We consider a SIR-type compartmental model divided into two age classes to explain the seasonal exacerbations of bacterial meningitis, especially among children outside of the meningitis belt. We describe the seasonal forcing through time-dependent transmission parameters that may represent the outbreak of the meningitis cases after the annual pilgrimage period (Hajj) or uncontrolled inflows of irregular immigrants. We present and analyse a mathematical model with time-dependent transmission. We consider not only periodic functions in the analysis but also general non-periodic transmission processes. We show that the long-time average values of transmission functions can be used as a stability marker of the equilibrium. Furthermore, we interpret the basic reproduction number in case of time-dependent transmission functions. Numerical simulations support and help visualize the theoretical results.

The emergence of epidemics has seriously threatened the running of human society, such as COVID-19. During the epidemics, some external factors usually have a non-negligible impact on the epidemic transmission. Therefore, we not only consider the interaction between epidemic-related information and infectious diseases, but also the influence of policy interventions on epidemic propagation in this work. We establish a novel model that includes two dynamic processes to explore the co-evolutionary spread of epidemic-related information and infectious diseases under policy intervention, one of which depicts information diffusion about infectious diseases and the other denotes the epidemic transmission. A weighted network is introduced into the epidemic spreading to characterize the impact of policy interventions on social distance between individuals. The dynamic equations are established to describe the proposed model according to the micro-Markov chain (MMC) method. The derived analytical expressions of the epidemic threshold indicate that the network topology, epidemic-related information diffusion and policy intervention all have a direct impact on the epidemic threshold. We use numerical simulation experiments to verify the dynamic equations and epidemic threshold, and further discuss the co-evolution dynamics of the proposed model. Our results show that strengthening epidemic-related information diffusion and policy intervention can significantly inhibit the outbreak and spread of infectious diseases. The current work can provide some valuable references for public health departments to formulate the epidemic prevention and control measures.

We investigate the dynamics of a SIRS epidemiological model taking into account cross-superdiffusion and delays in transmission, Beddington-DeAngelis incidence rate and Holling type II treatment. The superdiffusion is induced by inter-country and inter-urban exchange. The linear stability analysis for the steady-state solutions is performed, and the basic reproductive number is calculated. The sensitivity analysis of the basic reproductive number is presented, and we show that some parameters strongly influence the dynamics of the system. A bifurcation analysis to determine the direction and stability of the model is carried out using the normal form and center manifold theorem. The results reveal a proportionality between the transmission delay and the diffusion rate. The numerical results show the formation of patterns in the model, and their epidemiological implications are discussed.

The interplay between disease spreading and personal risk perception is of key importance for modelling the spread of infectious diseases. We propose a planar system of ordinary differential equations (ODEs) to describe the co-evolution of a spreading phenomenon and the average link density in the personal contact network. Contrary to standard epidemic models, we assume that the contact network changes based on the current prevalence of the disease in the population, i.e. the network adapts to the current state of the epidemic. We assume that personal risk perception is described using two functional responses: one for link-breaking and one for link-creation. The focus is on applying the model to epidemics, but we also highlight other possible fields of application. We derive an explicit form for the basic reproduction number and guarantee the existence of at least one endemic equilibrium, for all possible functional responses. Moreover, we show that for all functional responses, limit cycles do not exist. This means that our minimal model is not able to reproduce consequent waves of an epidemic, and more complex disease or behavioural dynamics are required to reproduce epidemic waves.

