importance of symmetry in both mathematical structure and biological behavior.