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Payoff matrix describing the Prisoner's Dilemma, a classic example from game theory. Players must choose whether to Cooperate (C) or Defect (D) against their opponent, with the payoffs for taking these two different courses of action also dependent on what their opponent chooses. The money values are the payoff to the focal player, and the payoffs for the opponent are symmetrical hence payoffs for the opponent are the same, if they are treated as the focal player instead.
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Historically, infectious diseases caused considerable damage to human societies, and they continue to do so today. To help reduce their impact, mathematical models of disease transmission have been studied to help understand disease dynamics and inform prevention strategies. Vaccination - one of the most important preventive measures of modern time...
Citations
... Furthermore, these methods can help public health decision-makers to enhance the adoption of public health interventions [22] like social distancing, vaccination, or behavior change campaigns, identifying those individuals most likely to get infected and spread an infectious disease or behavior (e.g., super-spreaders), or which places are more likely to be visited by those individuals [23]. This allows more efficient vaccination campaigns [24] when the vaccination of an entire population is not possible or recommended. ...
Detecting early signals of an outbreak in a viral process is challenging due to its exponential nature, yet crucial given the benefits to public health it can provide. If available, the network structure where infection happens can provide rich information about the very early stages of viral outbreaks. For example, more central nodes have been used as social network sensors in biological or informational diffusion processes to detect early contagious outbreaks. We aim to combine both approaches to detect early signals of a biological viral process (influenza-like illness, ILI), using its informational epidemic coverage in public social media. We use a large social media dataset covering three years in a country. We demonstrate that it is possible to use highly central users on social media, more precisely high out-degree users from Twitter, as sensors to detect the early signals of ILI outbreaks in the physical world without monitoring the whole population. We also investigate other behavioral and content features that distinguish those early sensors in social media beyond centrality. While high centrality on Twitter is the most distinctive feature of sensors, they are more likely to talk about local news, language, politics, or government than the rest of the users. Our new approach could detect a better and smaller set of social sensors for epidemic outbreaks and is more operationally efficient and privacy respectful than previous ones, not requiring the collection of vast amounts of data.
... A continuous velocity feedback protocol was initially proposed by Ren [24] to drive all oscillators to be asymptotically synchronised, at which point the directed communication topology was taken into account. In addressing the second-order consensus problem within MASs, coupled harmonic oscillators, a widely studied model in synchronization research [23,25], have been introduced. The synchronization speed states of these oscillators are characterized by time-varying and periodic oscillations, while the final consensus velocity states of first-order and secondorder integrator networks remain time-invariant [25][26][27]. ...
... In addressing the second-order consensus problem within MASs, coupled harmonic oscillators, a widely studied model in synchronization research [23,25], have been introduced. The synchronization speed states of these oscillators are characterized by time-varying and periodic oscillations, while the final consensus velocity states of first-order and secondorder integrator networks remain time-invariant [25][26][27]. However, to the best of our knowledge, there has been scarce exploration of applying reinforcement learning-based optimal control in MASs to coupled harmonic oscillator models. ...
This paper investigates the finite-time optimal synchronisation problem for leader-follower multi-agent systems and its application to the harmonic oscillator models. Neural networks are introduced to fit the nonlinear terms of multi-agent systems due to the existence of unknown dynamics. In our designed framework, by modelling each agent as a resonator, their interactions and environment can be shaped as a networked system. In pursuit of synchronized actions among adjacent agents, the actor-critic reinforcement learning algorithm is implemented. To simplify the algorithm and eliminate persistent incentive conditions simultaneously, gradient descent method is applied to a novel positive function. Furthermore, a finite-time control strategy, based on reinforcement learning algorithms, has been devised to ensure that the system not only achieves control objectives within finite time but also minimizes the energy consumption in the process. Finally, the validity of the theoretical method is proven by the Lyapunov stability theory and numerical simulation.
... Especially, popularizing media education for the susceptible individuals and punishing for the spreading individuals are terrific tactics to restrain rumor spreading [23]. Additionally, media coverage is similar to vaccination in infectious diseases [24]. In [25], authors indicated that the media could suppress the rumor propagation by reporting some truths. ...