This study aims at modeling the universal failure in preventing the outbreak of COVID-19 via real-world data from the perspective of complexity and network science. Through formalizing information heterogeneity and government intervention in the coupled dynamics of epidemic and infodemic spreading, first, we find that information heterogeneity and its induced variation in human responses significantly increase the complexity of the government intervention decision. The complexity results in a dilemma between the socially optimal intervention that is risky for the government and the privately optimal intervention that is safer for the government but harmful to the social welfare. Second, via counterfactual analysis against the COVID-19 crisis in Wuhan, 2020, we find that the intervention dilemma becomes even worse if the initial decision time and the decision horizon vary. In the short horizon, both socially and privately optimal interventions agree with each other and require blocking the spread of all COVID-19-related information, leading to a negligible infection ratio 30 days after the initial reporting time. However, if the time horizon is prolonged to 180 days, only the privately optimal intervention requires information blocking, which would induce a catastrophically higher infection ratio than that in the counterfactual world where the socially optimal intervention encourages early-stage information spread. These findings contribute to the literature by revealing the complexity incurred by the coupled infodemic-epidemic dynamics and information heterogeneity to the governmental intervention decision, which also sheds insight into the design of an effective early warning system against the epidemic crisis in the future.

The COVID-19 pandemic has created an urgent need for mathematical models that can project epidemic trends and evaluate the effectiveness of mitigation strategies. A major challenge in forecasting the transmission of COVID-19 is the accurate assessment of the multiscale human mobility and how it impacts infection through close contacts. By combining the stochastic agent-based modeling strategy and hierarchical structures of spatial containers corresponding to the notion of geographical places, this study proposes a novel model, Mob-Cov, to study the impact of human traveling behavior and individual health conditions on the disease outbreak and the probability of zero-COVID in the population. Specifically, individuals perform power law-type local movements within a container and global transport between different-level containers. It is revealed that frequent long-distance movements inside a small-level container (e.g., a road or a county) and a small population size reduce both the local crowdedness and disease transmission. It takes only half of the time to induce global disease outbreaks when the population increases from 150 to 500 (normalized unit). When the exponent of the long-tail distribution of distance moved in the same-level container, , increases, the outbreak time decreases rapidly from 75 to 25 (normalized unit). In contrast, travel between large-level containers (e.g., cities and nations) facilitates global spread of the disease and outbreak. When the mean traveling distance across containers increases from 0.5 to 1 (normalized unit), the outbreak occurs almost twice as fast. Moreover, dynamic infection and recovery in the population are able to drive the bifurcation of the system to a "zero-COVID" state or to a "live with COVID" state, depending on the mobility patterns, population number and health conditions. Reducing population size and restricting global travel help achieve zero-COVID-19. Specifically, when is smaller than 0.2, the ratio of people with low levels of mobility is larger than 80% and the population size is smaller than 400, zero-COVID can be achieved within fewer than 1000 time steps. In summary, the Mob-Cov model considers more realistic human mobility at a wide range of spatial scales, and has been designed with equal emphasis on performance, low simulation cost, accuracy, ease of use and flexibility. It is a useful tool for researchers and politicians to apply when investigating pandemic dynamics and when planning actions against disease.

This paper proposes a data-driven approximate Bayesian computation framework for parameter estimation and uncertainty quantification of epidemic models, which incorporates two novelties: (i) the identification of the initial conditions by using plausible dynamic states that are compatible with observational data; (ii) learning of an informative prior distribution for the model parameters via the cross-entropy method. The new methodology's effectiveness is illustrated with the aid of actual data from the COVID-19 epidemic in Rio de Janeiro city in Brazil, employing an ordinary differential equation-based model with a generalized SEIR mechanistic structure that includes time-dependent transmission rate, asymptomatics, and hospitalizations. A minimization problem with two cost terms (number of hospitalizations and deaths) is formulated, and twelve parameters are identified. The calibrated model provides a consistent description of the available data, able to extrapolate forecasts over a few weeks, making the proposed methodology very appealing for real-time epidemic modeling.

