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Mathematical Analysis of Diagnosis Rate Effects in Covid-19 Transmission Dynamics with Optimal Control

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Analysis of Infectious Disease Problems (Covid-19) and Their Global Impact

Part of the book series: Infosys Science Foundation Series ((ISFM))

Abstract

Many countries around the world are trying to fight Covid-19, and their main methods are lockdown, quarantine, isolation, and awareness programs to encourage people to adopt social distancing and maintain personal hygiene. The lockdown is aimed to restrict the movement of humans from or to certain places. Quarantine is aimed toward separating the susceptible humans from infected or exposed humans as much as possible, whereas isolation is aimed toward keeping the confirmed cases of infected humans away from the rest of the population. The confirmed cases are mainly identified through the diagnosis of individuals who showed symptoms of Covid-19 and sometimes through random checking of individuals hoping to identify either asymptomatic or pre-symptomatic cases, which is generally an expensive method. In this chapter, we develop a mathematical model to investigate the role of diagnosis rate in the transmission dynamics of Covid-19 together with the combined effects of quarantine and isolation. Our model will be fully analyzed both qualitatively and quantitatively in order to gain insight about the role of different model parameters in the disease transmission dynamics, especially those related to diagnosis and quarantine. The analysis will include the estimation of both the basic and the control reproduction numbers, and sensitivity analysis of the reproduction numbers to the corresponding model parameters. The optimal control theory will be also applied to the model to examine the role of some other optimal control strategies and to study the effect of diagnosis and quarantine rates in the effectiveness of these controls.

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References

  1. 4, S.R.N.B.: Covid-19: Public policies and societyś responses (2020). http://www.iea.usp.br/pesquisa/nucleos-de-apoio-a-pesquisa/observatorio-inovacao-competitividade/boletim-oic-4-en

  2. Agusto, F., ELmojtaba, I.: Optimal control and cost-effective analysis of malaria/visceral leishmaniasis co-infection. PLOS One 12(2) (2017)

    Google Scholar 

  3. Arons, M.M., Hatfield, K.M., Reddy, S.C., Kimball, A., James, A., Jacobs, J.R., Taylor, J., Spicer, K., Bardossy, A.C., Oakley, L.P., Tanwar, S., Dyal, J.W., Harney, J., Chisty, Z., Bell, J.M., Methner, M., Paul, P., Carlson, C.M., McLaughlin, H.P., Thornburg, N., Tong, S., Tamin, A., Tao, Y., Uehara, A., Harcourt, J., Clark, S., Brostrom-Smith, C., Page, L.C., Kay, M., Lewis, J., Montgomery, P., Stone, N.D., Clark, T.A., Honein, M.A., Duchin, J.S., Jernigan, J.A.: Presymptomatic sars-cov-2 infections and transmission in a skilled nursing facility. New Engl. J. Med. 382(22), 2081–2090 (2020)

    Google Scholar 

  4. Backer, J.A., Klinkenberg, D., Walling, J.: Incubation period of 2019 novel coronavirus (2019–ncov) infections among travellers from Wuhan, China, 20–28 January 2020. Eurosurveillance p. 25(5) (2020)

    Google Scholar 

  5. Bai, Y., Yao, L., Wei, T., Tian, F., Jin, D., Chen, L., Wang, M.: Presumed asymptomatic carrier transmission of covid-19. JAMA 323(14), 1406–1407 (2020)

    Article  Google Scholar 

  6. Blower, S., Dowlatabadi, H.: Sensitivity and uncertainty analysis of complex 448 models of disease transmission: an hiv model, as an example. Int. Stat. Rev., 229–243 (1994)

    Google Scholar 

  7. Carraturo, F., Del Giudice, C., Morelli, M., Cerullo, V., Libralato, G., Galdiero, E., Guida, M.: Persistence of sars-cov-2 in the environment and covid-19 transmission risk from environmental matrices and surfaces. Environmental pollution (Barking, Essex : 1987) (Pt B), 115010

