In this paper, we investigate and implement a numerical method that is based on the mimetic finite difference operator in order to solve the nonlinear Allen-Cahn equation with periodic and non-periodic boundary conditions. In addition, we also analyze the performance of this mimetic-based method by using the classical heat equation with a variety of boundary conditions. We assess the performance of the mimetic-based numerical method by comparing the errors of its solutions with those obtained by a classical finite difference method and the pdepde built-in Matlab function. We compute the errors by using the exact solutions when they are available or with reference solutions. We adapt and implement the mimetic-based numerical method by using the MOLE (Mimetic Operators Library Enhanced) library that includes some built-in functions that return representations of the curl, divergence and gradient operators, in order to deal with the Allen-Cahn and heat equations. We present several results with regard to errors and numerical convergence tests in order to provide insight into the accuracy of the mimetic-based numerical method. The results show that the numerical method based on the mimetic difference operator is a reliable method for solving the Allen-Cahn and heat equations with periodic and non-periodic boundary conditions. The numerical solutions generated by the mimetic-based method are relatively accurate. We also proposed a new method based on the mimetic finite difference operator and the convexity splitting approach to solve Allen-Cahn equation in 2D. We found that, for small time step sizes the solutions generated by the mimetic-based method are more accurate than the ones generated by the pdepe Matlab function and similar to the solutions given by a finite difference method.

A complex dynamic interplay exists between epidemic transmission and vaccination, which is significantly influenced by human behavioral responses. We construct a research framework combining both the function modeling of the cumulative global COVID-19 information and limited individuals' information processing capacity employing the Gompertz model for growing processes. Meanwhile, we built a function representing the decision to get vaccinated following benefit-cost analysis considered the choices made by people in each scenario have an influence from altruism, free-riding and immunity escaping capacity. Through the mean-field calculation analysis and using a fourth-order Runge-Kutta method with constant step size, we obtain plots from numerical simulations. We found that only when the total number of infectious individuals proves sufficient to reach and exceed a certain level will the individuals face a better trade-off in determining whether to get vaccinated against the diseases based on that information. Besides, authoritative media have a higher decisive influence and efforts should be focused on extending the duration of vaccine protection, which is beneficial to inhibit the outbreaks of epidemics. Our work elucidates that reducing the negative payoff brought about by the free-riding behavior for individuals or improving the positive payoff from the altruistic motivation helps to control the disease in cultures that value social benefits, vaccination willingness is generally stronger. We also note that at a high risk of infection, the decision of vaccination is highly correlated with global epidemic information concerning COVID-19 infection, while at times of lower risk, it depends on the game theoretic vaccine strategy. The findings demonstrate that improving health literacy, ensuring open and transparent information on vaccine safety and efficacy as a public health priority can be an effective strategy for mitigating inequalities in health education, as well as alleviating the phenomenon that immunity escaping abilities is more likely to panic by populations with high levels of education. In addition, prosocial nudges are great ways to bridge these immunity gaps that can contribute to implementing government public health control measures, creating a positive feedback loop.

In view of the spread of corona virus disease 2019 (COVID-19), this paper proposes a fractional-order generalized SEIR model. The non-negativity of the solution of the model is discussed. Based on the established threshold , the existence of the disease-free equilibrium and endemic equilibrium is analyzed. Then, sufficient conditions are established to ensure the local asymptotic stability of the equilibria. The parameters of the model are identified based on the statistical data of COVID-19 cases. Furthermore, the validity of the model for describing the COVID-19 outbreak is verified. Meanwhile, the accuracy of the relevant theoretical results are also verified. Considering the relevant strategies of COVID-19 prevention and control, the fractional optimal control problem (FOCP) is proposed. Numerical schemes for Riemann-Liouville (R-L) fractional-order adjoint system with transversal conditions is presented. Based on the relevant statistical data, the corresponding FOCP is numerically solved, and the control effect of the COVID-19 outbreak under the optimal control strategy is discussed.

During a pandemic event like the present COVID-19, self-quarantine, mask-wearing, hygiene maintenance, isolation, forced quarantine, and social distancing are the most effective nonpharmaceutical measures to control the epidemic when the vaccination and proper treatments are absent. In this study, we proposed an epidemiological model based on the SEIR dynamics along with the two interventions defined as self-quarantine and forced quarantine by human behavior dynamics. We consider a disease spreading through a population where some people can choose the self-quarantine option of paying some costs and be safer than the remaining ones. The remaining ones act normally and send to forced quarantine by the government if they get infected and symptomatic. The government pays the forced quarantine costs for individuals, and the government has a budget limit to treat the infected ones. Each intervention derived from the so-called behavior model has a dynamical equation that accounts for a proper balance between the costs for each case, the total budget, and the risk of infection. We show that the infection peak cannot be reduced if the authority does not enforce a proactive (quantified by a higher sensitivity parameter) intervention. While comparing the impact of both self- and forced quarantine provisions, our results demonstrate that the latter is more influential to reduce the disease prevalence and the social efficiency deficit (a gap between social optimum payoff and equilibrium payoff).

