A stochastically perturbed co-infection epidemic model for COVID-19 and hepatitis B virus.

A new co-infection model for the transmission dynamics of two virus hepatitis B (HBV) and coronavirus (COVID-19) is formulated to study the effect of white noise intensities. First, we present the model equilibria and basic reproduction number. The local stability of the equilibria points is proved. Moreover, the proposed stochastic model has been investigated for a non-negative solution and positively invariant region. With the help of Lyapunov function, analysis was performed and conditions for extinction and persistence of the disease based on the stochastic co-infection model were derived. Particularly, we discuss the dynamics of the stochastic model around the disease-free state. Similarly, we obtain the conditions that fluctuate at the disease endemic state holds if min ( R H s , R C s , R HC s ) > 1 . Based on extinction as well as persistence some conditions are established in form of expression containing white noise intensities as well as model parameters. The numerical results have also been used to illustrate our analytical results.

Global asymptotic stability, extinction and ergodic stationary distribution in a stochastic model for dual variants of SARS-CoV-2.

Several mathematical models have been developed to investigate the dynamics SARS-CoV-2 and its different variants. Most of the multi-strain SARS-CoV-2 models do not capture an important and more realistic feature of such models known as randomness. As the dynamical behavior of most epidemics, especially SARS-CoV-2, is unarguably influenced by several random factors, it is appropriate to consider a stochastic vaccination co-infection model for two strains of SARS-CoV-2. In this work, a new stochastic model for two variants of SARS-CoV-2 is presented. The conditions of existence and the uniqueness of a unique global solution of the stochastic model are derived. Constructing an appropriate Lyapunov function, the conditions for the stochastic system to fluctuate around endemic equilibrium of the deterministic system are derived. Stationary distribution and ergodicity for the new co-infection model are also studied. Numerical simulations are carried out to validate theoretical results. It is observed that when the white noise intensities are larger than certain thresholds and the associated stochastic reproduction numbers are less than unity, both strains die out and go into extinction with unit probability. More-over, it is observed that, for weak white noise intensities, the solution of the stochastic system fluctuates around the endemic equilibrium (EE) of the deterministic model. Frequency distributions are also studied to show random fluctuations due to stochastic white noise intensities. The results presented herein also reveal the impact of vaccination in reducing the co-circulation of SARS-CoV-2 variants within a given population.

A stochastic SIQR epidemic model with Lévy jumps and three-time delays.

Isolation and vaccination are the two most effective measures in protecting the public from the spread of illness. The SIQR model with vaccination is widely used to investigate the dynamics of an infectious disease at population level having the compartments: susceptible, infectious, quarantined and recovered. The paper mainly aims to extend the deterministic model to a stochastic SQIR case with Lévy jumps and three-time delays, which is more suitable for modeling complex and instable environment. The existence and uniqueness of the global positive solution are obtained by using the Lyapunov method. The dynamic properties of stochastic solution are studied around the disease-free and endemic equilibria of the deterministic model. Our results reveal that stochastic perturbation affect the asymptotic properties of the model. Numerical simulation shows the effects of interested parameters of theoretical results, including quarantine, vaccination and jump parameters. Finally, we apply both the stochastic and deterministic models to analyze the outbreak of mutant COVID-19 epidemic in Gansu Province, China.

Power Graphs of Finite Groups Determined by Hosoya Properties.

Suppose G is a finite group. The power graph represented by P(G) of G is a graph, whose node set is G, and two different elements are adjacent if and only if one is an integral power of the other. The Hosoya polynomial contains much information regarding graph invariants depending on the distance. In this article, we discuss the Hosoya characteristics (the Hosoya polynomial and its reciprocal) of the power graph related to an algebraic structure formed by the symmetries of regular molecular gones. As a consequence, we determined the Hosoya index of the power graphs of the dihedral and the generalized groups. This information is useful in determining the renowned chemical descriptors depending on the distance. The total number of matchings in a graph Γ is known as the Z-index or Hosoya index. The Z-index is a well-known type of topological index, which is popular in combinatorial chemistry and can be used to deal with a variety of chemical characteristics in molecular structures.

The stochastic bifurcation analysis and stochastic delayed optimal control for epidemic model with general incidence function.

In the history of the world, contagious diseases have been proved to pose serious threats to humanity that needs uttermost research in the field and its prompt implementations. With this motive, an attempt has been made to investigate the spread of such contagion by using a delayed stochastic epidemic model with general incidence rate, time-delay transmission, and the concept of cross immunity. It is proved that the system is mathematically and biologically well-posed by showing that there exist a positive and bounded global solution of the model. Necessary conditions are derived, which guarantees the permanence as well as extinction of the disease. The model is further investigated for the existence of an ergodic stationary distribution and established sufficient conditions. The non-zero periodic solution of the stochastic model is analyzed quantitatively. The analysis of optimality and time delay is used, and a proper strategy was presented for prevention of the disease. A scheme for the numerical simulations is developed and implemented in MATLAB, which reflects the long term behavior of the model. Simulation suggests that the noises play a vital role in controlling the spread of an epidemic following the proposed flow, and the case of disease extinction is directly proportional to the magnitude of the white noises. Since time delay reflects the dynamics of recurring epidemics, therefore, it is believed that this study will provide a robust basis for studying the behavior and mechanism of chronic infections.