    Google Scholar 

  8. Castillo-Chavez, C., Feng, Z., Huang, W.: On the computation of \(r_0\) and its role on global stability. In:Mathematical approaches for emerging and re emerging infectious 458 diseases. An Introduction. Castillo-Chavez, C., Blower, S., van den Driessche, P., Kirschner, 459 D. and Yakubu, A.-A. (Eds.) pp. 229–250 (2002)

    Google Scholar 

  9. Castillo-Chavez, C., Song, B.: Dynamical models of tuberculosis and their applications. Math. Biosci. Eng. 1(2), 361–404 (2004)

    Article  MathSciNet  Google Scholar 

  10. Center, J.H.U..M.C.R.: Cumulative cases (2020). https://coronavirus.jhu.edu/data/cumulative-cases

  11. Chang, D., Mo, G., Yuan, X., Tao, Y., Peng, X., Wang, F.S., Xie, L., Charles, S., Dela Cruz, L.S., Qin, E.: Mathematical modeling and epidemic prediction of covid-19 of the state of sao paulo, brazil. Int. J. Adv. Eng. Res. Sci. (IJAERS) 7, 338–347 (2020)

    Google Scholar 

  12. da Cruz, P.A., Cruz, L.C.C.: Time kinetics of viral clearance and resolution of symptoms in novel coronavirus infection. Am. J. Respir. Critical Care Med. 201, 1150–1152 (2020). https://ijaers.com/uploads/issue_files/41IJAERS-05202019-Mathematical.pdf

  13. Diekmann, O., Heesterbeek, J., Metz, J.: On the definition and the computation of the basic reproduction ratio r0 in models for infectious diseases in heterogeneous populations. J. Math. Biol. 28(4), 365–382 (1990)

    Article  MathSciNet  Google Scholar 

  14. van den Driessche, P., Watmough, J.: Reproduction numbers and the sub-threshold endemic equilibria for compartmental models of disease transmission. Math. Biosci. 180, 29–48 (2002)

    Article  MathSciNet  Google Scholar 

  15. Guan, W.j., Ni, Z.y., Hu, Y., Liang, W.h., Ou, C.q., He, J.x., Liu, L., Shan, H., Lei, C.l., Hui, D.S., Du, B., Li, L.j., Zeng, G., Yuen, K.Y., Chen, R.c., Tang, C.l., Wang, T., Chen, P.y., Xiang, J., Li, S.y., Wang, J.l., Liang, Z.j., Peng, Y.x., Wei, L., Liu, Y., Hu, Y.h., Peng, P., Wang, J.m., Liu, J.y., Chen, Z., Li, G., Zheng, Z.j., Qiu, S.q., Luo, J., Ye, C.j., Zhu, S.y., Zhong, N.s.: Clinical characteristics of coronavirus disease 2019 in china. New Engl. J. Med. 382(18), 1708–1720 (2020). https://doi.org/10.1056/NEJMoa2002032. https://doi.org/10.1056/NEJMoa2002032

  16. Jia, J., Ding, J., Liu, S., Liao, G., Li, J., Duan, B., Wang, G., Zhang, R.: Modeling the control of covid-19: impact of policy interventions and meteorological factors. Electron. J. Differ. Equ. 23, 1–24 (2020)

    MathSciNet  MATH  Google Scholar 

  17. Jung, E., Lenhart, S., Feng, Z.: Optimal control of treatments in a two-strain 490 tuberculosis model. Discrete Continuous Dyn. Syst.-Ser B 4(2), 473–482 (2002)

    Article  Google Scholar 

  18. Kong, W., Li, Y., Peng, M., Kong, D., Yang, X., Wang, L., Liu, M.: Sars-cov-2 detection in patients with influenza-like illness. Nat. Microbiol. 5, 675–678 (2020)

    Article  Google Scholar 

  19. Kucharski, A., Russell, T., Diamond, C., Liu, Y., Edmunds, J., Funk, S., Eggo, R., Sun, F., Jit, M., Munday, J., Davies, N., Gimma, A., van Zandvoort, K., Gibbs, H., Hellewell, J., Jarvis, C., Clifford, S., Quilty, B., Bosse, N., Abbott, S., Klepac, K., Flasche, F.: Early dynamics of transmission and control of covid-19: a mathematical modelling study. Lancet Infect. Dis. 3099(20), 1–7 (2020)