We investigate a class of iteratively regularized methods for finding a quasi-solution of a noisy nonlinear irregular operator equation in Hilbert space. The iteration uses an a priori stopping rule involving the error level in input data. In assumptions that the Frechet derivative of the problem operator at the desired quasi-solution has a closed range, and that the quasi-solution fulfills the standard source condition, we establish for the obtained approximation an accuracy estimate linear with respect to the error level. The proposed iterative process is applied to the parameter identification problem for a SEIR-like model of the COVID-19 pandemic.

COVID-19 has emphasized that a precise prediction of a disease spreading is one of the most pressing and crucial issues from a social standpoint. Although an ordinary differential equation (ODE) approach has been well established, stochastic spreading features might be hard to capture accurately. Perhaps, the most important factors adding such stochasticity are the effect of the underlying networks indicating physical contacts among individuals. The multi-agent simulation (MAS) approach works effectively to quantify the stochasticity. We systematically investigate the stochastic features of epidemic spreading on homogeneous and heterogeneous networks. The study quantitatively elucidates that a strong microscopic locality observed in one- and two-dimensional regular graphs, such as ring and lattice, leads to wide stochastic deviations in the final epidemic size (FES). The ensemble average of FES observed in this case shows substantial discrepancies with the results of ODE based mean-field approach. Unlike the regular graphs, results on heterogeneous networks, such as Erdős-Rényi random or scale-free, show less stochastic variations in FES. Also, the ensemble average of FES in heterogeneous networks seems closer to that of the mean-field result. Although the use of spatial structure is common in epidemic modeling, such fundamental results have not been well-recognized in literature. The stochastic outcomes brought by our MAS approach may lead to some implications when the authority designs social provisions to mitigate a pandemic of un-experienced infectious disease like COVID-19.

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.

Dimension governs dynamical processes on networks. The social and technological networks which we encounter in everyday life span a wide range of dimensions, but studies of spreading on finite-dimensional networks are usually restricted to one or two dimensions. To facilitate investigation of the impact of dimension on spreading processes, we define a flexible higher-dimensional small world network model and characterize the dependence of its structural properties on dimension. Subsequently, we derive mean field, pair approximation, intertwined continuous Markov chain and probabilistic discrete Markov chain models of a COVID-19-inspired susceptible-exposed-infected-removed (SEIR) epidemic process with quarantine and isolation strategies, and for each model identify the basic reproduction number , which determines whether an introduced infinitesimal level of infection in an initially susceptible population will shrink or grow. We apply these four continuous state models, together with discrete state Monte Carlo simulations, to analyse how spreading varies with model parameters. Both network properties and the outcome of Monte Carlo simulations vary substantially with dimension or rewiring rate, but predictions of continuous state models change only slightly. A different trend appears for epidemic model parameters: as these vary, the outcomes of Monte Carlo change less than those of continuous state methods. Furthermore, under a wide range of conditions, the four continuous state approximations present similar deviations from the outcome of Monte Carlo simulations. This bias is usually least when using the pair approximation model, varies only slightly with network size, and decreases with dimension or rewiring rate. Finally, we characterize the discrepancies between Monte Carlo and continuous state models by simultaneously considering network efficiency and network size.

The world is going through a critical period due to a new respiratory disease called coronavirus disease 2019 (COVID-19). This disease is caused by severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2). Mathematical modeling is one of the most important tools that can speed up finding a drug or vaccine for COVID-19. COVID-19 can lead to death especially for patients having chronic diseases such as cancer, AIDS, etc. We construct a new within-host SARS-CoV-2/cancer model. The model describes the interactions between six compartments: nutrient, healthy epithelial cells, cancer cells, SARS-CoV-2 virus particles, cancer-specific CTLs, and SARS-CoV-2-specific antibodies. We verify the nonnegativity and boundedness of its solutions. We outline all possible equilibrium points of the proposed model. We prove the global stability of equilibria by constructing proper Lyapunov functions. We do some numerical simulations to visualize the obtained results. According to our model, lymphopenia in COVID-19 cancer patients may worsen the outcomes of the infection and lead to death. Understanding dysfunctions in immune responses during COVID-19 infection in cancer patients could have implications for the development of treatments for this high-risk group.