Stationary distribution and extinction of stochastic coronavirus (COVID-19) epidemic model.

Similar to other epidemics, the novel coronavirus (COVID-19) spread very fast and infected almost two hundreds countries around the globe since December 2019. The unique characteristics of the COVID-19 include its ability of faster expansion through freely existed viruses or air molecules in the atmosphere. Assuming that the spread of virus follows a random process instead of deterministic. The continuous time Markov Chain (CTMC) through stochastic model approach has been utilized for predicting the impending states with the use of random variables. The proposed study is devoted to investigate a model consist of three exclusive compartments. The first class includes white nose based transmission rate (termed as susceptible individuals), the second one pertains to the infected population having the same perturbation occurrence and the last one isolated (quarantined) individuals. We discuss the model's extinction as well as the stationary distribution in order to derive the the sufficient criterion for the persistence and disease' extinction. Lastly, the numerical simulation is executed for supporting the theoretical findings.

Mathematical analysis of spread and control of the novel corona virus (COVID-19) in China.

Number of well-known contagious diseases exist around the world that mainly include HIV, Hepatitis B, influenzas etc., among these, a recently contested coronavirus (COVID-19) is a serious class of such transmissible syndromes. Abundant scientific evidence the wild animals are believed to be the primary hosts of the virus. Majority of such cases are considered to be human-to-human transmission, while a few are due to wild animals-to-human transmission and substantial burdens on healthcare system following this spread. To understand the dynamical behavior such diseases, we fitted a susceptible-infectious-quarantined model for human cases with constant proportions. We proposed a model that provide better constraints on understanding the climaxes of such unseen disastrous spread, relevant consequences, and suggesting future imperative strategies need to be adopted. The main features of the work include the positivity, boundedness, existence and uniqueness of solution of the model. The conditions were derived under which the COVID-19 may extinct or persist in the population. Sensitivity and estimation of those important parameters have been carried out that plays key role in the transmission mechanism. To optimize the spread of such disease, we present a control problem for further analysis using two control measures. The necessary conditions have been derived using the Pontryagin's maximum principle. Parameter values have been estimated from the real data and experimental numerical simulations are presented for comparison as well as verification of theoretical results. The obtained numerical results also present the verification, accuracy, validation, and robustness of the proposed scheme.

Linear approximations of global behaviors in nonlinear systems with moderate or strong noise.

While many physical or chemical systems can be modeled by nonlinear Langevin equations (LEs), dynamical analysis of these systems is challenging in the cases of moderate and strong noise. Here we develop a linear approximation scheme, which can transform an often intractable LE into a linear set of binomial moment equations (BMEs). This scheme provides a feasible way to capture nonlinear behaviors in the sense of probability distribution and is effective even when the noise is moderate or big. Based on BMEs, we further develop a noise reduction technique, which can effectively handle tough cases where traditional small-noise theories are inapplicable. The overall method not only provides an approximation-based paradigm to analysis of the local and global behaviors of nonlinear noisy systems but also has a wide range of applications.

Comprehensive analysis of a stochastic wireless sensor network motivated by Black-Karasinski process.

Wireless sensor networks (WSNs) encounter a significant challenge in ensuring network security due to their operational constraints. This challenge stems from the potential infiltration of malware into WSNs, where a single infected node can rapidly propagate worms to neighboring nodes. To address this issue, this research introduces a stochastic S E I R S model to characterize worm spread in WSNs. Initially, we established that our model possesses a globally positive solution. Subsequently, we determine a threshold value for our stochastic system and derive a set of sufficient conditions that dictate the persistence or extinction of worm spread in WSNs based on the mean behavior. Our study reveals that environmental randomness can impede the spread of malware in WSNs. Moreover, by utilizing various parameter sets, we obtain approximate solutions that showcase these precise findings and validate the effectiveness of the proposed S E I R S model, which surpasses existing models in mitigating worm transmission in WSNs.

The measles epidemic model assessment under real statistics: an application of stochastic optimal control theory.

A stochastic epidemic model with random noise transmission is taken into account, describing the dynamics of the measles viral infection. The basic reproductive number is calculated corresponding to the stochastic model. It is determined that, given initial positive data, the model has bounded, unique, and positive solution. Additionally, utilizing stochastic Lyapunov functional theory, we study the extinction of the disease. Stationary distribution and extinction of the infection are examined by providing sufficient conditions. We employed optimal control principles and examined stochastic control systems to regulate the transmission of the virus using environmental factors. Graphical representations have been offered for simplicity of comprehending in order to further verify the acquired analytical findings.