With the vigorous development of information technology, online rumors are spreading recklessly. In this article, by comprehensive consideration of the stochastic disturbance, generalized nonlinear incidence, education age and punishment age, the dynamical behaviors of a novel stochastic SEIFR (susceptible-educated-infected-forced silence-removed) rumor spreading model with class-age structure under the generalized nonlinear incidence is analyzed and discussed. Firstly, given that the ubiquity of stochastic disturbance in the process of rumor spreading, a stochastic rumor spreading model is established. Secondly, the existence of the unique global positive solution for the stochastic model is manifested. Thirdly, the sufficient condition of rumor extinction is attained by virtue of Itô’s formula and strong law of large numbers. Additionally, the existence of a unique stationary distribution implying the rumor persistence is investigated based on the method of Khasminskii. Finally, some simulations and a practical application are carried to validate the results.
... In the book Manfredi and d'Onofrio (2013) a vast literature on vaccination and other influences of human behavior on the spread of infectious diseases is presented. A detailed report reviewing models that account for behavioral feedback and / or the spatial / social structure of the population can be found in Wang et al. (2016). More recent publications among the same stream of literature are e.g. ...
... It can be interesting to analyze the stochastic evolution of the number of unvaccinated people. This goal could be obtained by introducing a stochastic effect in communication campaigns and changing the ordinary differential equation (3) into a stochastic one (see Wang et al. 2016). ...
Vaccination is one of the greatest discoveries of modern medicine, capable of defeating many diseases. However, misleading information on the effectiveness of vaccines has caused a decline in vaccination coverage in some countries, leading to the reappearance of related diseases. Therefore, a proper and well-planned pro-vax communication campaign may be effective in convincing people to get vaccinated. We formulate and solve a differential game with an infinite horizon played à la Nash. The players involved in the game are the national healthcare system and a pharmaceutical firm that produces and sells a certain type of vaccine. The former aims to minimize the healthcare costs that unvaccinated people would entail. In turn, the pharmaceutical firm wants to minimize the missed profits from unsold vaccines. The two players run suitable vaccination advertising campaigns to diminish the à-régime number of unvaccinated. The Hamilton-Jacobi-Bellman approach is used to determine a Markovian-Nash equilibrium, studying how communication strategies can be effective in reducing the strength of anti-vax word of mouth.
... The social dilemma of voluntary vaccination as a public goods dilemma was regarded in previous studies [17,18]. Inspired by other social dilemma studies [19,20], Bauch et al. [21] proposed an evolutionary game theory model that incorporated epidemiological dynamics, termed "evolutionary vaccination game", and applied it as a powerful framework for exploring voluntary behaviors [14,[22][23][24][25]. Since human interactions can be modeled as network structures, Fu et al. further combined network structures with vaccination games to analyze the impact of population structures on vaccination behaviors [22]. ...
Most individuals opt for vaccination to acquire immunity protection and prevent disease transmission. However, individuals cannot obtain perfect immunity protection after vaccination, due to various factors such as the limitation of vaccine itself, storage and transportation. Failed vaccination experiences can alter individuals' perception of vaccination behavior. To analyze the influence of vaccine efficacy on vaccination behavior with adaptive perception, we propose a novel vaccination game model. The results demonstrate that for the moderate vaccination cost, the introduction of adaptive perception can promote vaccination behavior, and the promoting effect becomes more pronounced in the population with smaller perception fluctuation. Nonetheless, vaccination behavior is still constrained by a significant number of free-riders when vaccine effectiveness is high. Analyzing the distribution of strategies among individuals with different degrees, it is revealed that the reduction in vaccinated individuals influenced by free-riders predominantly occurs in individuals with low-degree. Furthermore, we examine the coupled effects of vaccination cost and vaccine efficacy on vaccination behavior, considering various levels of perception fluctuations. The results indicate the crucial role of vaccination cost in enhancing vaccination behavior, and previous findings also are consistent across scenarios with diverse vaccination cost. Our work contributes to an improved comprehension of vaccination behavior considering vaccine efficacy and perception.