    Google Scholar 

  20. Lavezzo, E., Franchin, E., Ciavarella, C., Cuomo-Dannenburg, G., Barzon, L., Del Vecchio, C., Rossi, L., Manganelli, R., Loregian, A., Navarin, N., Abate, D., Sciro, M., Merigliano, S., De Canale, E., Vanuzzo, M.C., Besutti, V., Saluzzo, F., Onelia, F., Pacenti, M., Parisi, S.G., Carretta, G., Donato, D., Flor, L., Cocchio, S., Masi, G., Sperduti, A., Cattarino, L., Salvador, R., Nicoletti, M., Caldart, F., Castelli, G., Nieddu, E., Labella, B., Fava, L., Drigo, M., Gaythorpe, K.A.M., Ainslie, K.E.C., Baguelin, M., Bhatt, S., Boonyasiri, A., Boyd, O., Coupland, H.L., Cucunubá, Z., Djafaara, B.A., Donnelly, C.A., Dorigatti, I., van Elsland, S.L., FitzJohn, R., Flaxman, S., Gaythorpe, K.A.M., Green, W.D., Hallett, T., Hamlet, A., Haw, D., Imai, N., Jeffrey, B., Knock, E., Laydon, D.J., Mellan, T., Mishra, S., Nedjati-Gilani, G., Nouvellet, P., Okell, L.C., Parag, K.V., Riley, S., Thompson, H.A., Unwin, H.J.T., Verity, R., Vollmer, M.A.C., Walker, P.G.T., Walters, C.E., Wang, H., Wang, Y., Watson, O.J., Whittaker, C., Whittles, L.K., Xi, X., Ferguson, N.M., Brazzale, A.R., Toppo, S., Trevisan, M., Baldo, V., Donnelly, C.A., Ferguson, N.M., Crisanti, A., Team, I.C.C.R.: Suppression of a sars-cov-2 outbreak in the italian municipality of vo’. Nature (2020)

    Google Scholar 

  21. Lin, Q., Zhao, S., Gao, Z., Lou, Y., Yang, S., Musa, S., Wang, M., Cai, Y., Weiming Wang, W., Yang, L., He, D.: A conceptual model for the coronavirus disease 2019 (covid-19) outbreak in wuhan, china with individual reaction and governmental action. Int. J. Infect. Dis. 93, 211–216 (2020)

    Article  Google Scholar 

  22. Marino, S., Hogue, I., Ray, C., Kirschner, D.: A methodology for performing global uncertainty and sensitivity analysis in systems biology. J. Theorit. Biol. 254, 178–196 (2008)

    Article  MathSciNet  Google Scholar 

  23. McLeod, R., Brewster, J., Gumel, A., Slonowsky, D.: Sensitivity and uncertainty analyses for a sars model with time-varying inputs and outputs. Math. Biosci. Eng. 3, 527–544 (2006)

    Article  MathSciNet  Google Scholar 

  24. Moghadas, S.M., Fitzpatrick, M.C., Sah, P., Pandey, A., Shoukat, A., Singer, B.H., Galvani, A.P.: The implications of silent transmission for the control of covid-19 outbreaks. Proc. Natl. Acad. Sci. 117(30), 17513–17515 (2020)

    Google Scholar 

  25. Mwalili, S., Kimathi, M., Ojiambo, V., Gathungu, D., Mbogo, R.: Seir model for covid-19 dynamics incorporating the environment and social distancing. BMC Res. Notes 13(352)

    Google Scholar 

  26. Ndaïrou, F., Area, I., Nieto, J., Torres, D.: Mathematical modeling of covid-19 transmission dynamics with a case study of wuhan. Chaos, Solit. Fractals 135, 109846 (2020)

    Google Scholar 

  27. Oran, D.P., Topol., E.J.: Prevalence of asymptomatic sars-cov-2 infection: A narrative review. Ann. Intern. Med. (2020)

    Google Scholar 

  28. Peeling, R.W., Wedderburn, C.J., Garcia, P.J., Boeras, D., Fongwen, N., Nkengasong, J., Sall, A., Tanuri, A., Heymann, D.L.: Serology testing in the covid-19 pandemic response. The Lancet Infectious Diseases (2020)