For many applications, small-sample time series prediction based on grey forecasting models has become indispensable. Many algorithms have been developed recently to make them effective. Each of these methods has a specialized application depending on the properties of the time series that need to be inferred. In order to develop a generalized nonlinear multivariable grey model with higher compatibility and generalization performance, we realize the nonlinearization of traditional GM(1,N), and we call it NGM(1,N). The unidentified nonlinear function that maps the data into a better representational space is present in both the NGM(1,N) and its response function. The original optimization problem with linear equality constraints is established in terms of parameter estimation for the NGM(1,N), and two different approaches are taken to solve it. The former is the Lagrange multiplier method, which converts the optimization problem into a linear system to be solved; and the latter is the standard dualization method utilizing Lagrange multipliers, that uses a flexible estimation equation for the development coefficient. As the size of the training data increases, the estimation results of the potential development coefficient get richer and the final estimation results using the average value are more reliable. The kernel function expresses the dot product of two unidentified nonlinear functions during the solving process, greatly lowering the computational complexity of nonlinear functions. Three numerical examples show that the LDNGM(1,N) outperforms the other multivariate grey models compared in terms of generalization performance. The duality theory and framework with kernel learning are instructive for further research around multivariate grey models to follow.

The prediction of the temporal dynamics of chaotic systems is challenging because infinitesimal perturbations grow exponentially. The analysis of the dynamics of infinitesimal perturbations is the subject of stability analysis. In stability analysis, we linearize the equations of the dynamical system around a reference point and compute the properties of the tangent space (i.e. the Jacobian). The main goal of this paper is to propose a method that infers the Jacobian, thus, the stability properties, from observables (data). First, we propose the echo state network (ESN) with the Recycle validation as a tool to accurately infer the chaotic dynamics from data. Second, we mathematically derive the Jacobian of the echo state network, which provides the evolution of infinitesimal perturbations. Third, we analyse the stability properties of the Jacobian inferred from the ESN and compare them with the benchmark results obtained by linearizing the equations. The ESN correctly infers the nonlinear solution and its tangent space with negligible numerical errors. In detail, we compute from data only (i) the long-term statistics of the chaotic state; (ii) the covariant Lyapunov vectors; (iii) the Lyapunov spectrum; (iv) the finite-time Lyapunov exponents; (v) and the angles between the stable, neutral, and unstable splittings of the tangent space (the degree of hyperbolicity of the attractor). This work opens up new opportunities for the computation of stability properties of nonlinear systems from data, instead of equations.

In the classical infectious disease compartment model, the parameters are fixed. In reality, the probability of virus transmission in the process of disease transmission depends on the concentration of virus in the environment, and the concentration depends on the proportion of patients in the environment. Therefore, the probability of virus transmission changes with time. Then how to fit the parameters and get the trend of the parameters changing with time is the key to predict the disease course with the model. In this paper, based on the US COVID-19 epidemic statistics during calibration period, the parameters such as infection rate and recovery rate are fitted by using the linear regression algorithm of machine science, and the laws of these parameters changing with time are obtained. Then a SIR model with time delay and vaccination is proposed, and the optimal control strategy of epidemic situation is analyzed by using the optimal control theory and Pontryagin maximum principle, which proves the effectiveness of the control strategy in restraining the transmission of COVID-19. The numerical simulation results show that the time-varying law of the number of active cases obtained by our model basically conforms to the real changing law of the US COVID-19 epidemic statistics during calibration period. In addition, we have predicted the changes in the number of active cases in the COVID-19 epidemic in the USA over time in the future beyond the calibration cycle, and the predicted results are more in line with the actual epidemic data.

Compartmental models are commonly used in practice to investigate the dynamical response of infectious diseases such as the COVID-19 outbreak. Such models generally assume exponentially distributed latency and infectiousness periods. However, the exponential distribution assumption fails when the sojourn times are expected to distribute around their means. This study aims to derive a novel S (Susceptible)-E (Exposed)-P (Presymptomatic)-A (Asymptomatic)-D (Symptomatic)-C (Reported) model with arbitrarily distributed latency, presymptomatic infectiousness, asymptomatic infectiousness, and symptomatic infectiousness periods. The SEPADC model is represented by nonlinear Volterra integral equations that generalize ordinary differential equation-based models. Our primary aim is the derivation of a general relation between intrinsic growth rate and basic reproduction number with the help of the well-known Lotka-Euler equation. The resulting equation includes separate roles of various stages of the infection and their sojourn time distributions. We show that estimates are considerably affected by the choice of the sojourn time distributions for relatively higher values of . The well-known exponential distribution assumption has led to the underestimation of values for most of the countries. Exponential and delta-distributed sojourn times have been shown to yield lower and upper bounds of the values depending on the values. In quantitative experiments, values of 152 countries around the world were estimated through our novel formulae utilizing the parameter values and sojourn time distributions of the COVID-19 pandemic. The global convergence, , has been estimated through our novel formulation. Additionally, we have shown that increasing the shape parameter of the Erlang distributed sojourn times increases the skewness of the epidemic curves in entire dynamics.