... Such competition is usually observed in the epidemic trajectories of viruses with multiple strains circulating simultaneously, such as influenza [12] and DENV [13] or more recently in the complex landscape depicted by the different SARS-CoV-2 variants [14,15]. Outside the domain of interacting epidemics, another clear example of the relevance of interdependencies among simultaneous dynamics is the influence of social behavior on epidemic spreading [16,17]. For instance, the existence of mutual feedback between the individual adoption of preventive measures and the spread of a pathogen [18,19] provides a natural mechanism for the emergence of oscillations in the epidemic curves [20,21], even in the absence of seasonal effects shaping the intrinsic transmissibility of the virus. ...
Yet often neglected, dynamical interdependencies between concomitant contagion processes can alter their intrinsic equilibria and bifurcations. A particular case of interest for disease control is the emergence of discontinuous transitions in epidemic dynamics coming from their interactions with other simultaneous processes. To address this problem, here we propose a framework coupling a standard epidemic dynamics with another contagion process, presenting a tunable parameter shaping the nature of its transitions. Our model retrieves well-known results in the literature, such as the existence of first-order transitions arising from the mutual cooperation of epidemics or the onset of abrupt transitions when social contagions unidirectionally drive epidemics. We also reveal that negative feedback loops between simultaneous dynamical processes might suppress abrupt phenomena, thus increasing systems robustness against external perturbations. Our results render a general perspective towards finding different pathways to abrupt phenomena from the interaction of contagion processes.
... They presented the classical SIS and SIR models and defined the concept of infectious disease thresholds. Since then, several attempts have been made to consider more practical factors into epidemic models [4][5][6][7][8]. ...
In this study, we construct a network-based SAIS model that accounts for the interplay of human mobility and asymptomatic infected individuals. The stability analysis of equilibrium points is performed by constructing suitable Lyapunov functions. Based on the next generation matrix method, the basic reproduction number R0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathscr {R}_0$$\end{document} is attained in a form similar to the results derived in the two-strain epidemic model. Numerical results demonstrate the impact of R0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathscr {R}_0$$\end{document} on the final steady state of disease transmission and inspire us that the role of asymptomatic cases cannot be overlooked in the course of a disease. Further simulations reveal the influence mechanism of human mobility on disease spreading in population with heterogeneous contact modes, which will undergo four different stages as the degree of nodes increases. It is concluded that the optimal movement rate should be determined on the basis of human contact patterns, and that controlling population flow with this rate could significantly reduce the peak density of infected individuals.
... However, not every elementary constituent plays the same role in the structure or functionality of a system, with some constituents being more critical and "central" to ensure stability, resilience, or other desired global properties of the architecture [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17]. Identifying the most important nodes in a network architecture is indeed of paramount importance to ensure the integrity and functionality of transportation networks and critical infrastructures [18][19][20][21][22], as well as to allow users to retrieve an accurate list of webpages corresponding to an Internet query [23,24], or identify the most suitable receivers of a vaccine to mitigate a disease outbreak [25][26][27][28]. Our ability to exploit the advantages of living in a modern and interconnected society to the full heavily relies on preserving the integrity of crucial infrastructure such as the Internet and power grids [1, [29][30][31][32][33]. ...
... Solving the self-consistency equations (27) and (26) on the cavity graph and inserting the results into (30) and (29) provides the solution x ⋆ i = µ i of the linear system (7). The equations above are identical to those provided in [75], after some rewriting and rearrangements. ...
