EPIDEMIC SPREAD ANALYSIS IN SOCIAL COMMUNICATION NETWORKS WITH SIR MODEL

Abstract

Compartmental mathematical models are frequently used in epidemiology. These types of models rely on some assumptions, such as the homogeneity of the society and the equal contact ratio of everyone, to model real-life events mathematically. In real life, due to the heterogeneous nature of the social network that constitutes society, the contact rates and contact times of individuals vary. In sudden and new types of epidemics, solutions such as vaccines to slow down or end epidemics may be limited. In such cases, it becomes more important to use limited resources with maximum efficiency. In this study, the estimation results of disease spread in homogeneous and heterogeneous population structures were compared using the SIR compartment model. The dataset obtained from the science gallery in Dublin in 2009 was used to illustrate the heterogeneous community structure in real life. In the exhibition, the spread of the disease was simulated when individuals with different degrees of centrality in the network formed by the visitors who made face-to-face contacts were immunized. When the results obtained are compared, in the case of vaccination of individuals with high betweenness centrality, the spread of infection occurs 14,39% less than the homogeneous network structure accepted in SIR models.

Keywords

• graph
• epidemiology
• closeness
• betweenness
• sir model

SIR MODELİ İLE SOSYAL İLETİŞİM AĞLARINDA SALGIN YAYILIM ANALİZİ

Abstract

Epidemiyoloji alanında kompartıman tipi matematiksel modellerden sıklıkla yararlanılmaktadır. Bu tip modellerin birçoğu, gerçek hayattaki olayları matematiksel olarak modelleyebilmek amacıyla, toplumun homojen yapıda olduğu ve her bireyin temas oranının eşit olduğu gibi bazı varsayımlar üzerine kurulur. Gerçek hayatta ise toplumu oluşturan sosyal ağın heterojen yapıda olması nedeniyle bireylerin temas oranları ve temas süreleri farklılık göstermektedir. Ani ve yeni tip salgınlarda aşı gibi salgınları yavaşlatacak yada sonlandıracak çözümler sınırlı olabilmektedir. Bu tip durumlarda sınırlı kaynakları maksimum verim ile kullanmak daha önemli hale gelmektedir. Yapılan çalışmada, SIR kompartıman modeli kullanılarak, homojen ve heterojen toplum yapısındaki hastalık yayılımı tahmin sonuçları karşılaştırılmıştır. Gerçek hayattaki heterojen toplum yapısını örneklemek amacıyla 2009 yılında Dublin’deki bilim galerisini ziyaret eden ve yüz yüze temaslarda bulunan sergi ziyaretçilerine ait veri seti kullanılarak, ağdaki farklı merkeziyet derecelerine sahip bireylerin bağışıklık kazanması durumunda hastalık yayılımı simüle edilmiştir. Elde edilen sonuçlar karşılaştırıldığında, arasındalık merkezilik değeri yüksek olan bireylerin aşılanması durumunda enfeksiyon yayılımı SIR modellerinde kabul edilen homojen ağ yapısına göre % 14,39 daha az gerçekleşmektedir.

Keywords

• graph
• epidemiology
• closeness
• betweenness
• sir model

References

1. D. Bernoulli, S. Blower, "An attempt at a new analysis of the mortality caused by smallpox and of the advantages of inoculation to prevent it", Reviews in medical virology, vol. 14, no. 5, p. 275, 2004.
2. W. H. Hamer, "Epidemic disease in England: the evidence of variability and of persistency of type", Bedford Press, 1906.
3. W. O. Kermack, A. G. McKendrick, G. T. Walker, "A contribution to the mathematical theory of epidemics", Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character, vol. 115, no. 772, pp. 700-721, 1927.
4. R. Ross, "The prevention of malaria", John Murray, 1911.
5. W. O. Kermack, A. G. McKendrick, and G. T. Walker, "A contribution to the mathematical theory of epidemics", Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character, vol. 115, no. 772, pp. 700-721, 1927.
6. "Transmission Dynamics of the Etiological Agent of SARS in Hong Kong: Impact of Public Health Interventions Science" , https://science.sciencemag.org/content/300/5627/1961.abstract (accessed May 29, 2021).
7. "Measles outbreak", Netherlands, April 1999-January 2000,https://www.cabdirect.org/cabdirect/abstract/20002012656
8. "Identification of Severe Acute Respiratory Syndrome in Canada- NEJM." https://www.nejm.org/doi/full/10.1056/NEJMoa030634 (accessed May 29, 2021).