Due to the current COVID-19 pandemic, much effort has been put on studying the spread of infectious diseases to propose more adequate health politics. The most effective surveillance system consists of doing massive tests. Nonetheless, many countries cannot afford this class of health campaigns due to limited resources. Thus, a transmission model is a viable alternative to study the dynamics of the pandemic. The most used are the Susceptible, Infected and Removed type models (SIR). In this study, we tackle the population estimation problem of the A-SIR model, which takes into account asymptomatic or undetected individuals. By means of an algebraic differential approach, we design a model-free (no copy system) reduced-order estimation algorithm (observer) to determine the different non-measured population groups. We study two types of estimation algorithms: Proportional and Proportional-Integral. Both shown fast convergence speed, as well as a minimal estimation error. Additionally, we introduce random fluctuations in our analysis to represent changes in the external conditions and which result in poor measurements. The numerical results reveal that both model-free estimators are robust despite the presence of these fluctuations. As a point of reference, we apply the classical Luenberger type observer to our estimation problem and compare the results. Finally, we consider real data of infected individuals in Mexico City, reported from February 2020 to March 2021, and estimate the non-measured populations. Our work's main goal is to proportionate a simple and therefore, an accessible methodology to estimate the behavior of the COVID-19 pandemic from the available data, such that the competent authorities can propose more adequate health politics.

An outbreak of respiratory disease caused by a novel coronavirus is ongoing from December 2019. As of December 14, 2020, it has caused an epidemic outbreak with more than 73 million confirmed infections and above 1.5 million reported deaths worldwide. During this period of an epidemic when human-to-human transmission is established and reported cases of coronavirus disease 2019 (COVID-19) are rising worldwide, investigation of control strategies and forecasting are necessary for health care planning. In this study, we propose and analyze a compartmental epidemic model of COVID-19 to predict and control the outbreak. The basic reproduction number and the control reproduction number are calculated analytically. A detailed stability analysis of the model is performed to observe the dynamics of the system. We calibrated the proposed model to fit daily data from the United Kingdom (UK) where the situation is still alarming. Our findings suggest that independent self-sustaining human-to-human spread ( ) is already present. Short-term predictions show that the decreasing trend of new COVID-19 cases is well captured by the model. Further, we found that effective management of quarantined individuals is more effective than management of isolated individuals to reduce the disease burden. Thus, if limited resources are available, then investing on the quarantined individuals will be more fruitful in terms of reduction of cases.

The ongoing pandemic situation due to COVID-19 originated from the Wuhan city, China affects the world in an unprecedented scale. Unavailability of totally effective vaccination and proper treatment regimen forces to employ a non-pharmaceutical way of disease mitigation. The world is in desperate demand of useful control intervention to combat the deadly virus. This manuscript introduces a new mathematical model that addresses two different diagnosis efforts and isolation of confirmed cases. The basic reproductive number, is inspected, and the model's dynamical characteristics are also studied. We found that with the condition the disease can be eliminated from the system. Further, we fit our proposed model system with cumulative confirmed cases of six Indian states, namely, Maharashtra, Tamil Nadu, Andhra Pradesh, Karnataka, Delhi and West Bengal. Sensitivity analysis carried out to scale the impact of different parameters in determining the size of the epidemic threshold of . It reveals that unidentified symptomatic cases result in an underestimation of whereas, diagnosis based on new contact made by confirmed cases can gradually reduce the size of and hence helps to mitigate the ongoing disease. An optimal control problem is framed using a control variable projecting the effectiveness of diagnosis based on traced contacts made by a confirmed COVID patient. It is noticed that optimal contact tracing effort reduces effectively over time.

Nowadays, vaccination is the most effective way to control the epidemic spreading. In this paper, an epidemic SEIRV (susceptible-exposed-infected-removed -vaccinated) model and an evolutionary game model are established to analyze the difference between mandatory vaccination method and voluntary vaccination method on heterogeneous networks. Firstly, we divide the population into four categories, including susceptible individuals, exposed individuals, infected individuals and removed individuals. Based on the mean field approximation theory, differential equations are developed to characterize the changes of the proportions of the four groups over time under mandatory vaccination. Then through the analysis of the differential equations, the disease-free equilibrium point (DFE) and the endemic disease equilibrium point (EDE) are obtained. Also, the basic reproduction number is obtained by the next-generation matrix method and the stability analysis of the equilibrium points is performed. Next, by considering factors such as vaccination cost, treatment cost and government subsidy rate, differential equations are established to represent the change of vaccination rate over time. By analyzing the final vaccination coverage rate, we can get the minimum vaccination cost to make infectious disease disappear. Finally, the Monte Carlo method is used for numerical simulation to verify the results obtained from the theoretical analysis. Using the SARS-Cov-2 pandemic data from Wuhan, China, the experimental results show that when the effectiveness rate of vaccination is 0.75, the vaccination cost is not higher than 0.886 so that the vaccination strategy can be spread among the population. If mandatory vaccination is adopted, the minimum vaccination rate is 0.146.

