Discrete spread model for COVID-19: the case of Lebanon

doi:10.15302/J-QB-022-0292

Quant. Biol.

2022, Vol. 10

Issue (2) : 157-171 https://doi.org/10.15302/J-QB-022-0292

RESEARCH ARTICLE

Discrete spread model for COVID-19: the case of Lebanon

Ayman Mourad¹, Fatima Mroue²(

)

¹. Department of Mathematics, Faculty of Sciences, Lebanese University, Hadat 1500, Lebanon
². Department of Mathematics, American University of Beirut, Beirut 1107 2020, Lebanon

Download: PDF(4917 KB) HTML
Export: BibTeX | EndNote | Reference Manager | ProCite | RefWorks

Abstract

Background: Mathematical models are essential to predict the likely outcome of an epidemic. Various models have been proposed in the literature for disease spreads. Some are individual based models and others are compartmental models. In this study, discrete mathematical models are developed for the spread of the coronavirus disease 2019 (COVID-19).

Methods: The proposed models take into account the known special characteristics of this disease such as the latency and incubation periods, and the different social and infectiousness conditions of infected people. In particular, they include a novel approach that considers the social structure, the fraction of detected cases over the real total infected cases, the influx of undetected infected people from outside the borders, as well as contact-tracing and quarantine period for travelers. The first model is a simplified model and the second is a complete model.

Results: From a numerical point of view, the particular case of Lebanon has been studied and its reported data have been used to estimate the complete discrete model parameters using optimization techniques. Moreover, a parameter analysis and several prediction scenarios are presented in order to better understand the role of the parameters.

Conclusions: Understanding the role of the parameters involved in the models help policy makers in deciding the appropriate mitigation measures. Also, the proposed approach paves the way for models that take into account societal factors and complex human behavior without an extensive process of data collection.

Keywords discrete stochastic modeling COVID-19 numerical simulation

Corresponding Author(s): Fatima Mroue

Online First Date: 17 May 2022 Issue Date: 07 July 2022

Cite this article:

Ayman Mourad,Fatima Mroue. Discrete spread model for COVID-19: the case of Lebanon[J]. Quant. Biol., 2022, 10(2): 157-171.

URL:

https://academic.hep.com.cn/qb/EN/10.15302/J-QB-022-0292
https://academic.hep.com.cn/qb/EN/Y2022/V10/I2/157

Fig.1 (A) The household size distribution. (B) The frequencies of the number of days corresponding to the number of infected travelers per day (obtained from data over a period of 188 days from the beginning of the epidemic). (C) The number of known contacts encountered per day. (D) The number of known contacts encountered at a single day

Fig.2 Schematic of the infection process between the categories N, F and C.

Fig.3 Schematic of the infection process for each category: N, F, C and U.

Fig.4 Real data of Lebanon: Daily new cases from February 21, 2020 till June 2, 2021.

Date	$β$	$γ$	q_max
Feb 21 to Mar 21	0.065	0.020	10
Mar 22 to Apr 10	0.025	0.020	8
Apr 11 to Apr 30	0.030	0.020	8
May 1 to June 30	0.030	0.025	5
July 1 to August 3	0.044	0.030	5
Aug 4 to Aug 15	0.045	0.035	5
Aug 16 to Oct 31	0.037	0.032	5
Nov 1 to Dec 21	0.035	0.030	5
Dec 22 to Jan 5	0.050	0.040	5
Jan 6 to Feb 8	0.035	0.030	5

Tab.1 Parameters for Lebanon

Fig.5 Real data of Lebanon and average simulation results with parameters as in Tab.1

Fig.6 Using constant values β = 0.040 and γ = 0.035 for the period following February 8, the parameters’ analysis is done using 4 scenarios in which either β or γ or both are reduced: (I): [0.98β], (II): [0.98β, 0.90γ], (III): [0.95β], (IV): [0.90γ]

Fig.7 The parameters are chosen to be constant for the period following February 8: β = 0.040 and γ = 0.035

