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Hussain SA, Meine DCA, Vvedensky DD. Integrate-and-fire model of disease transmission. Phys Rev E 2024; 110:014305. [PMID: 39160983 DOI: 10.1103/physreve.110.014305] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/27/2024] [Accepted: 07/04/2024] [Indexed: 08/21/2024]
Abstract
We create an epidemiological susceptible-infected-susceptible model of disease transmission using integrate-and-fire nodes on a network, allowing memory of previous interactions and infections. Agents in the network sum infectious matter from their nearest neighbors at every time step, until they exceed their infection threshold, at which point they "fire" and become infected for as long as the recovery time. The model has memory of previous interactions by tracking the amount of infectious matter carried by agents as well as just binary infected or susceptible states, and the model has memory of previous infections by modeling immunity as increasing the infection threshold after recovery. Creating a simulation of the model on networks with a power-law degree distribution and homogeneous agent parameters, we find a single strain version of the model matches well with the England COVID-19 case data, with a root-mean-squared error of 0.014%. A simulation of a multistrain version of the model (where there is cross-strain immunity) matches well with the influenza strain A and strain B case numbers in Canada, with a root-mean-squared error of 0.002% and 0.0012%, respectively, though due to the coupling in the model, both strains peak in phase. Since the dynamics of the model successfully capture real-life transmission dynamics, we test interventions to study their effect on case numbers, with both quarantining and social gathering restrictions lowering the peak. Since the model has memory, the stricter the intervention, the higher the secondary peak when the restriction is removed, showing that interventions change only the shape of the curves and not the overall number infected in the population.
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2
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Granger T, Michelitsch TM, Bestehorn M, Riascos AP, Collet BA. Stochastic Compartment Model with Mortality and Its Application to Epidemic Spreading in Complex Networks. ENTROPY (BASEL, SWITZERLAND) 2024; 26:362. [PMID: 38785610 PMCID: PMC11120256 DOI: 10.3390/e26050362] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/18/2024] [Revised: 04/21/2024] [Accepted: 04/23/2024] [Indexed: 05/25/2024]
Abstract
We study epidemic spreading in complex networks by a multiple random walker approach. Each walker performs an independent simple Markovian random walk on a complex undirected (ergodic) random graph where we focus on the Barabási-Albert (BA), Erdös-Rényi (ER), and Watts-Strogatz (WS) types. Both walkers and nodes can be either susceptible (S) or infected and infectious (I), representing their state of health. Susceptible nodes may be infected by visits of infected walkers, and susceptible walkers may be infected by visiting infected nodes. No direct transmission of the disease among walkers (or among nodes) is possible. This model mimics a large class of diseases such as Dengue and Malaria with the transmission of the disease via vectors (mosquitoes). Infected walkers may die during the time span of their infection, introducing an additional compartment D of dead walkers. Contrary to the walkers, there is no mortality of infected nodes. Infected nodes always recover from their infection after a random finite time span. This assumption is based on the observation that infectious vectors (mosquitoes) are not ill and do not die from the infection. The infectious time spans of nodes and walkers, and the survival times of infected walkers, are represented by independent random variables. We derive stochastic evolution equations for the mean-field compartmental populations with the mortality of walkers and delayed transitions among the compartments. From linear stability analysis, we derive the basic reproduction numbers RM,R0 with and without mortality, respectively, and prove that RM1, the healthy state is unstable, whereas for zero mortality, a stable endemic equilibrium exists (independent of the initial conditions), which we obtained explicitly. We observed that the solutions of the random walk simulations in the considered networks agree well with the mean-field solutions for strongly connected graph topologies, whereas less well for weakly connected structures and for diseases with high mortality. Our model has applications beyond epidemic dynamics, for instance in the kinetics of chemical reactions, the propagation of contaminants, wood fires, and others.
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Affiliation(s)
- Téo Granger
- Sorbonne Université, Institut Jean le Rond d’Alembert, CNRS UMR 7190, 4 Place Jussieu, 75252 Paris, Cedex 05, France (B.A.C.)
| | - Thomas M. Michelitsch
- Sorbonne Université, Institut Jean le Rond d’Alembert, CNRS UMR 7190, 4 Place Jussieu, 75252 Paris, Cedex 05, France (B.A.C.)
| | - Michael Bestehorn
- Institut für Physik, Brandenburgische Technische Universität Cottbus-Senftenberg, Erich-Weinert-Straße 1, 03046 Cottbus, Germany;
| | | | - Bernard A. Collet
- Sorbonne Université, Institut Jean le Rond d’Alembert, CNRS UMR 7190, 4 Place Jussieu, 75252 Paris, Cedex 05, France (B.A.C.)