Social distancing can be divided into two categories: spontaneous social distancing adopted by the individuals themselves, and public social distancing promoted by the government. Both types of social distancing have been proved to suppress the spread of infectious disease effectively. While previous studies examined the impact of each social distancing separately, the simultaneous impacts of them are less studied. In this research, we develop a mathematical model to analyze how spontaneous social distancing and public social distancing simultaneously affect the outbreak threshold of an infectious disease with asymptomatic infection. A communication-contact two-layer network is constructed to consider the difference between spontaneous social distancing and public social distancing. Based on link overlap of the two layers, the two-layer network is divided into three subnetworks: communication-only network, contact-only network, and overlapped network. Our results show that public social distancing can significantly increase the outbreak threshold of an infectious disease. To achieve better control effect, the subnetwork of higher infection risk should be more targeted by public social distancing, but the subnetworks of lower infection risk shouldn't be overlooked. The impact of spontaneous social distancing is relatively weak. On the one hand, spontaneous social distancing in the communication-only network has no impact on the outbreak threshold of the infectious disease. On the other hand, the impact of spontaneous social distancing in the overlapped network is highly dependent on the detection of asymptomatic infection sources. Moreover, public social distancing collaborates with infection detection on controlling an infectious disease, but their impacts can't add up perfectly. Besides, public social distancing is slightly less effective than infection detection, because infection detection can also promote spontaneous social distancing.

This paper investigates the problem of defining an optimal long-term investment strategy, where the investor can exit the investment before maturity without severe loss. Our setting is a multi-period one, where the aim is to make a plan for allocating all of wealth among the assets within a time horizon of periods. In addition, the investor can rebalance the portfolio at the beginning of each period. We develop a model in Markowitz context, based on a fused lasso approach. According to it, both wealth and its variation across periods are penalized using the norm, so to produce sparse portfolios, with limited number of transactions. The model leads to a non-smooth constrained optimization problem, where the inequality constraints are aimed to guarantee at least a minimum level of expected wealth at each date. We solve it by using split Bregman method, that has proved to be efficient in the solution of this type of problems. Due to the additive structure of the objective function, the alternating split Bregman at each iteration yields to easier subproblems to be solved, which either admit closed form solutions or can be solved very quickly. Numerical results on data sets generated using real-world price values show the effectiveness of the proposed model.

Aggregation of proteins towards amyloid formation is a significant event in many neurodegenerative diseases. Low-molecular weight oligomers are considered to be the primary toxic agents in many of these maladies. Therefore, there is an increasing interest in understanding their formation and behavior. In this paper, we build on our previously established theoretical investigations on the interactions between A and lipids (L) that adopt off-pathway fibril formation under the control of L concentrations. Our previously developed competing game theoretic framework between the on- and off-pathway dynamics has been expanded to understand the underlying network topological structures in the reaction kinetics of amyloid formation. The mass-action based dynamical systems are solved to identify dominant pathways in the system with fixed initial conditions, and variations in the occurrence of these dominant pathways are identified as a function of various seeding conditions. The mechanistic approach is supported by thermodynamic free energy computations which helps identify stable reactions. The resulting analysis provides possible intervention strategies that can draw the dynamics away from the off-pathways and potential toxic intermediates. We also draw upon the classic literature on network thermodynamics to suggest new approaches to better understand such complex systems.

This work applies a novel geometric criterion for global stability of nonlinear autonomous differential equations generalized by Lu and Lu (2017) to establish global threshold dynamics for several SVEIS epidemic models with temporary immunity, incorporating saturated incidence and nonmonotone incidence with psychological effect, and an SVEIS model with saturated incidence and partial temporary immunity. Incidentally, global stability for the SVEIS models with saturated incidence in Cai and Li (2009), Sahu and Dhar (2012) is completely solved. Furthermore, employing the DEDiscover simulation tool, the parameters in Sahu and Dhar'model are estimated with the 2009-2010 pandemic H1N1 case data in Hong Kong China, and it is validated that the vaccination programme indeed avoided subsequent potential outbreak waves of the pandemic. Finally, global sensitivity analysis reveals that multiple control measures should be utilized jointly to cut down the peak of the waves dramatically and delay the arrival of the second wave, thereinto timely vaccination is particularly effective.