Fig.8 The parameters vary according to a significant increase or decrease in the reported number of cases

Fig.9 Simulation results for the scenario with variable coefficients, vaccination and waning immunity (immunity lasts for 6 months)

1	Health Organization. Statement on the second meeting of the international health regulations World. Emergency committee regarding the outbreak of novel coronavirus (2019-ncov). . Accessed: December 1, 2020
2	Health Organization World. Director-general’s opening remarks at the media briefing on covid-19—11 march 2020. . Accessed: December 1, 2020
3	E. Bonabeau. ( 2002) Agent-based modeling: Methods and techniques for simulating human systems. In: Proceedings of the National Academy of Sciences, 99(suppl 3), 7280‒ 7287
4	M., Ajelli, B., alves, D., Balcan, V., Colizza, H., Hu, J. J., Ramasco, S. Merler, ( 2010). Comparing large-scale computational approaches to epidemic modeling: agent-based versus structured metapopulation models. BMC Infect. Dis., 10 : 190 https://doi.org/10.1186/1471-2334-10-190
5	H. Hethcote, ( 2000). The mathematics of infectious diseases. SIAM Review, 42 : 599– 653
6	O., Diekmann H. Heesterbeek. ( 2012) Mathematical Tools for Understanding Infectious Disease Dynamics, volume 7. Princeton University Press
7	F. Brauer. ( 2012) Mathematical Models in Population Biology and Epidemiology, volume 2. Springer
8	W. O. Kermack A. McKendrick. ( 1927) A contribution to the mathematical theory of epidemics. In: Proceedings of the royal society of london. Series A, Containing papers of a mathematical and physical character, 115, 700‒ 721
9	I. sell, ( 1996) The quasi-stationary distribution of the closed endemic sis model. Adv. Appl. Probab., 895‒ 932
10	M., Hurley, G. Jacobs, ( 2006). The basic SI model. In: New Directions for Teaching and Learning, 2006 : 11– 22
11	Y. Jin W. Wang. ( 2007) An sirs model with a nonlinear incidence rate. Chaos, Solitons, Fract., 34, 1482‒ 1497
12	J. Epstein, ( 2009). Modelling to contain pandemics. Nature, 460 : 687– 687
13	E., Hunter, B. M. Namee, ( 2017). A taxonomy for agent-based models in human infectious disease epidemiology. J. Artif. Soc. Social Simul., 20 : 2
14	E., Hunter, B. Mac Namee, ( 2018). An open-data-driven agent-based model to simulate infectious disease outbreaks. PLoS One, 13 : e0208775 https://doi.org/10.1371/journal.pone.0208775
15	M., Tracy, M. Cerda, K. Keyes, ( 2018). Agent-based modeling in public health: current applications and future directions. Annu. Rev. Publ. Health, 39 : 77– 94
16	G. De Kai A., Morgunov V. Nangalia. ( 2020) Universal masking is urgent in the COVID-19 pandemic: Seir and agent based models, empirical validation, policy recommendations. arXiv, 2004.13553
17	J. M., Epstein, D. M., Goedecke, F., Yu, R. J., Morris, D. K. Wagener, G. Bobashev, ( 2007). Controlling pandemic flu: the value of international air travel restrictions. PLoS One, 2 : e401
18	C., Anastassopoulou, L., Russo, A. Tsakris, ( 2020). Data-based analysis, modelling and forecasting of the COVID-19 outbreak. PLoS One, 15 : e0230405 https://doi.org/10.1371/journal.pone.0230405
19	F. Casella. ( 2020) Can the COVID-19 epidemic be managed on the basis of daily data? arXiv, 2003.06967
20	J. T., Wu, K., Leung, M., Bushman, N., Kishore, R., Niehus, P. M., de Salazar, B. J., Cowling, M. Lipsitch, G. Leung, ( 2020). Estimating clinical severity of COVID-19 from the transmission dynamics in Wuhan, China. Nat. Med, 26 : 506– 510 https://doi.org/10.1038/s41591-020-0822-7
