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(Colour online) Schematic illustration of the adaptive SIV model. The model evolves concurrently according to the node dynamics rule (a) and the link dynamics rule (b) with adaptive rewiring of SI-links. At each time step, susceptible nodes are infected by their infected neighbours at a transmission rate α per SI-link and are vaccinated at a per capita rate ϕ. Infected nodes recover and return to being susceptible at a per capita rate β. Vaccinated nodes become susceptible at a per capita rate ψ and are infected at a vaccinereduced transmission rate δα per IV-link due to imperfect immunity. With the background demographic changes taken into consideration, we assume that each node can give birth to a susceptible node with a birth rate η 1 , and each node dies at a natural death rate η 2. In addition, the infected nodes die with an extra rate µ because of infection, namely, each infected node will die at a rate η 2 + µ. In the adaptive SIV model, we assume that only susceptible nodes are permitted to rewire connections away from their infected neighbours at a rate ω per SI-link. In other words, with rate ω, for each SI-link, the susceptible node breaks the link to the infected node and forms a new link to a randomly selected susceptible or vaccinated node.
Source publication
In the face of serious infectious diseases, governments endeavour to implement containment measures such as public vaccination at a macroscopic level. Meanwhile, individuals tend to protect themselves by avoiding contacts with infections at a microscopic level. However, a comprehensive understanding of how such combined strategy influences epidemic...
Contexts in source publication
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... as a public intervention in reaction to an infectious disease. The nodes of the network represent individuals in the population and links are potentially infectious contacts among them. In the adaptive SIV model, each node may have only one of the three possible states: susceptible (S), infected (I) and vaccinated (V ). As demon-strated in Fig. 1(a), the transition probabilities for node states are as follows: a sus- ceptible node becomes infected with the transmission rate α per SI-link (where the parameters in the model definition are listed in Table 1); an infective node recovers Table 1 Notation used in the model ...
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... to a susceptible node with a birth rate η 1 , and each node (irrespective of its state) dies at a natural death rate η 2 . In addition, the infected nodes die with an extra rate µ because of infection. That is to say, each in- fected node will die at a rate η 2 + µ. Meanwhile, the network is rewired adaptively during the disease propagation [see Fig. 1(b) as an illustration]. For simplicity, only susceptible nodes are allowed to rewire their connections from infected neighbours to randomly selected noninfected (either susceptible or vaccinated) nodes with the rewiring rate ω. Self links and multiple links are prohibited. The adaptive SIV model therefore consists of two processes ( ...
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... [see Fig. 1(b) as an illustration]. For simplicity, only susceptible nodes are allowed to rewire their connections from infected neighbours to randomly selected noninfected (either susceptible or vaccinated) nodes with the rewiring rate ω. Self links and multiple links are prohibited. The adaptive SIV model therefore consists of two processes ( Fig. 1): one is the node dynamics due to the epi- demic spreading and the birth and death process, the other is the link dynamics due to the rewiring of SI-links of the underlying ...
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... F ( ϕ=0.0010, ω=0.010) S F ( ϕ=0.0008, ω=0.010) S F ( ϕ=0.0008, ω=0.008) the rewiring rate causes M SI to decrease, whereas enhancing the vaccination rate increases M SI . This case is in sharp contrast to the model without demographic pro- cesses, in which both the rewiring and vaccination contribute to decrease M SI and hence N I , as shown in Fig. 10. These counterintuitive results imply that in the case of a varying population with demographic effects (births, deaths and migration), sus- ceptible individuals' protective behavior and the public vaccination interventions that aim to suppress the epidemic, would make the situation even worse. These phenom- ena can be explained as ...
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... our model is able to capture the topological structure of the network in the stationary state of the epidemic. In particular, the degree distribution in each class of nodes is shown in Fig. 11, where the black, red, and blue colours correspond to the susceptible, infected, and vaccinated nodes, respectively. To be precise, the sym- ...