In this paper we study bifurcations in mass-action networks with two chemical species and reactant complexes of molecularity no more than two. We refer to these as planar, quadratic networks as they give rise to (at most) quadratic differential equations on the nonnegative quadrant of the plane. Our aim is to study bifurcations in networks in this class with the fewest possible reactions, and the lowest possible product molecularity. We fully characterise generic bifurcations of positive equilibria in such networks with up to four reactions, and product molecularity no higher than three. In these networks we find fold, Andronov-Hopf, Bogdanov-Takens and Bautin bifurcations, and prove the non-occurrence of any other generic bifurcations of positive equilibria. In addition, we present a number of results which go beyond planar, quadratic networks. For example, we show that mass-action networks without conservation laws admit no bifurcations of codimension greater than , where is the number of reactions; we fully characterise quadratic, rank-one mass-action networks admitting fold bifurcations; and we write down some necessary conditions for Andronov-Hopf and cusp bifurcations in mass-action networks. Finally, we draw connections with a number of previous results in the literature on nontrivial dynamics, bifurcations, and inheritance in mass-action networks.

This study develops a dynamics model of a microrobot vibrating in a blood vessel aiming to detect potential cancer metastasis. We derive an analytical solution for microrobot's motion, considering interactions with the vessel walls modelled by a linear spring-dashpot and a constant damping value for blood viscosity. The model facilitates instantaneous state transitions of the microrobot, such as contact with the vessel wall and free motion within the fluid. Amplitudes and phase angles from the transient solutions of dynamics model of the microrobot are solved at arbitrary moments, providing insights into its transient dynamics. The analytical solution of the proposed system is validated by experimental data, serving as a benchmark to examine the influence of pertinent parameters on microrobot's dynamic response. It is found that the contact force transmitted to the vessel wall, assessed by system's transmissibility function dependent on damping and frequency ratios, decreases with increasing damping ratio and intensifies when the frequency ratio is below . At the frequency ratio is equal to 1, resonance phenomenon is dominated by the magnification factor linked to the damping ratio, increasing the amplitude of resonance as damping decreases. Finally, different sets of system parameters, including excitation frequency and magnitude, fluid damping, vessel wall's stiffness and damping, reveal multi-periodic motions and fake collision of the microrobot with the vessel wall. Simulation results imply that these phenomena are minimally affected by vessel wall's stiffness but are significantly influenced by other parameters, such as fluid damping coefficient and damping coefficient of the blood vessel wall. This research provides a robust theoretical foundation for developing control strategies for microrobots aimed at detecting cancer metastasis.