Impact of Levy noise on a stochastic Norovirus epidemic model with information intervention.

In this article, we study the dynamics of Norovirus infection by developing a stochastic epidemic model having Levy noise. The study shows that Levy noise and informative interventions have more influence on the said dynamics. Firstly, we show that the model has a unique global positive solution. After this, we formulated a stochastic threshold value as a necessary condition for the extinction and persistence in the mean of the proposed epidemic model. Finally, numerical simulation are drawn to verify our obtained results of the dynamics. The perturbed stochastic approach may affect the dynamical properties of the model and large noises will be greatly significant to minimize its transmission.

Mathematical analysis of a new nonlinear stochastic hepatitis B epidemic model with vaccination effect and a case study.

This work present a detailed analysis of a stochastic delayed model which governs the transmission mechanism of the Hepatitis B virus (HBV) while considering the white noises and the effect of vaccinations. It is assumed that the perturbations are nonlinear and an individual may lose his/her immunity after the vaccination, that is, the vaccination can produce temporal immunity. Based on the characteristics of the disease and the underlying assumptions, we formulated the associated deterministic model for which the threshold parameter R 0 D is calculated. The model was further extended to a stochastic model and it is well-justified that the model is both mathematically and biologically feasible by showing that the model solution exists globally, bounded stochastically and is positive. By utilizing the concepts of stochastic theory and by constructing appropriate Lyapunov functions, we developed the theory for the extinction and persistence of the disease. Further, it is shown that the model is ergodic and has a unique stationary distribution. The stochastic bifurcation theory is utilized and a detailed bifurcation analysis of the model is presented. By using the standard curve fitting tools, we fitted the model against the available HBV data in Pakistan from March 2018 to February 2019 and accordingly the parameters of the model were estimated. These estimated values were used in simulating the model, theoretical findings of the study are validated through simulations and predictions were drawn. Simulations suggest that for a complete understanding of HBV dynamics, one must include time delay into such studies, and improvements in every vaccination program are unavoidable.

Fractal fractional based transmission dynamics of COVID-19 epidemic model.

We investigate the dynamical behavior of Coronavirus (COVID-19) for different infections phases and multiple routes of transmission. In this regard, we study a COVID-19 model in the context of fractal-fractional order operator. First, we study the COVID-19 dynamics with a fractal fractional-order operator in the framework of Atangana-Baleanu fractal-fractional operator. We estimated the basic reduction number and the stability results of the proposed model. We show the data fitting to the proposed model. The system has been investigated for qualitative analysis. Novel numerical methods are introduced for the derivation of an iterative scheme of the fractal-fractional Atangana-Baleanu order. Finally, numerical simulations are performed for various orders of fractal-fractional dimension.

Impact of information and Lévy noise on stochastic COVID-19 epidemic model under real statistical data.

In this paper, we consider the dynamical behaviour of a stochastic coronavirus (COVID-19) susceptible-infected-removed epidemic model with the inclusion of the influence of information intervention and Lévy noise. The existence and uniqueness of the model positive solution are proved. Then, we establish a stochastic threshold as a sufficient condition for the extinction and persistence in mean of the disease. Based on the available COVID-19 data, the parameters of the model were estimated and we fit the model with real statistics. Finally, numerical simulations are presented to support our theoretical results.

Theoretical and numerical analysis of COVID-19 pandemic model with non-local and non-singular kernels.

The global consequences of Coronavirus (COVID-19) have been evident by several hundreds of demises of human beings; hence such plagues are significantly imperative to predict. For this purpose, the mathematical formulation has been proved to be one of the best tools for the assessment of present circumstances and future predictions. In this article, we propose a fractional epidemic model of coronavirus (COVID-19) with vaccination effects. An arbitrary order model of COVID-19 is analyzed through three different fractional operators namely, Caputo, Atangana-Baleanu-Caputo (ABC), and Caputo-Fabrizio (CF), respectively. The fractional dynamics are composed of the interaction among the human population and the external environmental factors of infected peoples. It gives an extra description of the situation of the epidemic. Both the classical and modern approaches have been tested for the proposed model. The qualitative analysis has been checked through the Banach fixed point theory in the sense of a fractional operator. The stability concept of Hyers-Ulam idea is derived. The Newton interpolation scheme is applied for numerical solutions and by assigning values to different parameters. The numerical works in this research verified the analytical results. Finally, some important conclusions are drawn that might provide further basis for in-depth studies of such epidemics.

Stochastic modeling of the Monkeypox 2022 epidemic with cross-infection hypothesis in a highly disturbed environment.

