Narrow Results

This paper incorporates generalized nonlinear vaccination rate and temporary immunity into the infectious disease models and presents the corresponding system. The qualitative results show that the disease-free equilibrium point and the endemic equilibrium point are globally asymptotically stable under some certain conditions. The local sensitivity analysis reveals that the importance of the critical parameters to the effective reproduction number $R_v$ in the order from high to low is $\mu$, $\mathrm{\Lambda }$, $𝜃$, $\gamma$ and $\xi$, respectively. The global sensitivity analysis shows that the parameters which impact on the effective reproduction number $R_v$ are $\alpha$, $𝜃$, $\mu$, $\beta$, $\mathrm{\Lambda }$, $p$, $\alpha$, $\xi$ and $\gamma$ in the descending order. Based on the dynamical behavior and the sensitivity analysis of the considered system, the controlling measures are presented to prevent and control the infectious diseases: (1) increasing the nonlinear vaccination rate of the susceptible subpopulation, (2) controlling the number of immigrants to infectious areas, (3) increasing vaccination coverage of immigrants and newborns and (4) improving the effectiveness of vaccines and reducing the rate of immune loss.

In this paper, we study the dynamical behavior of solutions of nonlinear Schrödinger equations with quadratic interaction and $L^2$-critical growth. We give sharp conditions under which the existence of global and blow-up solutions are deduced. We also show the existence, stability, and blow-up behavior of normalized solutions of this system.

Cellular automata are very simple systems that can exhibit complex dynamics on its time evolution. Over the last decade there have been many applications of cellular automata to modeling of biological systems. Those applications have been stimulated by the study of complex systems which has brought many insights into the cooperative and global behavior of the biological systems. Along with this discussion we present two different applications of deterministic and also of probabilistic cellular automata that are used to model the dynamics involved in cooperative and collective behavior of the immune system. In the first example, we use a deterministic cellular automata to model the time evolution of the immune repertoire, as a network, according to the Jerne's theory. Using this model we could reproduce some recent experimental results about immunization and aging of the immune system. In the second example, we use a probabilistic cellular automata model to study the evolution of HIV infection and the onset of AIDS. The results are in excellent agreement with experimental data obtained from infected patients. Besides the examples above, other interesting applications, such as models for cancer and recurrent epidemics, are being considered in the present framework.

Usually, in order to investigate the evolution of a theory, one may find the critical points of the system and then perform perturbations around these critical points to see whether they are stable or not. This local method is very useful when the initial values of the dynamical variables are not far away from the critical points. Essentially, the nonlinear effects are totally neglected in such kind of approach. Therefore, one cannot tell whether the dynamical system will evolute to the stable critical points or not when the initial values of the variables do not close enough to these critical points. Furthermore, when there are two or more stable critical points in the system, local analysis cannot provide the information on which the system will finally evolute to. In this paper, we have further developed the nullcline method to study the bifurcation phenomenon and global dynamical behavior of the f(T) theory. We overcome the shortcoming of local analysis. And, it is very clear to see the evolution of the system under any initial conditions.

In this paper, we investigate the complex dynamical behaviors of a biological network that is derived from innate immune responses and that couples positive and negative feedback loops. The stability conditions of the non-negative equilibrium points (EPs) of the system are obtained, using the theory of dynamical systems, and we deduce that no more than three stable EPs exist in this system. Through bifurcation analysis and numerical simulations, we find that the system presents rich dynamical behaviors, such as monostability, bistability and oscillations. These results reveal how positive and negative feedback cooperatively regulate the dynamical behavior of the system.

It is a significant and challenging task to detect both the coexistence of singular cycles, mainly homoclinic and heteroclinic cycles, and chaos induced by the coexistence in nonsmooth systems. By analyzing the dynamical behaviors on manifolds, this paper proposes some criteria to accurately locate the coexistence of homoclinic cycles and of heteroclinic cycles in a class of three-dimensional (3D) piecewise affine systems (PASs), respectively. It further establishes the existence conditions of chaos arising from such coexistence, and presents a mathematical proof by analyzing the constructed Poincaré map. Finally, the simulations for two numerical examples are provided to validate the established results.

