Concept controlling model for arresting epidemics, including COVID-19

G. D. Kaminskiy   ; Yu. I. Prostov   ; D. A. Semenova   ; M. Yu. Prostov   ; N. N. Pimenov   ; E. I. Veselova   ; A. S. Vinokurov   ; E. V. Karamov   ; A. E. Panova   ; A. S. Turgiev   ; V. V. Chernetsova   ; A. E. Lomovtsev

Sibirskie èlektronnye matematičeskie izvestiâ, Tome 21 (2024) no. 1, pp. 125-163

Résumé

Concept controlling model for arresting epidemics (further on - the model) of emerging, new and re-emerging infections has been developed. Epidemic force parameters are defined: high values of contact rate of infection in acute ($ R_1 $) and chronic ($ R_2 $) forms of disease, high frequency of chronization $\gamma_2 $ with pathogen excretion, high rate of loss of natural immunity $ k_1 $, high inflow of susceptible population $\mu $. Control targets have been identified: infected persons (detection, isolation and treatment $\delta $), transmission mechanism (regime-restrictive measures, sanitary and hygienic procedures $ r $), the decrease in susceptibility (vaccination, pre- and post-exposure prophylaxis $\lambda $). Critical interdependencies between epidemic force parameters and control coefficients were studied. We obtained threshold conditions for "zero infection" asymptotic stability. In order to achieve the target result more quickly, the use of "supercritical" control levels is proposed, with the model determining the time to achieve the result. The need to affect both acute and chronic forms of infection has been proven. The model allows to solve direct and inverse problems.

Détail

Keywords: control of communicable diseases, threshold, intervention campaign, parameters of the epidemic process, mathematical model.

Bibliographie

[1] H. Gao, W. Li, M. Pan, Z. Han, H.V. Poor, “Modeling COVID-19 with mean field evolutionary dynamics: Social distancing and seasonality”, J. Commun. Networks, 23:5 (2021), 314–325 | DOI

[2] WHO, International health regulations, second ed., World Health Organization, 2005

[3] Zatsepin, O.V. et al, “A COVID-19 agent-based model”, Zababakhin scientific readings, a collection of materials of the XV International Conference (Snezhinsk), 2021, 302

[4] K.P. Hadeler, “Parameter identification in epidemic models”, Math. Biosci., 229:2 (2011), 185–189 | DOI | MR | Zbl

[5] Z. Fodor, S.D. Katz, T.G. Kovacs, Why integral equations should be used instead of differential equations to describe the dynamics of epidemics, 2020, arXiv: 2004.07208

[6] R.M. Anderson, “Transmission dynamics and control of infectious disease agents”, Population Biology of Infectious Diseases, Dahlem Workshop Reports, 25, eds. Anderson, R.M., May, R.M., Springer, Berlin–Heidelberg, 1982, 149–176

[7] Cano, C., The SIR models, their applications, and approximations of their rates, Louisiana Tech University, 2020

[8] N. Ferguson, et al, Report 9: Impact of non-pharmaceutical interventions (NPIs) to reduce COVID19 mortality and healthcare demand, Imperial College, London, 2020

[9] R. Feßler, A General Integral Equation Model for Epidemics, Fraunhofer Institute for Industrial Mathematics ITWM, 2020

[10] R.M. Anderson, The population dynamics of infectious diseases: theory and applications, Springer, 2013

[11] D.L. Heymann, Control of communicable diseases manual, American Public Health Association, 2008

[12] S. Alizon, M. van Baalen, “Emergence of a convex trade-off between transmission and virulence”, Am. Nat., 165:6 (2005), E155–E167 | DOI

[13] R.M. Anderson, C. Fraser, A.C. Ghani, C.A. Donnelly, S. Riley, N.M. Ferguson, G.M. Leung, T.H. Lam, A.J. Hedley, “Epidemiology, transmission dynamics and control of SARS: the 2002-2003 epidemic”, Philos. Trans. R. Soc. Lond. B Biol. Sci., 359:1447 (2004), 1091–1105 | DOI | MR

[14] R.M. Anderson, R.M. May, Infectious diseases of humans: dynamics and control, Oxford University Press, Oxford, 1992

[15] S.M.N. Arifin, C. Zimmer, C. Trotter, A. Colombini, E. Sidikou, F.M. LaForce, T. Cohen, R. Yaesoubi, “Cost-effectiveness of alternative uses of polyvalent meningococcal vaccines in Niger: An agent-based transmission modeling study”, Med. Decis. Making, 39:5 (2019), 553–567 | DOI

[16] M.S. Aronna, R. Guglielmi, L.M. Moschen, “A model for COVID-19 with isolation, quarantine and testing as control measures”, Epidemics, 34 (2021), 100437 | DOI

[17] K.E. Atkins, A.F. Read, S.W. Walkden-Brown, N.J. Savill, M.E. Woolhouse, “The effectiveness of mass vaccination on Marek's disease virus (MDV) outbreaks and detection within a broiler barn: A modeling study”, Epidemics, 5:4 (2013), 208–217 | DOI