21	G., Giordano, F., Blanchini, R., Bruno, P., Colaneri, A., Di Filippo, A. Di Matteo, ( 2020). Modelling the COVID-19 epidemic and implementation of population-wide interventions in Italy. Nat. Med, 26 : 855– 860 https://doi.org/10.1038/s41591-020-0883-7
22	R. Sameni. ( 2020) Mathematical modeling of epidemic diseases; a case study of the COVID-19 coronavirus. arXiv, 2003.11371
23	R. Goel, ( 2020). Mobility based SIR model for pandemics−with case study of COVID-19. In: 2020 IEEE/ACM International Conference on Advances in Social Networks Analysis and Mining (ASONAM), pp. 110– 117
24	J., Hellewell, S., Abbott, A., Gimma, N. I., Bosse, C. I., Jarvis, T. W., Russell, J. D., Munday, A. J., Kucharski, W. J., Edmunds, S. Funk, et al.. ( 2020). Feasibility of controlling COVID-19 outbreaks by isolation of cases and contacts. Lancet Glob. Health, 8 : e488– e496 https://doi.org/10.1016/S2214-109X(20)30074-7
25	A. J., Kucharski, T. W., Russell, C., Diamond, Y., Liu, J., Edmunds, S., Funk, R. M., Eggo, F., Sun, M., Jit, J. D. Munday, et al.. ( 2020). Early dynamics of transmission and control of COVID-19: a mathematical modelling study. Lancet Infect. Dis., 20 : 553– 558 https://doi.org/10.1016/S1473-3099(20)30144-4
26	S. Chang, N., Harding, C., Zachreson, O. Cliff, ( 2020). Modelling transmission and control of the COVID-19 pandemic in Australia. Nat. Commun., 11 : 5710 https://doi.org/10.1038/s41467-020-19393-6
27	C. Varotsos, V. Krapivin, ( 2020). A new model for the spread of COVID-19 and the improvement of safety. Saf. Sci., 132 : 104962 https://doi.org/10.1016/j.ssci.2020.104962
28	A., Comunian, R. Gaburro, ( 2020). Inversion of a SIR-based model: A critical analysis about the application to COVID-19 epidemic. Physica D, 413 : 132674 https://doi.org/10.1016/j.physd.2020.132674
29	G. C., Calafiore, C. Novara, ( 2020). A modified SIR model for the COVID-19 contagion in Italy. In: 2020 59th IEEE Conference on Decision and Control (CDC), pp. 3889– 3894
30	D., Calvetti, A., Hoover, J. Rose, ( 2020). Bayesian dynamical estimation of the parameters of an SE(A)IR COVID-19 spread model. arXiv, 2005.04365
31	M. T., Rouabah, A. Tounsi, N. Belaloui, ( 2020). Early dynamics of COVID-19 in Algeria: a model-based study. ArXiv, 2005.13516
32	R. Pastor-Satorras, ( 2001). Epidemic spreading in scale-free networks. Phys. Rev. Lett., 86 : 3200– 3203 https://doi.org/10.1103/PhysRevLett.86.3200
33	T. Boulmezaoud, ( 2020). A discrete epidemic model and a zigzag strategy for curbing the COVID-19 outbreak and for lifting the lockdown. Math. Model. Nat. Phenom., 15 : 75
34	S., He, S. Tang, ( 2020). A discrete stochastic model of the COVID-19 outbreak: Forecast and control. Math. Biosci. Eng, 17 : 2792– 2804
35	D., Klinkenberg, C. Fraser, ( 2006). The effectiveness of contact tracing in emerging epidemics. PLoS One, 1 : e12 https://doi.org/10.1371/journal.pone.0000012
36	Q., Li, X., Guan, P., Wu, X., Wang, L., Zhou, Y., Tong, R., Ren, K. S. M., Leung, E. H. Y., Lau, J. Y. Wong, et al.. ( 2020). Early transmission dynamics in Wuhan, China, of novel coronavirus-infected pneumonia. N. Engl. J. Med., 382 : 1199– 1207 https://doi.org/10.1056/NEJMoa2001316
37	N. M., Linton, T., Kobayashi, Y., Yang, K., Hayashi, A. R., Akhmetzhanov, S. M., Jung, B., Yuan, R. Kinoshita, ( 2020). Incubation period and other epidemiological characteristics of 2019 novel coronavirus infections with right truncation: A statistical analysis of publicly available case data. J. Clin. Med., 9 : 538 https://doi.org/10.3390/jcm9020538