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Huang Z, Shu X, Xuan Q, Ruan Z. Epidemic spreading under game-based self-quarantine behaviors: The different effects of local and global information. CHAOS (WOODBURY, N.Y.) 2024; 34:013112. [PMID: 38198677 DOI: 10.1063/5.0180484] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/10/2023] [Accepted: 12/12/2023] [Indexed: 01/12/2024]
Abstract
During the outbreak of an epidemic, individuals may modify their behaviors in response to external (including local and global) infection-related information. However, the difference between local and global information in influencing the spread of diseases remains inadequately explored. Here, we study a simple epidemic model that incorporates the game-based self-quarantine behavior of individuals, taking into account the influence of local infection status, global disease prevalence, and node heterogeneity (non-identical degree distribution). Our findings reveal that local information can effectively contain an epidemic, even with only a small proportion of individuals opting for self-quarantine. On the other hand, global information can cause infection evolution curves shaking during the declining phase of an epidemic, owing to the synchronous release of nodes with the same degree from the quarantined state. In contrast, the releasing pattern under the local information appears to be more random. This shaking phenomenon can be observed in various types of networks associated with different characteristics. Moreover, it is found that under the proposed game-epidemic framework, a disease is more difficult to spread in heterogeneous networks than in homogeneous networks, which differs from conventional epidemic models.
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Affiliation(s)
- Zegang Huang
- Institute of Cyberspace Security, Zhejiang University of Technology, Hangzhou 310023, China
- Binjiang Cyberspace Security Institute of ZJUT, Hangzhou 310051, China
| | - Xincheng Shu
- Institute of Cyberspace Security, Zhejiang University of Technology, Hangzhou 310023, China
- Binjiang Cyberspace Security Institute of ZJUT, Hangzhou 310051, China
| | - Qi Xuan
- Institute of Cyberspace Security, Zhejiang University of Technology, Hangzhou 310023, China
- Binjiang Cyberspace Security Institute of ZJUT, Hangzhou 310051, China
| | - Zhongyuan Ruan
- Institute of Cyberspace Security, Zhejiang University of Technology, Hangzhou 310023, China
- Binjiang Cyberspace Security Institute of ZJUT, Hangzhou 310051, China
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Peyrard M. What can we learn from the dynamics of the Covid-19 epidemic ? CHAOS (WOODBURY, N.Y.) 2023; 33:103101. [PMID: 37782831 DOI: 10.1063/5.0161222] [Citation(s) in RCA: 2] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/08/2023] [Accepted: 09/13/2023] [Indexed: 10/04/2023]
Abstract
We investigate the mechanisms behind quasi-periodic outbursts on the Covid-19 epidemics. Data for France and Germany show that the patterns of outbursts exhibit a qualitative change in early 2022, which appears in a change in their average period and which is confirmed by the time-frequency analysis. This provides a signal that can be used to discriminate among several mechanisms. Two main ideas have been proposed to explain periodicity in epidemics. One involves memory effects and another considers exchanges between epidemic clusters and a reservoir of population. We test these two approaches in the particular case of the Covid-19 epidemics and show that the "cluster model" is the only one that appears to be able to explain the observed pattern with realistic parameters. The last section discusses our results in the context of early studies of epidemics, and we stress the importance to work with models with a limited number of parameters, which moreover can be sufficiently well estimated, to draw conclusions on the general mechanisms behind the observations.
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Affiliation(s)
- M Peyrard
- Ecole Normale Supérieure de Lyon, Laboratoire de Physique CNRS UMR 5672, 46 allée d'Italie, F-69364 Lyon Cedex 7, France
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Zhu Y, Shen R, Dong H, Wang W. Spatial heterogeneity and infection patterns on epidemic transmission disclosed by a combined contact-dependent dynamics and compartmental model. PLoS One 2023; 18:e0286558. [PMID: 37310972 DOI: 10.1371/journal.pone.0286558] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/11/2023] [Accepted: 05/18/2023] [Indexed: 06/15/2023] Open
Abstract
Epidemics, such as COVID-19, have caused significant harm to human society worldwide. A better understanding of epidemic transmission dynamics can contribute to more efficient prevention and control measures. Compartmental models, which assume homogeneous mixing of the population, have been widely used in the study of epidemic transmission dynamics, while agent-based models rely on a network definition for individuals. In this study, we developed a real-scale contact-dependent dynamic (CDD) model and combined it with the traditional susceptible-exposed-infectious-recovered (SEIR) compartment model. By considering individual random movement and disease spread, our simulations using the CDD-SEIR model reveal that the distribution of agent types in the community exhibits spatial heterogeneity. The estimated basic reproduction number R0 depends on group mobility, increasing logarithmically in strongly heterogeneous cases and saturating in weakly heterogeneous conditions. Notably, R0 is approximately independent of virus virulence when group mobility is low. We also show that transmission through small amounts of long-term contact is possible due to short-term contact patterns. The dependence of R0 on environment and individual movement patterns implies that reduced contact time and vaccination policies can significantly reduce the virus transmission capacity in situations where the virus is highly transmissible (i.e., R0 is relatively large). This work provides new insights into how individual movement patterns affect virus spreading and how to protect people more efficiently.