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... Fig. 12 (Colour online) Mean nearest-neighbour degree knn as a function of any given degree k in each class of nodes in the state at t = 1000. The results on the left (right) column are obtained from an initial ER (SF) network, with an average degree k = 3. The empty boxes (black) "", circles (red) "" and triangles (blue) "" represent ...
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... and the solid lines are the analytical solutions to system equations. As shown in Figs. 11(a) and 11(b), without the birth and death processes, the network evolves to the same degree distribution due to the rewiring mechanism. On the contrary, the situation is quite different with the introduction of birth and death processes. For instance, in Figs. 11(c) and 11(d) we exhibit the degree distribution of each class of nodes at the final time step t = 1000, for the settings in Figs. 7(e) and 7(f), respectively. In the presence of node birth, different degree dis- tributions of the newborns will lead to different network structures. It is clear that the final network is narrowly distributed if nodes ...
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... each class of nodes at the final time step t = 1000, for the settings in Figs. 7(e) and 7(f), respectively. In the presence of node birth, different degree dis- tributions of the newborns will lead to different network structures. It is clear that the final network is narrowly distributed if nodes are born with a poisson degree dis- tribution [ Fig. 11(c)], while the network evolves to a broader degree distribution if the newborns follow a power-law degree distribution [ Fig. 11(d)]. It is worth noting that by the mathematical model of Eqs. (10)- (18), the theoretical results of degree distribution can be obtained not only for the model with demography, but also for the case without ...
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... node birth, different degree dis- tributions of the newborns will lead to different network structures. It is clear that the final network is narrowly distributed if nodes are born with a poisson degree dis- tribution [ Fig. 11(c)], while the network evolves to a broader degree distribution if the newborns follow a power-law degree distribution [ Fig. 11(d)]. It is worth noting that by the mathematical model of Eqs. (10)- (18), the theoretical results of degree distribution can be obtained not only for the model with demography, but also for the case without demography where the birth and death rates are set to zero, as il- lustrated in the Figs. 11(a) and 11(b). Nevertheless, under the ...
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... follow a power-law degree distribution [ Fig. 11(d)]. It is worth noting that by the mathematical model of Eqs. (10)- (18), the theoretical results of degree distribution can be obtained not only for the model with demography, but also for the case without demography where the birth and death rates are set to zero, as il- lustrated in the Figs. 11(a) and 11(b). Nevertheless, under the modelling framework of Eqs. (1)-(9) where network properties are not included, the theoretical results of degree distribution cannot be ...
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... mentioned above, since the birth process dominates in affecting the degree distribution of nodes, the network structure is degree uncorrelated, which is shown in Fig. 12. Although the individual rewiring changes the local connectivity pattern among nodes, the continuous input of the newborns that randomly link to old ones breaks the degree correlation that emerges in the scenario without birth and death processes (Fig. 6). Note that the average nearest-neighbour degree is close to 4, k nn 4, which is ...
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... processes (Fig. 6). Note that the average nearest-neighbour degree is close to 4, k nn 4, which is larger than the average degree of the network k, which is around 2.9 in our simulations. This is consistent with the friendship paradox. Similar results are observed for different parameter values among each class of nodes in both ER and SF networks (Fig. 12). It is worth noting that although newborn nodes arrive with mean degree 3, it will not still be the mean degree of the network due to a more rapid death of infectives. Actually, the mean degree of the entire network is parameter ...
Citations
... Historically, great victories in the prevention of disease spread have been achieved through the distribution of vaccines [28]. Therefore it makes sense to include vaccines in the modelling [29,30]. ...
We numerically investigate the dynamics of an SIR model with infection level-based lockdowns on Small-World networks. Using a large-deviation approach, namely the Wang-Landau algorithm, we study the distribution of the cumulative fraction of infected individuals. We are able to resolve the density of states for values as low as 10 ⁻⁸⁵ . Hence, we measure the distribution on its full support giving a complete characterization of this quantity. The lockdowns are implemented by severing a certain fraction of the edges in the Small-World network, and are initiated and released at different levels of infection, which are varied within this study. We observe points of non-analytical behaviour for the pdf and discontinuous transitions for correlations with other quantities such as the maximum fraction of infected and the duration of outbreaks. Further, empirical rate functions were calculated for different system sizes, for which a convergence is clearly visible indicating that the large-deviation principle is valid for the system with lockdowns.