This paper deals with a kind of reaction–diffusion–advection model which depicts a predator–prey ecosystem in rivers or streams. We obtain a complete classification on the dynamical behavior of the system in the parameter space of the predator’s mortality rate $\mu$ and the prey’s intrinsic growth rate $r$. More precisely, both species fail to survive when $r$ is small. With the increase of $r$, there exist two critical values $r^{\ast }$ and ${\sigma }^{\ast }$ such that both species can coexist in the long run when $r>r^{\ast }$ and $0<\mu <{\sigma }^{\ast }$, otherwise the prey survives alone. Finally, with the aid of numerical simulations, we investigate the effects of diffusion and advection on the global dynamics of the system by computing these two critical values. Numerical simulations illustrate that the diffusion of both predators and prey would benefit the invasion of the predators, and the advection would be unfavorable to the survival of the predators. Moreover, numerical simulations also suggest that the unique positive steady state is globally asymptotically stable among all non-negative and nontrivial initial data.

The infection of coronavirus (COVID-19) is a dangerous and life-threatening disease which spread to almost all parts of the globe. We present a mathematical model for the transmission of COVID-19 with vaccination effects. The basic properties of fractional calculus are presented for the inspection of the model. We calculate the equilibria of the model and determined the reproduction number ${{ℛ}}_0$. Local asymptotic stability conditions for the disease-free are obtained which determines the conditions to stabilize the exponential spread of the disease. The nonlinear least-square procedure is utilized to parameterize the model from actual cases reported in Pakistan. By fixed point theory, we prove the existence of a unique solution. We also present numerical results to simulate virus transmission and compare the results with those of the Caputo derivative. We study the solution pathways of the COVID-19 system to provide effective control policies for the infection. Significant changes have been noticed by lowering the order of fractional derivative.

Although the therapy of chronic myelogenous leukemia (CML) has progressed because of imatinib (IM) and other tyrosine kinase inhibitors (TKIs), the majority of patients still do not recover. To better regulate the remaining leukemic cell population, TKI combo therapy may be improved with a deeper understanding of the underlying mechanisms. We employed a mathematical system which incorporated the intricate phenomena of immune system to CML. We use a fractional derivative framework in this work to understand the dynamics of CML. Additionally, in our work, we concentrate on the qualitative characterization and dynamical behavior of CML interactions. For the proposed model, we examine the singularity and existence using fixed point theorems by Banach and Schaefer. We provide the necessary criteria for our suggested fractional model’s Ulam–Hyers stability. The influence of the factors on the dynamics of CML is highlighted by closely examining the solution paths by using a numerical scheme. To be more precise, we emphasized how the suggested system’s dynamic and chaotic behavior varied depending on the fractional order and other system factors. Policymakers are advised to consider the most crucial elements of CML dynamics. In order to inform policymakers and health authorities about the systems essential for control and treatment, it is crucial to investigate the dynamic characteristics of CML disease.

In this research work, we offer an epidemic model for monkeypox virus infection with the help of non-integer derivative as well as classical ones. The model takes into account every potential connection that can aid in the spread of infection among the people. We look into the model’s endemic equilibrium, disease-free equilibrium, and reproduction number ${{ℛ}}_0$. In addition to this, we concentrated on the qualitative analysis and dynamic behavior of the monkeypox virus. Through fixed point theorem, Banach’s and Schaefer’s are applied to study the existence and uniqueness of the solution of the suggested system of the monkeypox virus infection. We provide the necessary criteria for the recommended fractional system’s Ulam–Hyers stability. Furthermore, a numerical approach is used to study the solution routes and emphasize how the parameters affect the dynamics of the monkey pox virus. The most crucial features of the dynamics of the monkeypox virus are noticed and suggested to decision-makers.

In this paper, we analyzed the chaotic complexity of a financial mathematical model in terms of a new generalized Caputo fractional derivative. There are three components in this nonlinear financial model: price indexes, interest rates, and investment demand. Our analysis is based on applying the fixed point hypothesis to determine the existence and uniqueness of the solutions. The bifurcation of the proposed financial system has been analyzed at various parameters of the system. Dynamical phase portraits of the proposed financial model are demonstrated at various fractional-order values. We investigated the possibility of finding new complex dynamical behavior with generalized Caputo fractional derivative. This economic model is solved numerically using a predictor–corrector (PC) algorithm with a generalized Caputo derivative. This algorithm can be viewed as a non-integer extension of the classical Adams–Bashforth–Moulton (ABM) algorithm. Additionally, this numerical algorithm has been studied for stability. A number of diverse dynamic behaviors have been observed in numerical simulations of the system, including chaos. Over a broad range of system parameters, bifurcation diagrams indicate that the system behaves chaotically.