Monkeypox 2022, a new re-emerging disease, is caused by the Monkeypox virus. Structurally, this virus is related to the smallpox virus and infects the host in a similar way; however, the symptoms of Monkeypox are more severe. In this research work, a mathematical model for understanding the dynamics of Monkeypox 2022 is suggested that takes into account two modes of transmission: horizontal human dissemination and cross-infection between animals and humans. Due to lack of substantial knowledge about the virus diffusion and the effect of external perturbations, the model is extended to the probabilistic formulation with Lévy jumps. The proposed model is a two block compartmental system that requires the form of Itô-Lévy stochastic differential equations. Based on some assumptions and nonstandard analytical techniques, two principal asymptotic properties are proved: the eradication and continuation in the mean of Monkeypox 2022. The outcomes of the study reveals that the dynamical behavior of the proposed Monkeypox 2022 system is chiefly governed by some parameters that are precisely correlated with the noise intensities. To support the obtained theoretical finding, examples based on numerical simulations and real data are presented at the end of the study. The numerical simulations also exhibit the impact of the innovative adopted mathematical techniques on the findings of this work.

Impact of information intervention on stochastic hepatitis B model and its variable-order fractional network.

This paper aims at analyzing the dynamical behavior of a SIR hepatitis B epidemic stochastic model via a novel approach by incorporating the effect of information interventions and random perturbations. Initially, we demonstrate the positivity and global existence of the solutions. Afterward, we derive the stochastic threshold parameter R s , followed by the fact that this number concludes the transmission of hepatitis B from the population. By increasing the intensity of noise, we get R s less than one, inferring that ultimately hepatitis B will lapse. While decreasing the intensity of noise to a sufficient level, we have R s > 1 . For the case R s > 1 , adequate results for the presence of stationary distribution are achieved, showing the prevalence of hepatitis B. The present study also involves the derivation of the necessary conditions for the persistence of the epidemic. Finally, the main theoretical solutions are plotted through simulations. Discussion on theoretical and numerical results shows that utilizing random perturbations and information interventions have a pronounced impact on the syndrome's dynamics. Furthermore, since most communities interact with each other, and the disease spread rate is affected by this factor, a new variable-order fractional network of the stochastic hepatitis B model is offered. Subsequently, this study will provide a robust theoretical basis for comprehending worldwide SIR stochastic and variable-order fractional network-related case studies.

Fractal-fractional and stochastic analysis of norovirus transmission epidemic model with vaccination effects.

In this paper, we investigate an norovirus (NoV) epidemic model with stochastic perturbation and the new definition of a nonlocal fractal-fractional derivative in the Atangana-Baleanu-Caputo (ABC) sense. First we present some basic properties including equilibria and the basic reproduction number of the model. Further, we analyze that the proposed stochastic system has a unique global positive solution. Next, the sufficient conditions of the extinction and the existence of a stationary probability measure for the disease are established. Furthermore, the fractal-fractional dynamics of the proposed model under Atangana-Baleanu-Caputo (ABC) derivative of fractional order "[Formula: see text]" and fractal dimension "[Formula: see text]" have also been addressed. Besides, coupling the non-linear functional analysis with fixed point theory, the qualitative analysis of the proposed model has been performed. The numerical simulations are carried out to demonstrate the analytical results. It is believed that this study will comprehensively strengthen the theoretical basis for comprehending the dynamics of the worldwide contagious diseases.

The Complex Dynamics of Hepatitis B Infected Individuals with Optimal Control.

This paper proposes various stages of the hepatitis B virus (HBV) besides its transmissibility and nonlinear incidence rate to develop an epidemic model. The authors plan the model, and then prove some basic results for the well-posedness in term of boundedness and positivity. Moreover, the authors find the threshold parameter R 0, called the basic/effective reproductive number and carry out local sensitive analysis. Furthermore, the authors examine stability and hence condition for stability in terms of R 0. By using sensitivity analysis, the authors formulate a control problem in order to eradicate HBV from the population and proved that the control problem actually exists. The complete characterization of the optimum system was achieved by using the 4th-order Runge-Kutta procedure.

Stochastic COVID-19 SEIQ epidemic model with time-delay.

In this work, we consider an epidemic model for corona-virus (COVID-19) with random perturbations as well as time delay, composed of four different classes of susceptible population, the exposed population, the infectious population and the quarantine population. We investigate the proposed problem for the derivation of at least one and unique solution in the positive feasible region of non-local solution. For one stationary ergodic distribution, the necessary result of existence is developed by applying the Lyapunov function in the sense of delay-stochastic approach and the condition for the extinction of the disease is also established. Our obtained results show that the effect of Brownian motion and noise terms on the transmission of the epidemic is very high. If the noise is large the infection may decrease or vanish. For validation of our obtained scheme, the results for all the classes of the problem have been numerically simulated.