... The second motivation of our work is that most recent works address the roles of vaccination and quarantine 2 J o u r n a l P r e -p r o o f Journal Pre-proof (isolation) for preventing and controlling epidemic outbreak. Some works have introduced vaccination and quarantine into mathematical models to study the role of these two controls [23][24][25][26], which provides understanding of the impact of control measures on infectious disease transmission mechanisms. In addition, the development and improvement of related control theories offer sufficient theoretical support for us to carry out in-depth analysis on the prevention of infectious diseases using optimal control theory [27,28]. ...
We present a novel SIRS model on scale-free networks that takes into account behavioral memory and time delayto depict an adaptive behavioral feedback mechanism, which can better characterize the actual spread of epidemics.We conduct rigorous analysis on the dynamics of the model, including the basic reproduction number R0, uniformpersistence and the global asymptotic stability of equilibria. The model has the sharp threshold property, namely,if R0 is less than 1 then the disease-free equilibrium is globally asymptotically stable while if R0 is larger than 1then the endemic equilibrium is globally asymptotically stable. We further perform an optimal control study forthe model to seek effective vaccination and treatment strategies. The existence and uniqueness of these optimalcontrol strategies are demonstrated. Finally, we perform some stochastic network simulations that yield quantita-tive agreement with the deterministic mean-field approach. Our findings indicate that time delay does not affectR0, but behavioral memory does.
... In this work, we consider a discrete epidemic spreading model consisting of three states: susceptible (S), infectious (I ), and vaccinated (V ), for which the changes of states between S and I or S and V take into account the limited medical resources. The continuous S − (I , V ) − S models (Kribs-Zaleta and Valesco-Hernández 2000; Gumel and Moghadas 2003;Knipl et al. 2015;Peng et al. 2013Peng et al. , 2016Lv et al. 2020) with limited medical resources have been extensively studied partly because there are more tractable mathematically. However, a good reason for studying discrete models is that data are collected at discrete times and hence it may be easier to compare data with the output of a discrete model. ...
A discrete epidemic model with vaccination and limited medical resources is proposed to understand its underlying dynamics. The model induces a nonsmooth two dimensional map that exhibits a surprising array of dynamical behavior including the phenomena of the forward-backward bifurcation and period doubling route to chaos with feasible parameters in an invariant region. We demonstrate, among other things, that the model generates the above described phenomena as the transmission rate or the basic reproduction number of the disease gradually increases provided that the immunization rate is low, the vaccine failure rate is high and the medical resources are limited. Finally, the numerical simulations are provided to illustrate our main results.
... Models have been developed for HIV/AIDS [33], SARS [34], malaria [35], cholera [36], tuberculosis [37], and COVID-19 [38]. Additionally, Magori and Park [39] studied the consequences of imperfect vaccines, Peng et al. [40] studied susceptible-infected-susceptible epidemic model with imperfect vaccination on dynamic contact networks, Abboubakar et al. [41] explored imperfect vaccines as a way to control arboviral diseases, and Arino and Milliken [42] related imperfect vaccines to the treatment of COVID- 19. There are two main mechanisms of imperfect vaccines: ''all-ornothing'' and ''leaky'' vaccine [43]. ...