38	E., Lavezzo, E., Franchin, C., Ciavarella, G., Cuomo-Dannenburg, L., Barzon, C., Del Vecchio, L., Rossi, R., Manganelli, A., Loregian, N. Navarin, et al.. ( 2020). Suppression of a SARS-CoV-2 outbreak in the Italian municipality of Vo’. Nature, 584 : 425– 429
39	H., Nishiura, M. Natalie, A. Akhmetzhanov, ( 2020). Serial interval of novel coronavirus (COVID-19) infections. Int. J. Infect. Dis. 93, 284– 286
40	Z., Du, X., Xu, Y., Wu, L., Wang, B. J. Cowling, L. Meyers, ( 2020). Serial interval of COVID-19 among publicly reported confirmed cases. Emerg. Infect. Dis., 26 : 1341– 1343 https://doi.org/10.3201/eid2606.200357
41	D. Long S., Gombar C. Hogan A. Greninger S. V., OReilly C., Bryson-Cahn B., Stevens A., Rustagi K. Jerome C. Kong. ( 2020) Occurrence and timing of subsequent SARS-CoV-2 RT-PCR positivity among initially negative patients. medRxiv, 20089151
42	I., Arevalo-Rodriguez, D., Buitrago-Garcia, D., Simancas-Racines, P., Zambrano-Achig, R., Del Campo, A., Ciapponi, O., Sued, A. W., Rutjes, N. Low, et al.. ( 2020). False-negative results of initial RT-PCR assays for COVID-19: A systematic review. PLoS One, 15 : e0242958 https://doi.org/10.1371/journal.pone.0242958
43	L. Kucirka, S. Lauer, O., Laeyendecker, D., Boon, ( 2020). Variation in false-negative rate of reverse transcriptase polymerase chain reaction-based SARS-CoV-2 tests by time since exposure. Ann. Intern. Med, 173 : 262– 267
44	Y., Yang, M., Yang, C., Shen, F., Wang, J., Yuan, J., Li, M., Zhang, Z., Wang, L., Xing, J. Wei, et al.. ( 2020). Laboratory diagnosis and monitoring the viral shedding of 2019-ncov infections. Innovation (N Y), 1 : 100061 https://doi.org/10.1016/j.xinn.2020.100061
45	Y., Pan, L., Long, D., Zhang, T., Yuan, S., Cui, P., Yang, Q. Wang, ( 2020). Potential false-negative nucleic acid testing results for severe acute respiratory syndrome coronavirus 2 from thermal inactivation of samples with low viral loads. Clin. Chem., 66 : 794– 801 https://doi.org/10.1093/clinchem/hvaa091
46	X., He, E. Lau, P., Wu, X., Deng, J., Wang, X., Hao, Y. Lau, J. Wong, Y., Guan, X. Tan, et al.. ( 2020). Temporal dynamics in viral shedding and transmissibility of COVID-19. Nat. Med., 26 : 672– 675
47	C. Dugdale, M. Anahtar, J. Chiosi, J. Lazarus, S. McCluskey, A. Ciaranello, T., Gogakos, B. Little, J. Branda, E. Shenoy, et al.. ( 2020). Clinical, laboratory, and radiologic characteristics of patients with initial false-negative severe acute respiratory syndrome coronavirus 2 nucleic acid amplification test results. Open Forum Infect Dis, 8 : ofaa559
48	N. Kinloch, G., Ritchie, C. Brumme, W., Dong, W., Dong, T., Lawson, R. Jones, J. Montaner, V., Leung, M. Romney, et al.. ( 2020). Suboptimal biological sampling as a probable cause of false-negative COVID-19 diagnostic test results. J. Infect. Dis., 222 : 899– 902
49	H., Zhu, Y., Li, X., Jin, J., Huang, X., Liu, Y. Qian, ( 2021). Transmission dynamics and control methodology of COVID-19: a modeling study. Appl. Math. Model, 89 : 1983– 1998
50	Z., Liu, P., Magal, O. Seydi, ( 2020). A COVID-19 epidemic model with latency period. Infect. Dis. Model, 5 : 323– 337
51	of Public Health Ministry. . Accessed: December 1, 2020
52	HOUSSARI Najia. Lebanon reinstates lockdown measures after virus rebound. Accessed: December 1, 2020
53	Madi Masri Nayla. The development and state of the art of adult learning and education (ale): National report of Lebanon. National Committee for Illiteracy and Adult Education, Ministry of Social Affairs, 2008. Accessed: December 1, 2020
54	Yaacoub Najwa Badre Lara. Population & housing in Lebanon, 2012. Accessed: December 1, 2020