    Google Scholar 

  29. Pontryagin, L.: Mathematical Theory of Optimal Processes. CRC Press (1987)

    Google Scholar 

  30. Qian G, Y.N., AHY, M., et al.: Covid-19 transmission within a family cluster by presymptomatic carriers in china. Clin. Infect Dis. 71(15) (2020). https://doi.org/10.1093/cid/ciaa316

  31. Studdert, D.M., Hall, M.A.: Disease control, civil liberties, and mass testing - calibrating restrictions during the covid-19 pandemic. New Engl. J. Med. 383(2), 102–104 (2020)

    Google Scholar 

  32. Tang, B., Wang, X., Li, Q., Bragazzi, N.L., Tang, S., Xiao, Y., Wu, J.: Estimation of the transmission risk of the 2019-ncov and its implication for public health interventions. J. Clin. Med. 9(2), 462 (2020)

    Article  Google Scholar 

  33. Wang, W., Tang, J., Wei, F.: Updated understanding of the outbreak of 2019 novel coronavirus (2019-ncov) in wuhan, china. J. Med. Virol. 92(4), 441–447 (2020)

    Article  Google Scholar 

  34. WHO: What are the symptoms of covid-19? (2020). https://www.who.int/emergencies/diseases/novel-coronavirus-2019/question-and-answers-hub/q-a-detail/q-a-coronaviruses

  35. WHO: Who coronavirus disease (covid-19) dashboard (2020). https://covid19.who.int

  36. Wu, J., Leung, K., Leung, G.: Nowcasting and forecasting the potential domestic and international spread of the 2019-ncov outbreak originating in wuhan, china: a modelling study. Lancet 6736(20), 675–678 (2020)

    Google Scholar 

  37. Yang, C., Wang, J.: A mathematical model for the novel coronavirus epidemic in wuhan, china. Math. Biosci. Eng. 17(3), 2708–2724 (2020)

    Article  MathSciNet  Google Scholar 

  38. Zou, L., Ruan, F., Huang, M., Liang, L., Huang, H., Hong, Z., Yu, J., Kang, M., Song, Y., Xia, J., Guo, Q., Song, T., He, J., Yen, H., Peiris, M., Wu, J.: Sars-cov-2 viral load in upper respiratory specimens of infected patients. N. Engl. J. Med 382, 1177–1179 (2020)

    Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Mathematics, Sultan Qaboos University, Al-Khoudh 123, Oman

    Nasser Al-Salti, Ibrahim M. Elmojtaba & Maryam Al-Yahyai

  2. Department of Mathematics, Universidade de Brasília, Brasília, Brazil

    Jaqueline Mesquita

  3. Department of Mathematics, Centro Federal de Educaçåo Tecnológica Celso Suckow da Fonseca, Rio de Janeiro, Brazil

    Dayse Pastore

Authors
  1. Nasser Al-Salti
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  2. Ibrahim M. Elmojtaba
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  3. Jaqueline Mesquita
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  4. Dayse Pastore
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  5. Maryam Al-Yahyai
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Editor information

Editors and Affiliations

  1. Anand International College of Engineering, Jaipur, Rajasthan, India

    Praveen Agarwal

  2. Institute of Mathematics, University of Santiago de Compostela, Santiago, Spain

    Juan J. Nieto

  3. Mathematical Sciences, Queen Mary University of London, London, UK

    Michael Ruzhansky

  4. Department of Mathematics, University of Aveiro, Aveiro, Portugal

    Delfim F. M. Torres

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Al-Salti, N., Elmojtaba, I.M., Mesquita, J., Pastore, D., Al-Yahyai, M. (2021). Mathematical Analysis of Diagnosis Rate Effects in Covid-19 Transmission Dynamics with Optimal Control. In: Agarwal, P., Nieto, J.J., Ruzhansky, M., Torres, D.F.M. (eds) Analysis of Infectious Disease Problems (Covid-19) and Their Global Impact. Infosys Science Foundation Series(). Springer, Singapore. https://doi.org/10.1007/978-981-16-2450-6_11

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