9. H. W. Hethcote and J. A. Yorke, "Gonorrhea Transmission Dynamics and Control", Springer, 2014. [10] M. E. J. Newman, "The structure and function of complex networks", SIAM Review, vol. 45, no. 2, pp. 167–256, 2003.
10. S. Jain, S. Kumar, "Dynamical analysis of SEIS model with nonlinear innate immunity and saturated treatment", The European Physical Journal Plus, 136(9), 952, 2021.
11. S. Annas, M. I. Pratama, M. Rifandi, W. Sanusi, S. Side, "Stability analysis and numerical simulation of SEIR model for pandemic COVID-19 spread in Indonesia", Chaos, Solitons & Fractals, 139, 110072, 2020.
12. S. He, Y. Peng, K. Sun, "SEIR modeling of the COVID-19 and its dynamics", Nonlinear Dynamics, vol. 3, no. 101, pp. 1667–1680, 2020.
13. P. E. Lekone, B. F. Finkenstädt, "Statistical inference in a stochastic epidemic SEIR model with control intervention: Ebola as a case study", Biometrics, vol. 4, no. 62, pp. 1170–1177, 2006.
14. M. B. Hooten, J. Anderson, & L. A. Waller, "Assessing North American influenza dynamics with a statistical SIRS model. Spatial and Spatio-Temporal", Epidemiology, vol. 1, no. 2-3, pp. 177–185, 2010.
15. A. Lahrouz, L. Omari, D. Kiouach, & A. Belmaâti, "Complete global stability for an SIRS epidemic model with generalized non-linear incidence and vaccination", Applied Mathematics and Computation, 218(11), pp. 6519–6525, 2012.
16. C.-H. Li, C.-C. Tsai, S.-Y. Yang, "Analysis of epidemic spreading of an SIRS model in complex heterogeneous network", Communications in Nonlinear Science and Numerical Simulation, 19(4), pp. 1042–1054, 2014.
17. C. Vargas-De-León, "On the global stability of SIS, SIR and SIRS epidemic models with standard incidence", Chaos, Solitons & Fractals, 44(12), pp. 1106–1110, 2011.
18. K. L. Cooke, P. Van Den Driessche, "Analysis of an SEIRS epidemic model with two delays", Journal of Mathematical Biology, 35(2), pp. 240–260, 1996.
19. B. K. Mishra, D. K. Saini, "SEIRS epidemic model with delay for transmission of malicious objects in computer network", Applied Mathematics and Computation, 188(2), pp. 1476–1482, 2007.
20. W. Wang, "Global behavior of an SEIRS epidemic model with time delays", Applied Mathematics Letters, 15(4), pp. 423–428, 2002.
21. D. Bichara, A. Iggidr, G. Sallet, "Global analysis of multi-strains SIS, SIR and MSIR epidemic models", Journal of Applied Mathematics and Computing, 44(1), pp. 273–292, 2014.
22. A. Menon, N. K. Rajendran, A. Chandrachud, & G. Setlur, "Modelling and simulation of COVID-19 propagation in a large population with specific reference to India", MedRxiv, 2020.
23. B. Rahman, E. Sadraddin, A. Porreca, "The basic reproduction number of SARS-CoV-2 in Wuhan is about to die out, how about the rest of the world? " Reviews in Medical Virology, 30(4), e2111, 2020.
24. X. Chen, A. Zhang, H. Wang, A. Gallaher, X. Zhu, "Compliance and containment in social distancing: Mathematical modeling of COVID-19 across townships", International Journal of Geographical Information Science, 35(3), pp. 446–465, 2021.
25. J. Huo, H. Zhao, "Dynamical analysis of a fractional SIR model with birth and death on heterogeneous complex networks", Physica A: Statistical Mechanics and Its Applications, 448, pp. 41–56, 2016.
26. S. Bansal, B. T. Grenfell, L. A. Meyers, "When individual behaviour matters: homogeneous and network models in epidemiology", Journal of The Royal Society Interface, vol. 4, pp. 879-891, 2007.
27. L. Pellis et al., "Eight challenges for network epidemic models", Epidemics, vol. 10, pp. 58-62, 2015.
28. L. Isella, J. Stehlé, A. Barrat, C. Cattuto, J.-F. Pinton, and W. V. den Broeck, “What's in a crowd? Analysis of face-to-face behavioral networks", Journal of Theoretical Biology, vol. 271, no. 1, pp. 166-180, 2011.
29. A. d’Onofrio, "A note on the global behaviour of the network-based SIS epidemic model", Nonlinear Analysis: Real World Applications, vol. 9, no. 4, pp. 1567–1572, 2008.
30. M. Saeedian, M. Khalighi, N. Azimi-Tafreshi, G. R. Jafari, and M. Ausloos, "Memory effects on epidemic evolution: The susceptible-infected-recovered epidemic model", Phys. Rev. E, vol. 95, no. 2-1, p. 022409, 2017.