As infectious diseases continue to threaten communities across the globe, people are faced with a choice to vaccinate, or not. Many factors influence this decision, such as the cost of the disease, the chance of contracting the disease, the population vaccination coverage, and the efficacy of the vaccine. While the vaccination games in which individuals decide whether to vaccinate or not based on their own interests are gaining in popularity in recent years, the vaccine imperfection has been an overlooked aspect so far. In this paper we investigate the effects of an imperfect vaccine on the outcomes of a vaccination game. We use a simple SIR compartmental model for the underlying model of disease transmission. We model the vaccine imperfection by adding vaccination at birth and maintain a possibility for the vaccinated individual to become infected. We derive explicit conditions for the existence of different Nash equilibria, the solutions of the vaccination game. The outcomes of the game depend on the complex interplay between disease transmission dynamics (the basic reproduction number), the relative cost of the infection, and the vaccine efficacy. We show that for diseases with relatively low basic reproduction numbers (smaller than about 2.62), there is a little difference between outcomes for perfect or imperfect vaccines and thus the simpler models assuming perfect vaccines are good enough. However, when the basic reproduction number is above 2.62, then, unlike in the case of a perfect vaccine, there can be multiple equilibria. Moreover, unless there is a mandatory vaccination policy in place that would push the vaccination coverage above the value of unstable Nash equilibrium, the population could eventually slip to the "do not vaccinate" state. Thus, for diseases that have relatively high basic reproduction numbers, the potential for the vaccine not being perfect should be explicitly considered in the models.
... Knowledge of the characteristics of society is an important element in the fight against pandemics (Peng et al., 2016). It is important that the strategies used in different countries to combat the pandemic match the profile of the country and the characteristics of the population, and determining which strategy is implemented where, when, and by whom is important to achieving successful results (Weng et al., 2020). ...
The aim of the study is to examine personality traits and COVID-19 related factors as predictors of attitudes toward the COVID-19 vaccine. Fear of COVID-19 and perceived causes of COVID-19 are investigated as COVID-19 related factors. The sample consists of 1697 Turkish emerging adults (50.1% female; ages 18-25). The result of T-test for gender difference shows that females (M=35.19, SD=8.27) have significantly more positive attitude towards COVID-19 vaccine than males (M=33.86, SD=8.76). Attitude towards COVID-19 vaccine is predicted by conspiracy, environment and faith as causes of COVID-19, fear of COVID-19 and openness to experience as personality traits. All of these predictors explained 29% of the total variance in attitudes toward the vaccine. Results are discussed in light of previous research.
... In the first, the models assume vaccination as a treatment, whereby individuals are moved from the infected to the recovered population. [28][29][30][31] In the second category, the models consider the vaccinated individuals as members of a separate population, [32][33][34][35] which is modeled by its own ODE. According to the models in the first category, it is assumed that the population is vaccinated at a constant rate. ...
The slogan “nobody is safe until everybody is safe” is a dictum to raise awareness that in an interconnected world, pandemics, such as COVID-19, require a global approach. Motivated by the ongoing COVID-19 pandemic, we model here the spread of a virus in interconnected communities and explore different vaccination scenarios, assuming that the efficacy of the vaccination wanes over time. We start with susceptible populations and consider a susceptible–vaccinated–infected–recovered model with unvaccinated (“Bronze”), moderately vaccinated (“Silver”), and very-well-vaccinated (“Gold”) communities, connected through different types of networks via a diffusive linear coupling for local spreading. We show that when considering interactions in “Bronze”–“Gold” and “Bronze”–“Silver” communities, the “Bronze” community is driving an increase in infections in the “Silver” and “Gold” communities. This shows a detrimental, unidirectional effect of non-vaccinated to vaccinated communities. Regarding the interactions between “Gold,” “Silver,” and “Bronze” communities in a network, we find that two factors play a central role: the coupling strength in the dynamics and network density. When considering the spread of a virus in Barabási–Albert networks, infections in “Silver” and “Gold” communities are lower than in “Bronze” communities. We find that the “Gold” communities are the best in keeping their infection levels low. However, a small number of “Bronze” communities are enough to give rise to an increase in infections in moderately and well-vaccinated communities. When studying the spread of a virus in dense Erdős–Rényi and sparse Watts–Strogatz and Barabási–Albert networks, the communities reach the disease-free state in the dense Erdős–Rényi networks, but not in the sparse Watts–Strogatz and Barabási–Albert networks. However, we also find that if all these networks are dense enough, all types of communities reach the disease-free state. We conclude that the presence of a few unvaccinated or partially vaccinated communities in a network can increase significantly the rate of infected population in other communities. This reveals the necessity of a global effort to facilitate access to vaccines for all communities.