[1]	Pavel Grinchuk, Sergey Fisenko. Power-law multi-wave model for COVID-19 propagation in countries with nonuniform population density[J]. Quant. Biol., 2022, 10(2): 150-156.
[2]	Georgios D. Barmparis, Giorgos P. Tsironis. Physics-informed machine learning for the COVID-19 pandemic: Adherence to social distancing and short-term predictions for eight countries[J]. Quant. Biol., 2022, 10(2): 139-149.
[3]	HyeongChan Jo, Juhyun Kim, Tzu-Chen Huang, Yu-Li Ni. condLSTM-Q: A novel deep learning model for predicting COVID-19 mortality in fine geographical scale[J]. Quant. Biol., 2022, 10(2): 125-138.
[4]	Rudra Banerjee, Srijit Bhattacharjee, Pritish Kumar Varadwaj. A study of the COVID-19 epidemic in India using the SEIRD model[J]. Quant. Biol., 2021, 9(3): 317-328.
[5]	Shilei Zhao, Tong Sha, Chung-I Wu, Yongbiao Xue, Hua Chen. Will the large-scale vaccination succeed in containing the COVID-19 pandemic and how soon?[J]. Quant. Biol., 2021, 9(3): 304-316.
[6]	Yuehui Zhang, Lin Chen, Qili Shi, Zhongguang Luo, Libing Shen. Forecasting the development of the COVID-19 epidemic by nowcasting: when did things start to get better?[J]. Quant. Biol., 2021, 9(1): 93-99.
[7]	Saroj K Biswas, Nasir U Ahmed. Mathematical modeling and optimal intervention of COVID-19 outbreak[J]. Quant. Biol., 2021, 9(1): 84-92.
[8]	Cecylia S. Lupala, Xuanxuan Li, Jian Lei, Hong Chen, Jianxun Qi, Haiguang Liu, Xiao-Dong Su. Computational simulations reveal the binding dynamics between human ACE2 and the receptor binding domain of SARS-CoV-2 spike protein[J]. Quant. Biol., 2021, 9(1): 61-72.
[9]	Wanyu Tao, Zhengqing Yu, Jing-Dong J. Han. A digitized catalog of COVID-19 epidemiology data[J]. Quant. Biol., 2021, 9(1): 23-46.
[10]	Hanshuang Pan, Nian Shao, Yue Yan, Xinyue Luo, Shufen Wang, Ling Ye, Jin Cheng, Wenbin Chen. Multi-chain Fudan-CCDC model for COVID-19 -- a revisit to Singapore’s case[J]. Quant. Biol., 2020, 8(4): 325-335.
[11]	Xiaofei Yang, Tun Xu, Peng Jia, Han Xia, Li Guo, Lei Zhang, Kai Ye. Transportation, germs, culture: a dynamic graph model of COVID-19 outbreak[J]. Quant. Biol., 2020, 8(3): 238-244.

Viewed

Full text

Abstract

Cited

Shared

Discussed