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Affiliation(s)
- Youyuan Zhu
- Kuang Yaming Honors School, Nanjing University, Nanjing, China
- Collaborative Innovation Center of Advanced Microstructures, National Laboratory of Solid State Microstructure, Nanjing, China
- Department of Physics, Nanjing University, Nanjing, China
| | - Ruizhe Shen
- Kuang Yaming Honors School, Nanjing University, Nanjing, China
- Collaborative Innovation Center of Advanced Microstructures, National Laboratory of Solid State Microstructure, Nanjing, China
- Department of Physics, Nanjing University, Nanjing, China
| | - Hao Dong
- Kuang Yaming Honors School, Nanjing University, Nanjing, China
- Department of Physics, Nanjing University, Nanjing, China
- State Key Laboratory of Analytical Chemistry for Life Science, Nanjing University, Nanjing, China
| | - Wei Wang
- Collaborative Innovation Center of Advanced Microstructures, National Laboratory of Solid State Microstructure, Nanjing, China
- Department of Physics, Nanjing University, Nanjing, China
- Institute for Brain Sciences, Nanjing University, Nanjing, China
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Granger T, Michelitsch TM, Bestehorn M, Riascos AP, Collet BA. Four-compartment epidemic model with retarded transition rates. Phys Rev E 2023; 107:044207. [PMID: 37198844 DOI: 10.1103/physreve.107.044207] [Citation(s) in RCA: 2] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/21/2022] [Accepted: 03/17/2023] [Indexed: 05/19/2023]
Abstract
We study an epidemic model for a constant population by taking into account four compartments of the individuals characterizing their states of health. Each individual is in one of the following compartments: susceptible S; incubated, i.e., infected yet not infectious, C; infected and infectious I; and recovered, i.e., immune, R. An infection is visible only when an individual is in state I. Upon infection, an individual performs the transition pathway S→C→I→R→S, remaining in compartments C, I, and R for a certain random waiting time t_{C}, t_{I}, and t_{R}, respectively. The waiting times for each compartment are independent and drawn from specific probability density functions (PDFs) introducing memory into the model. The first part of the paper is devoted to the macroscopic S-C-I-R-S model. We derive memory evolution equations involving convolutions (time derivatives of general fractional type). We consider several cases. The memoryless case is represented by exponentially distributed waiting times. Cases of long waiting times with fat-tailed waiting-time distributions are considered as well where the S-C-I-R-S evolution equations take the form of time-fractional ordinary differential equations. We obtain formulas for the endemic equilibrium and a condition of its existence for cases when the waiting-time PDFs have existing means. We analyze the stability of healthy and endemic equilibria and derive conditions for which the endemic state becomes oscillatory (Hopf) unstable. In the second part, we implement a simple multiple-random-walker approach (microscopic model of Brownian motion of Z independent walkers) with random S-C-I-R-S waiting times in computer simulations. Infections occur with a certain probability by collisions of walkers in compartments I and S. We compare the endemic states predicted in the macroscopic model with the numerical results of the simulations and find accordance of high accuracy. We conclude that a simple random-walker approach offers an appropriate microscopic description for the macroscopic model. The S-C-I-R-S-type models open a wide field of applications allowing the identification of pertinent parameters governing the phenomenology of epidemic dynamics such as extinction, convergence to a stable endemic equilibrium, or persistent oscillatory behavior.