There exist extensive studies on periodic and random perturbations of various smooth maps investigating their dynamics. Unlike smooth maps, non-smooth maps are yet to be studied extensively under a stochastic regime. This paper presents a stochastic piecewise-smooth map derived from a simple inductorless switching circuit. The stochasticity is introduced in parameter values. The distribution of the parameter values is bounded and randomly selected from uniform and triangular distributions and ranges between high and low bifurcation parameter values of the deterministic map. Due to this inherent stochasticity in parameter values, the time evolution of the state variable cannot be predicted at a specific time instant. We observe that the state variable exhibits completely ergodic behavior when the minimum value of the parameter is the same as the minimum bifurcation parameter of the deterministic system. However, the ensemble average of the state variable converges to a fixed value. The system demonstrates nonchaotic behavior for a particular range of parameter values but the deterministic map in that bifurcation range shows interplay between chaos and periodic orbits. The values of Lyapunov exponents decrease monotonically with increased asymmetry of the distribution from which the bifurcation parameter values are chosen. We determine the probability density function of the stochastic map and verify its invariance under initial conditions. The most noteworthy result is the disappearance of chaotic behavior when the lower range of the distribution is varied while maintaining a fixed upper threshold for a particular distribution, even though the deterministic map exhibits an array of periodic and chaotic behaviors within the range. As the period-incrementing cascade with chaotic inclusion only occurs in nonsmooth maps, this paper numerically shows the stochasticity of a piecewise-smooth map obtained from a practical system for the first time where randomness is introduced in the parameter space.

A very important area where COVID-19 has seriously disrupted is the global financial markets, where stock markets have experienced great turmoil. To shed light on the nature of this turmoil and to characterize nonlinear dynamics in inter-market risk transmission, we formally test the existence of inter-stock market contagion, identify the main channel once the presence of contagion has been established, and assess the upside and downside risk spillovers dynamically focusing on complexity during pre-COVID-19 and post-COVID-19 periods. Applying multiple measures including time-varying conditional value-at-risk based on copula theory, and sample entropy methods, considering a sample covering seven countries (USA, UK, France, Germany, Japan, Brazil, China) during the period from 4 January 2019 to 30 December 2020, we show that contagion is widely present among analysed stock markets with only a few exceptions and that "portfolio rebalancing" as opposed to "wealth constraint" occurs more as the main channel of transmission. All market pairings exhibit significant bilateral upside and downside spillovers after the outbreak of COVID-19. A significant shift in complexity of risk spillover dynamics is evident for most recipient countries following the shock of COVID-19, among which all but China display a downward shift. The findings of this paper could help regulators, politicians, and portfolio risk managers amid the uncertainty created by the COVID-19 pandemic.

Dynamic mode decomposition (DMD) and its variants, such as extended DMD (EDMD), are broadly used to fit simple linear models to dynamical systems known from observable data. As DMD methods work well in several situations but perform poorly in others, a clarification of the assumptions under which DMD is applicable is desirable. Upon closer inspection, existing interpretations of DMD methods based on the Koopman operator are not quite satisfactory: they justify DMD under assumptions that hold only with probability zero for generic observables. Here, we give a justification for DMD as a local, leading-order reduced model for the dominant system dynamics under conditions that hold with probability one for generic observables and non-degenerate observational data. We achieve this for autonomous and for periodically forced systems of finite or infinite dimensions by constructing linearizing transformations for their dominant dynamics within attracting slow spectral submanifolds (SSMs). Our arguments also lead to a new algorithm, data-driven linearization (DDL), which is a higher-order, systematic linearization of the observable dynamics within slow SSMs. We show by examples how DDL outperforms DMD and EDMD on numerical and experimental data.

Since the Leibniz rule for integer-order derivatives of the product of functions, which includes a finite number of terms, is not true for fractional-order (FO) derivatives of that, all sliding mode control (SMC) methods introduced in the literature involved a very limited class of FO nonlinear systems. This article presents a solution for the unsolved problem of SMC of a class of FO nonstrict-feedback nonlinear systems with uncertainties. Using the Leibniz rule for the FO derivative of the product of two functions, which includes an infinite number of terms, it is shown that only one of these terms is needed to design a SMC law. Using this point, an algorithm is given to design the controller for reference tracking, that significantly reduces the number of design parameters, compared to the literature. Then, it is proved that the algorithm has a closed-form solution which presents a straightforward tool to the designer to obtain the controller. The solution is applicable to the systems with a mixture of integer-order and FO dynamics. Stability and finite-time convergence of the offered control law are also demonstrated. In the end, the availability of the suggested SMC is illustrated through a numerical example arising from a real system.