... In the first, the models assume vaccination as a treatment, whereby individuals are moved from the infected to the recovered population [7,22,52,76]. In the second category, the models consider the vaccinated individuals as members of a separate population [34,51,50,61], which is modelled by its own ODE. According to the models in the first category, it is assumed that the population is vaccinated at a constant rate. ...
The slogan "nobody is safe until everybody is safe" is a dictum to raise awareness that in an interconnected world, pandemics such as COVID-19, require a global approach. Motivated by the ongoing COVID-19 pandemic, we model here the spread of a virus in interconnected communities and explore different vaccination scenarios, assuming that the efficacy of the vaccination wanes over time. We start with susceptible populations and consider a susceptible-vaccinated-infected-recovered model with unvaccinated ("Bronze"), moderately vaccinated ("Silver") and very well vaccinated ("Gold") communities, connected through different types of networks via a diffusive linear coupling for local spreading. We show that when considering interactions in "Bronze"-"Gold" and "Bronze"-"Silver" communities, the "Bronze" community is driving an increase in infections in the "Silver" and "Gold" communities. This shows a detrimental, unidirectional effect of non-vaccinated to vaccinated communities. Regarding the interactions between "Gold", "Silver" and "Bronze" communities in a network, we find that two factors play central role: the coupling strength in the dynamics and network density. When considering the spread of a virus in Barab\'asi-Albert networks, infections in "Silver" and "Gold" communities are lower than in "Bronze" communities. We find that the "Gold" communities are the best in keeping their infection levels low. However, a small number of "Bronze" communities are enough to give rise to an increase in infections in moderately and well-vaccinated communities. When studying the spread of a virus in a dense Erd\H{o}s-R\'enyi, and sparse Watts-Strogatz and Barab\'asi-Albert networks, the communities reach the disease-free state in the dense Erd\H{o}s-R\'enyi networks, but not in the sparse Watts-Strogatz and Barab\'asi-Albert networks. However, we also find that if all these networks...
... Today, the COVID-19 pandemic has spread all over the world and, as of June 2021, SARS-CoV-2 has infected more than 175 million people, causing around 3.8 million deaths. Typically, vaccination is one of the most important preventive measures to prevent or reduce virus propagation and, when its availability is limited, one of the most important issues to study is the effectiveness of different kinds of vaccination strategies aimed at cutting off potential chains of transmission and avoiding as many potential deaths as possible [5][6][7][8][9][10][11][12][13][14]. However, when vaccines are unavailable or scarce, it is social behaviour combined with preventive measures that is the most effective way to reduce and control the spread of the disease [15][16][17]. ...
The behaviour of individuals is a main actor in the control of the spread of a communicable disease and, in turn, the spread of an infectious disease can trigger behavioural changes in a population. Here, we study the emergence of individuals’ protective behaviours in response to the spread of a disease by considering two different social attitudes within the same population: concerned and risky. Generally speaking, concerned individuals have a larger risk aversion than risky individuals. To study the emergence of protective behaviours, we couple, to the epidemic evolution of a susceptible-infected-susceptible model, a decision game based on the perceived risk of infection. Using this framework, we find the effect of the protection strategy on the epidemic threshold for each of the two subpopulations (concerned and risky), and study under which conditions risky individuals are persuaded to protect themselves or, on the contrary, can take advantage of a herd immunity by remaining healthy without protecting themselves, thanks to the shield provided by concerned individuals.
This article is part of the theme issue ‘Emergent phenomena in complex physical and socio-technical systems: from cells to societies’.