31. A. Azizi, C. Montalvo, B. Espinoza, Y. Kang, and C. Castillo-Chavez, "Epidemics on networks: Reducing disease transmission using health emergency declarations and peer communication", Infectious Disease Modelling, vol. 5, pp. 12-22, 2020.
32. M. Nadini, A. Rizzo, M. Porri, "Epidemic Spreading in Temporal and Adaptive Networks with Static Backbone", IEEE Transactions on Network Science and Engineering, vol. 7, no. 1, pp. 549-561, 2020.
33. R. Olinky, L. Stone, "Unexpected epidemic thresholds in heterogeneous networks: The role of disease transmission", Physical Review E, 70(3), 030902, 2004.
34. M. Barthélemy, A. Barrat, R. Pastor-Satorras, & A. Vespignani, "Dynamical patterns of epidemic outbreaks in complex heterogeneous networks", Journal of Theoretical Biology, 235(2), 275–288, 2005.
35. S. P. Borgatti, M. G. Everett, "A Graph-theoretic perspective on centrality", Social Networks, 28(4), 466–484, 2006.
36. K. Faust, "Centrality in affiliation networks", Social Networks, 19(2), 157–191, 1997.
37. S. Wang, Y. Du, Y. Deng, "A new measure of identifying influential nodes: Efficiency centrality", Communications in Nonlinear Science and Numerical Simulation, 47, 151–163, 2007.
38. C. Zimmer, J. Corum, S.-L. Wee, "Coronavirus Vaccine Tracker", The New York Times, Jun. 10, 2020. Erişim tarihi: Sep. 21, 2021, Erişim linki: https://www.nytimes.com/interactive/2020/science/coronavirus-vaccine-tracker.html
39. L. E. C. Rocha, V. D. Blondel, "Bursts of Vertex Activation and Epidemics in Evolving Networks", PLOS Computational Biology, vol. 9, no.3, e1002974, 2013.
40. M. J., Keeling, K. T. D. Eames, "Networks and epidemic models", Journal of The Royal Society Interface, vol. 2, no. 4, pp. 295–307, 2005.
41. "Infectious" veri seti, http://konect.cc/networks/sociopatterns-infectious, (erişim tarihi: 05.06.2021).
42. M. C. Golumbic, "Algorithmic graph theory and perfect graphs", Elsevier, 2004.
43. B. Bollobás, "Modern graph theory, vol. 184. Springer Science Business Media, 2013.
44. S. Wasserman, K. Faust, "Social network analysis: Methods and applications", 1994.
45. N. Katz, D. Lazer, H. Arrow, N. Contractor, "Network theory and small groups", Small group research, vol. 35, no. 3, pp. 307-332, 2004.
46. L. C. Freeman, "A Set of Measures of Centrality Based on Betweenness," Sociometry, vol. 40, no. 1, pp. 35-41, 1977.
47. L. C. Freeman, "Centrality in social networks conceptual clarification", Social networks, vol. 1, no. 3, pp. 215-239, 1978.
48. L. C. Freeman, D. Roeder, R. R. Mulholland, "Centrality in social networks: ii. experimental results", Social Networks, vol. 2, no. 2, pp. 119-141, 1979.
49. J. Zhang, Y. Luo, "Degree Centrality, Betweenness Centrality, and Closeness Centrality in Social Network," pp. 300-303, 2017.
50. J. M. Bolland, "Sorting out centrality: An analysis of the performance of four centrality models in real and simulated networks", Social Networks, vol. 10, no. 3, pp. 233-253, 1988.
51. M. E. Shaw, "Group Structure and the Behavior of Individuals in Small Groups", The Journal of Psychology, vol. 38, no. 1, pp. 139-149,1954.
52. M. A. Beauchamp, "An improved index of centrality", Behavioral Science, vol. 10, no. 2, pp. 161-163, 1965.
53. G. Sabidussi, "The centrality index of a graph", Psychometrika, vol. 31, no. 4, pp. 581–603. Scopus, 1966.
54. N. Chen, M. Zhou, X. Dong, J. Qu, F. Gong, Y. Han, Y. Qiu, J. Wang, Y. Liu, Y. Wei, J. Xia, T. Yu, X. Zhang, L. Zhang, "Epidemiological and clinical characteristics of 99 cases of 2019 novel coronavirus pneumonia in Wuhan, China: a descriptive study", Lancet (London, England), vol. 395, no. 10223, pp. 507–513, 2020.
55. E. Prompetchara, C. Ketloy, T. Palaga, "Immune responses in COVID-19 and potential vaccines: Lessons learned from SARS and MERS epidemic", Asian Pacific Journal of Allergy and Immunology, vol. 38, no. 1, pp. 1–9, 2020.
56. F. Wu, S. Zhao, B. Yu, Y. -M. Chen, W. Wang, Z.-G. Song, Y. Hu, Z.-W. Tao, J.-H. Tian, Y.-Y. Pei, "A new coronavirus associated with human respiratory disease in China", Nature, vol. 579, no.7798, pp. 265–269, 2020.