A competitive system on the n-rectangle: {x ∈ Rⁿ: 0 ≤ x_i ≤ l_i, i = 1, …, n} was considered, each species of which, in isolation, admits logistic growth with the hyperbolic structure saturation. It has an (n - 1)-dimensional invariant surface called carrying simplex Σ as a globe attractor, hence the long term dynamics of the system is completely determined by the dynamics on Σ. For the three-dimensional system, the whole dynamical behavior was presented. It has a unique positive equilibrium point and any limit set is either an equilibrium point or a limit cycle. The system is permanent and it is proved that the number of periodic orbits is finite and non-periodic oscillation the May–Leonard phenomenon does not exist. A criterion for the positive equilibrium to be globally asymptotically stable is provided. Whether there exist limit cycles or not remains open.

First, we show that the theorem by Hirsch which guarantees the existence of carrying simplex for competitive system on any n-rectangle: {x ∈ Rⁿ : 0 ≤ x_i ≤ k_i, i = 1, …, n} still holds. Next, based on the theorem a competitive system with the linear structure saturation defined on the n-rectangle is investigated, which admits a unique (n - 1)-dimensional carrying simplex as a global attractor. Further, we focus on the whole dynamical behavior of the three-dimensional case, which has a unique locally asymptotically stable positive equilibrium. Hopf bifurcations do not occur. We prove that any limit set is either this positive equilibrium or a limit cycle. If limit cycles exist, the number of them is finite. We also give a criterion for the positive equilibrium to be globally asymptotically stable.

In this paper, we formulate the transmission phenomena of Hand–Foot–Mouth Disease (HFMD) through non-integer derivative. We interrogate the biological meaningful results of the recommended system of HFMD. The basic reproduction number is determined through next generation method and the impact of different parameters on the reproduction number is examined with the help of partial rank correlation coefficient (PRCC) technique. In addition, we concentrated on qualitative analysis and dynamical behavior of HFMD dynamics. Banach’s and Schaefer’s fixed-point theorems are used to analyze the uniqueness and existence of the solution of the proposed HFMD model. The HFMD system’s Ulam–Hyers stability has been confirmed to be sufficient. To highlight the impact of the parameters on the dynamics of HFMD, we performed several simulations through numerical scheme to conceptualize the transmission route of the infection. To be more specific, numerical simulations are used to visualize the effect of input parameters on the systems dynamics. We have shown the key input parameters of the system for the control of infection in the society.

Norovirus infection has been documented to have a significant economic impact in different regions of the world. Even though young children bear the greatest economic burden, older age groups in some locations have the highest costs per illness. Most of these costs result from lost production caused by acute illnesses. This viral infection causes inflammation of the intestines and stomach, also called stomach flu and food poisoning. Our research work constructs a new compartmental model of norovirus infection based on contaminated water and food contamination to conceptualize how norovirus spreads. The proposed dynamics of norovirus have been presented in the fractional Caputo framework. We present the Caputo operator’s rudimentary results for the system’s examination. By applying the fixed point theory to the system, the existence theory has been investigated. To inspect the solution pathways of norovirus infection, we introduce a novel numerical scheme to explore the dynamical behavior of the system. Finally, a numerical investigation has been done to show the impact of different factors of the system for the control of norovirus illness. We suggested the most critical factors of the system to the policymakers to control and prevent infection in the community.

In this paper, we develop a conceptual framework of a innovation diffusion dynamics model in which multiple parallel innovations are effecting each other during the diffusion process. A mathematical model is proposed to explore the interaction and diffusion of three innovations simultaneously available in market. The stability analysis is carried out for various types of diffusions on such system both analytically and numerically. It is observed that the association between innovations in product market could be complementary, substitute, independent or competitive. The co-existence and extinction of innovation depends on the level of diffusion between the innovations and it may or may not be sensitive to initial distribution of innovations.