... The PLOS series and Hindawi publishers published a special collection of articles on this topic lately. Third, the vast development of vaccines for many infectious diseases enabled health authorities to carry out mass vaccination/immunization and could change the pattern of infectious disease dynamics [57,58]. Fourth, the recent advancement of technology, computer programing enabled mathematicians to use complex equations to predict future patterns based on certain early epidemiologic data [59,60]. ...
Background
Mathematical analysis and modeling allow policymakers to understand and predict the dynamics of an infectious disease under several different scenarios. The current study aimed to analyze global research activity on mathematical modeling of transmission and control of several infectious diseases with a known history of serious outbreaks.
Methods
Relevant publications were retrieved using a comprehensive validated search query. The database used was SciVerse Scopus. Indicators related to evolution, growth of publications, infectious diseases encountered, key players, citations, and international research collaboration were presented.
Results
The search strategy found 5606. The growth of publications started in 1967 and showed a sharp rise in 2020 and 2021. The retrieved articles received relatively high citations (h-index = 158). Despite being multidisciplinary, Plos One journal made the highest contribution to the field. The main findings of the study are summarized as follows: (a) COVID-19 had a strong impact on the number of publications in the field, specifically during the years 2020 and 2021; (b) research in the field was published in a wide range of journals, mainly those in the field of infectious diseases and mathematical sciences; (c) research in the field was mainly published by scholars in the United States and the United Kingdom; (d) international research collaboration between active countries and less developed countries was poor; (e) research activity relied on research groups with a large number of researchers per group indicative of good author-author collaboration; (f) HIV/AIDS, coronavirus disease, influenza, and malaria were the most frequently researched diseases; (g) recently published articles on COVID-19 received the highest number of citations; and (h) researchers in the Eastern Mediterranian and South-East Asian regions made the least contribution to the retrieved articles.
Conclusion
Mathematical modeling is gaining popularity as a tool for understanding the dynamics of infectious diseases. The application of mathematical modeling on new emerging infectious disease outbreaks is a priority. Research collaboration with less developed countries in the field of mathematical epidemiology is needed and should be prioritized and funded.
... They believed that the contact between individuals and the extension of the quarantine period was the most effective strategies to combat infectious diseases. Quarantine measures play an essential role in preventing human diseases and epidemics, such as smallpox, tuberculosis, SARS, and the current outbreak of COVID- 19. ...
... Therefore, vaccination is one of the most effective public policies to prevent the spread of infectious diseases [18]. Peng et al. [19] built the SIV model, which combined public vaccination with personal protection, and provided a more comprehensive approach to eradicate infectious diseases. Lv et al. [20] established a SIVS model based on the variability of population structure and calculated the optimal control strategy for vaccination. ...
To contain the novel SARS-CoV-2 (COVID-19) spreading worldwide, governments generally adopt two measures: quarantining the infected people and vaccinating the susceptible people. To investigate the disease latency's influence on the transmission characteristics of the system, we establish a new SIQR-V (susceptible-infective-quarantined-recovered-vaccinated) dynamic model that focus on the effectiveness of quarantine and vaccination measures in the scale-free network. We use theoretical analysis and numerical simulation to explore the evolution trend of different nodes and factors influencing the system stability. The study shows that both the complexity of the network and latency delay can affect the evolution trend of the infected nodes in the system. Still, only latency delay can destroy the stability of the system. In addition, through the parameter sensitivity analysis of the basic reproduction number, we find that the effect of the vaccination parameter α on the basic reproduction number R0 is more significant than that of transmission rate β and quarantine parameter σ . It shows that vaccination is one of the most effective public policies to prevent infectious diseases’ spread. Finally, we calculate the basic reproduction numbers that are greater than one for Germany and Pakistan under COVID-19 and validate the model’s effectiveness based on the disease data of COVID-19 in Germany. The results show that the changing trend of the infected population in Germany based on the SIQR-V model is roughly the same as that reflected by the actual epidemic data in Germany. Therefore, providing suggestions and guidance for treating infectious diseases based on this model can effectively reduce the harm caused by the outbreak of contagious diseases.