The interaction between epidemic spreading and information diffusion is an interdisciplinary research problem. During an epidemic, people tend to take self-protective measures to reduce the infection risk. However, with the diffusion of rumor, people may be difficult to make an appropriate choice. How to reduce the negative impact of rumor and to control epidemic has become a critical issue in the social network. Elaborate mathematical model is instructive to understand such complex dynamics. In this paper, we develop a two-layer network to model the interaction between the spread of epidemic and the competitive diffusions of information. The results show that knowledge diffusion can eradicate both rumor and epidemic, where the penetration intensity of knowledge into rumor plays a vital role. Specifically, the penetration intensity of knowledge significantly increases the thresholds for rumor and epidemic to break out, even when the self-protective measure is not perfectly effective. But eradicating rumor shouldn't be equated with eradicating epidemic. The epidemic can be eradicated with rumor still diffusing, and the epidemic may keep spreading with rumor being eradicated. Moreover, the communication-layer network structure greatly affects the spread of epidemic in the contact-layer network. When people have more connections in the communication-layer network, the knowledge is more likely to diffuse widely, and the rumor and epidemic can be eradicated more efficiently. When the communication-layer network is sparse, a larger penetration intensity of knowledge into rumor is required to promote the diffusion of knowledge.

Recent studies have demonstrated that the allocation of individual resources has a significant influence on the dynamics of epidemic spreading. In the real scenario, individuals have a different level of awareness for self-protection when facing the outbreak of an epidemic. To investigate the effects of the heterogeneous self-awareness distribution on the epidemic dynamics, we propose a resource-epidemic coevolution model in this paper. We first study the effects of the heterogeneous distributions of node degree and self-awareness on the epidemic dynamics on artificial networks. Through extensive simulations, we find that the heterogeneity of self-awareness distribution suppresses the outbreak of an epidemic, and the heterogeneity of degree distribution enhances the epidemic spreading. Next, we study how the correlation between node degree and self-awareness affects the epidemic dynamics. The results reveal that when the correlation is positive, the heterogeneity of self-awareness restrains the epidemic spreading. While, when there is a significant negative correlation, strong heterogeneous or strong homogeneous distribution of the self-awareness is not conducive for disease suppression. We find an optimal heterogeneity of self-awareness, at which the disease can be suppressed to the most extent. Further research shows that the epidemic threshold increases monotonously when the correlation changes from most negative to most positive, and a critical value of the correlation coefficient is found. When the coefficient is below the critical value, an optimal heterogeneity of self-awareness exists; otherwise, the epidemic threshold decreases monotonously with the decline of the self-awareness heterogeneity. At last, we verify the results on four typical real-world networks and find that the results on the real-world networks are consistent with those on the artificial network.

This work focuses on the definition and study of the -dimensional -vector, an algorithm devised to perform orthogonal range searching in static databases with multiple dimensions. The methodology first finds the order in which to search the dimensions, and then, performs the search using a modified projection method. In order to determine the dimension order, the algorithm uses the -vector, a range searching technique for one dimension that identifies the number of elements contained in the searching range. Then, using this information, the algorithm predicts and selects the best approach to deal with each dimension. The algorithm has a worst case complexity of , where is the number of elements retrieved, is the number of elements in the database, and is the number of dimensions of the database. This work includes a detailed description of the methodology as well as a study of the algorithm performance.

Traditional infectious diseases models on metapopulation networks focus on direct transportations (e.g., direct flights), ignoring the effect of indirect transportations. Based on global aviation network, we turn the problem of indirect flights into a question of second neighbors, and propose a susceptible-infectious-susceptible model to study disease transmission on a connected metapopulation network coupled with its second-neighbor network (SNN). We calculate the basic reproduction number, which is independent of human mobility, and we prove the global stability of disease-free and endemic equilibria of the model. Furthermore, the study shows that the behavior that all travelers travel along the SNN may hinder the spread of disease if the SNN is not connected. However, the behavior that individuals travel along the metapopulation network coupled with its SNN contributes to the spread of disease. Thus for an emerging infectious disease, if the real network and its SNN keep the same connectivity, indirect transportations may be a potential threat and need to be controlled. Our work can be generalized to high-speed train and rail networks, which may further promote other research on metapopulation networks.