In this paper, a computational approach based on numerical dissipation is proposed to simulate rocking blocks. A rocking block is idealized as a solid body interacting with its foundation through a contact-based formulation. An implicit time integration scheme with numerical dissipation, set to optimally treat dissipation in contact problems, is employed. The numerical dissipation is ruled by the time step and the rocking dissipative phenomenon at impacts is accurately predicted without any damping model. A broad numerical campaign is conducted to define a regression law in analytic form for the setting of the time step, depending on the block size and aspect ratio, the contact stiffness, as well as the coefficient of restitution selected. The so-obtained regression law appears accurate and an a posteriori validation with cases not in the training dataset confirms the effectiveness of the approach. Finally, the comparison with available experimental tests highlights the approach efficacy for free rocking and harmonic loading cases (in a deterministic sense), and for earthquake-like loading cases (in a statistical sense). It is found that rocking blocks with sizes of interest for structural engineering (e.g., cultural heritage structures) can be simulated with time steps within 10 ÷ 10 s, so allowing very fast computations.

The ultimate isolation offered by levitation provides new opportunities for studying fundamental science and realizing ultra-sensitive floating sensors. Among different levitation schemes, diamagnetic levitation is attractive because it allows stable levitation at room temperature without a continuous power supply. While the dynamics of diamagnetically levitating objects in the linear regime are well studied, their nonlinear dynamics have received little attention. Here, we experimentally and theoretically study the nonlinear dynamic response of graphite resonators that levitate in permanent magnetic traps. By large amplitude actuation, we drive the resonators into nonlinear regime and measure their motion using laser Doppler interferometry. Unlike other magnetic levitation systems, here we observe a resonance frequency reduction with amplitude in a diamagnetic levitation system that we attribute to the softening effect of the magnetic force. We then analyze the asymmetric magnetic potential and construct a model that captures the experimental nonlinear dynamic behavior over a wide range of excitation forces. We also investigate the linearity of the damping forces on the levitating resonator, and show that although eddy current damping remains linear over a large range, gas damping opens a route for tuning nonlinear damping forces via the squeeze-film effect.

A major constraint of the behavioral epidemiological models is the assumption that human behavior is static; however, it is highly dynamic, especially in uncertain circumstances during a pandemic. To incorporate the dynamicity of human nature in the existing epidemiological models, we propose a population-wide multi-time-scale theoretical framework that assimilates neuronal plasticity as the basis of altering human emotions and behavior. For that, variable connection weights between different brain regions and their firing frequencies are coupled with a compartmental susceptible-infected-recovered model to incorporate the intrinsic dynamicity in the contact transmission rate ( ). As an illustration, a model of fear conditioning in conjunction with awareness campaigns is developed and simulated. Results indicate that in the presence of fear conditioning, there exists an optimum duration of daily broadcast time during which awareness campaigns are most effective in mitigating the pandemic. Further, global sensitivity analysis using the Morris method highlighted that the learning rate and firing frequency of the unconditioned circuit are crucial regulators in modulating the emergent pandemic waves. The present study makes a case for incorporating neuronal dynamics as a basis of behavioral immune response and has further implications in designing awareness campaigns.

Time series is a data structure prevalent in a wide range of fields such as healthcare, finance and meteorology. It goes without saying that analyzing time series data holds the key to gaining insight into our day-to-day observations. Among the vast spectrum of time series analysis, time series classification offers the unique opportunity to classify the sequences into their respective categories for the sake of automated detection. To this end, two types of mainstream approaches, recurrent neural networks and distance-based methods, have been commonly employed to address this specific problem. Despite their enormous success, methods like Long Short-Term Memory networks typically require high computational resources. It is largely as a consequence of the nature of backpropagation, driving the search for some backpropagation-free alternatives. Reservoir computing is an instance of recurrent neural networks that is known for its efficiency in processing time series sequences. Therefore, in this article, we will develop two reservoir computing based methods that can effectively deal with regular and irregular time series with minimal computational cost, both while achieving a desirable level of classification accuracy.