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Affiliation(s)
- Téo Granger
- Sorbonne Université, Institut Jean le Rond d'Alembert, CNRS UMR 7190, 4 Place Jussieu, 75252 Paris Cedex 05, France
| | - Thomas M Michelitsch
- Sorbonne Université, Institut Jean le Rond d'Alembert, CNRS UMR 7190, 4 Place Jussieu, 75252 Paris Cedex 05, France
| | - Michael Bestehorn
- Institut für Physik, Brandenburgische Technische Universität Cottbus-Senftenberg, Erich-Weinert-Straße 1, 03046 Cottbus, Germany
| | - Alejandro P Riascos
- Instituto de Física, Universidad Nacional Autónoma de México, Apartado Postal 20-364, 01000 Ciudad de México, Mexico
| | - Bernard A Collet
- Sorbonne Université, Institut Jean le Rond d'Alembert, CNRS UMR 7190, 4 Place Jussieu, 75252 Paris Cedex 05, France
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Libál A, Forgács P, Néda Á, Reichhardt C, Hengartner N, Reichhardt CJO. Transition from susceptible-infected to susceptible-infected-recovered dynamics in a susceptible-cleric-zombie-recovered active matter model. Phys Rev E 2023; 107:024604. [PMID: 36932562 DOI: 10.1103/physreve.107.024604] [Citation(s) in RCA: 1] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/13/2022] [Accepted: 12/18/2022] [Indexed: 06/18/2023]
Abstract
The susceptible-infected (SI) and susceptible-infected-recovered (SIR) models provide two distinct representations of epidemic evolution, distinguished by whether or not the number of susceptibles always drops to zero at long times. Here we introduce a new active matter epidemic model, the "susceptible-cleric-zombie-recovered" (SCZR) model, in which spontaneous recovery is absent but zombies can recover with probability γ via interaction with a cleric. Upon colliding with a zombie, both susceptibles and clerics enter the zombie state with probability β and α, respectively. By changing the initial fraction of clerics or their healing ability rate γ, we can tune the SCZR model between SI dynamics, in which no susceptibles or clerics remain at long times, and SIR dynamics, in which a finite number of clerics and susceptibles survive at long times. The model is relevant to certain real world diseases such as HIV where spontaneous recovery is impossible but where medical interventions by a limited number of caregivers can reduce or eliminate the spread of infection.
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Affiliation(s)
- A Libál
- Mathematics and Computer Science Department, Babeş-Bolyai University, Cluj-Napoca 400084, Romania
| | - P Forgács
- Mathematics and Computer Science Department, Babeş-Bolyai University, Cluj-Napoca 400084, Romania
| | - Á Néda
- Mathematics and Computer Science Department, Babeş-Bolyai University, Cluj-Napoca 400084, Romania
| | - C Reichhardt
- Theoretical Division and Center for Nonlinear Studies, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA
| | - N Hengartner
- Theoretical Division and Center for Nonlinear Studies, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA
| | - C J O Reichhardt
- Theoretical Division and Center for Nonlinear Studies, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA
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Gostiaux L, Bos WJT, Bertoglio JP. Periodic epidemic outbursts explained by local saturation of clusters. Phys Rev E 2023; 107:L012201. [PMID: 36797932 DOI: 10.1103/physreve.107.l012201] [Citation(s) in RCA: 4] [Impact Index Per Article: 4.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/17/2022] [Accepted: 12/19/2022] [Indexed: 06/18/2023]
Abstract
Adding the notion of spatial locality to the susceptible-infected-recovered (or SIR) model, allows to capture local saturation of an epidemic. The resulting minimum model of an epidemic, consisting of five ordinary differential equations with constant model coefficients, reproduces slowly decaying periodic outbursts, as observed in the COVID-19 or Spanish flu epidemic. It is shown that if immunity decays, even slowly, the model yields a fully periodic dynamics.
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Affiliation(s)
- Louis Gostiaux
- Univ Lyon, École Centrale de Lyon, INSA Lyon, Université Claude Bernard Lyon 1, CNRS, Laboratoire de Mécanique des Fluides et d'Acoustique, UMR 5509, 36 Avenue Guy de Collongue, F-69134 Ecully, France
| | - Wouter J T Bos
- Univ Lyon, École Centrale de Lyon, INSA Lyon, Université Claude Bernard Lyon 1, CNRS, Laboratoire de Mécanique des Fluides et d'Acoustique, UMR 5509, 36 Avenue Guy de Collongue, F-69134 Ecully, France
| | - Jean-Pierre Bertoglio
- Univ Lyon, École Centrale de Lyon, INSA Lyon, Université Claude Bernard Lyon 1, CNRS, Laboratoire de Mécanique des Fluides et d'Acoustique, UMR 5509, 36 Avenue Guy de Collongue, F-69134 Ecully, France
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