The Katz centrality of a node in a complex network is a measure of the node's importance as far as the flow of information across the network is concerned. For ensembles of locally tree-like and undirected random graphs, this observable is a random variable. Its full probability distribution is of interest but difficult to handle analytically because of its "global" character and its definition in terms of a matrix inverse. Leveraging a fast Gaussian Belief Propagation-cavity algorithm to solve linear systems on a tree-like structure, we show that (i) the Katz centrality of a single instance can be computed recursively in a very fast way, and (ii) the probability P (K) that a random node in the ensemble of undirected random graphs has centrality K satisfies a set of recursive distributional equations, which can be analytically characterized and efficiently solved using a population dynamics algorithm. We test our solution on ensembles of Erdős-Rényi and scale-free networks in the locally tree-like regime, with excellent agreement. The distributions display a crossover between multimodality and unimodality as the mean degree increases, where distinct peaks correspond to the contribution to the centrality coming from nodes of different degrees. We also provide an approximate formula based on a rank-1 projection that works well if the network is not too sparse, and we argue that an extension of our method could be efficiently extended to tackle analytical distributions of other centrality measures such as PageRank for directed networks in a transparent and user-friendly way.
... In the mathematical epidemiology community, the behavioural epidemiology of infectious diseases (BEID) [15,22] represents a new discipline that models the evolution of a disease spreading by taking into account spontaneous behavioural responses of individuals to epidemics. The BEID made great use of the evolutionary game theory [21], and in particular, of the imitation game dynamics to model human decisionmaking on the adoption of disease-protective strategies, in terms of ODE and PDE models [1,2,5,14,18], network and individual-based models [10,17]. ...
... The BEID made great use of the evolutionary game theory [21], and in particular, of the imitation game dynamics to model human decisionmaking on the adoption of disease-protective strategies, in terms of ODE and PDE models [1,2,5,14,18], network and individual-based models [10,17]. In particular, the behaviours of individuals are modelled as different strategies whose convenience is defined by a cost-benefit balance [22]. ...
... The switches between the compartments S p and S a are driven by an imitation game dynamics [18,22]. Specifically, let π p , π a be the net costs at population level induced by the pro-vaccine and the anti-vaccine attitude, respectively (the costs associated to, e.g., disease safety, social and psychological well-being, …). ...
Spontaneous behavioural responses of individuals to epidemics are a relevant factor in the understanding of infection dynamics. In this work, we consider a vaccine–preventable infectious disease spreading within a population, where vaccination is on a voluntary basis and individuals can conform to either the pro–vaccine or the anti–vaccine group. A switch of vaccinating attitude may occur following an imitation game dynamics. In particular, we incorporate the role of individuals’ opinion flexibility , that is a measure of the personal propensity to change opinion, in the switch of vaccinating attitude. We consider a disease dynamics of Susceptible–Infected–Removed type. Then, we use the tools of kinetic theory to describe the overall system at microscopic, mesoscopic and macroscopic scale. Finally, the role of flexibility of opinion on the vaccination choice during an epidemic is shown by providing some numerical simulations.
... The expression of the first Lyapunov coefficient 29 is If the environmental factor is in the leading position in the process of the transmission, the endemic will replace the periodic behaviour [ Fig.2(a)]. Meanwhile, a 1 , a 3 , a 4 show their function in the dynamical behaviour of infectious diseases when a 2 = 0.1[ Fig.2(b,c,d)]. As the susceptibility increases, the number of infected individuals will also increase, which is accompanied by the periodic behaviour [ Fig.2(b)]. ...
... and (u, v) is the periodic solution of system (4). ...
The outbreak of infectious diseases often exhibits periodicity, and this periodic behavior can be mathematically represented as a limit cycle. However, the periodic behavior has rarely been considered in demonstrating the cluster phenomenon of infection induced by diffusion (the instability modes) in the SIR model. We investigate the emergence of Turing instability from a stable equilibrium and a limit cycle to illustrate the dynamical and biological mechanisms of pattern formation. We identify the Hopf bifurcation to demonstrate the existence of a stable limit cycle using First Lyapunov coefficient in our spatiotemporal diffusion-driven SIR model. The competition between different instability modes induces different types of patterns and eventually spot patterns emerge as stable patterns. We investigate the impact of susceptible, infected, and recovered individuals on the type of patterns. Interestingly, these instability modes play a vital role in selecting the pattern formations, which is directly related to the number of observed spot patterns. Subsequently, we explain the dynamical and biological mechanisms of spot patterns to develop an effective epidemic prevention strategy.