[18] V.D. Beliakov, “Revoliutsiia v epidemiologii”, Zhurnal Mikrobiologii Epidemiologii i Immunobiologii, 12 (1974), 20–24

[19] A.S. Benenson, Control of communicable diseases in man thirteenth edition, American Public Health Association, 1981

[20] A. Bensoussan, J. Frehse, P. Yam, Mean field games and mean field type control theory, Springer, New York, 2013 | MR | Zbl

[21] M. Biggerstaff, et al, “Early insights from statistical and mathematical modeling of key epidemiologic parameters of COVID-19”, Emerg. Infect. Dis., 26:11 (2020), e201074 | DOI

[22] S. Cauchemez, N.M. Ferguson, “Likelihood-based estimation of continuous-time epidemic models from time-series data: application to measles transmission in London”, J. R. Soc. Interface, 5:25 (2008), 885–897 | DOI

[23] M. Cevik, M. Tate, O. Lloyd, A.E. Maraolo, J. Schafers, “SARS-CoV-2, SARS-CoV, and MERS-CoV viral load dynamics, duration of viral shedding, and infectiousness: a systematic review and meta-analysis”, Lancet Microbe, 2:1 (2021), e13–e22 | DOI

[24] B.M. Chen-Charpentier, D. Stanescu, “Epidemic models with random coefficients”, Math. Comput. Modelling, 52:7-8 (2010), 1004–1010 | DOI | MR | Zbl

[25] F.T. Cutts el al, “Using models to shape measles control and elimination strategies in low- and middle-income countries: A review of recent applications”, Vaccine, 38:5 (2020), 979–992 | DOI

[26] A. Datta, P. Winkelstein, S. Sen, “An agent-based model of spread of a pandemic with validation using COVID-19 data from New York State”, Physica A: Statistical Mechanics and its Applications, 585 (2022), 126401 | DOI

[27] J. Demongeot, Q. Griette, P. Magal, “SI epidemic model applied to COVID-19 data in mainland China”, Royal Society Open Science, 7:12 (2020), 201878 | DOI

[28] L. Ferretti et al, “Quantifying SARS-CoV-2 transmission suggests epidemic control with digital contact tracing”, Science, 368:6491 (2020), eabb6936 | DOI

[29] C. Fraser, S. Riley, R.M. Anderson, N.M. Ferguson, “Factors that make an infectious disease outbreak controllable”, Proc. Nat. Acad. Sci., 101:16 (2004), 6146–6151 | DOI

[30] F.M. Guerra, S. Bolotin, G. Lim, et al., “The basic reproduction number ({$R_0$}) of measles: a systematic review”, Lancet Infectious Diseases, 17:12 (2017), e420–e428 | DOI

[31] E.S. Knock et al, “Key epidemiological drivers and impact of interventions in the 2020 SARS-CoV-2 epidemic in England”, Sci. Transl. Med., 13:602 (2021), eabg4262 | DOI

[32] M. Koivu-Jolma, A. Annila, “Epidemic as a natural process”, Math. Biosci., 299 (2018), 97–102 | DOI | MR | Zbl

[33] A. Matsuki, G. Tanaka, “Intervention threshold for epidemic control in susceptible-infected-recovered metapopulation models”, Phys. Rev. E, 100 (2019), 022302 | DOI

[34] D.M. Morens, G.K. Folkers, A.S. Fauci, “The challenge of emerging and re-emerging infectious diseases”, Nature, 430 (2004), 242–249 | DOI

[35] R. Padmanabhan, H.S. Abed, N. Meskin, T. Khattab, M. Shraim, M.A. Al-Hitmi, “A review of mathematical model-based scenario analysis and interventions for COVID-19”, Comput. Methods Programs Biomed., 209 (2021), 106301 | DOI

[36] Y.S. Popkov, Y.A. Dubnov, A.Y. Popkov, “Forecasting development of COVID-19 epidemic in European Union using entropy-randomized approach”, IA, 20:5 (2021), 1010–1033 | DOI | MR

[37] K. Shah, D. Saxena, D. Mavalankar, “Secondary attack rate of COVID-19 in household contacts: a systematic review”, QJM, 113:12 (2020), 841–850 | DOI

[38] Y. Shao, J. Wo, “IDM editorial statement on the 2019-nCoV”, Infect. Dis. Model., 5 (2020), 233–234 | MR

[39] P.C.L. Silva, P.V.C. Batista, H.S. Lima, M.A. Alves, F.G. Guimarães, R.C.P. Silva, “COVID-ABS: An agent-based model of COVID-19 epidemic to simulate health and economic effects of social distancing interventions”, Chaos Solitons Fractals, 139 (2020), 110088 | DOI | MR

[40] S. Sturniolo, W. Waites, T. Colbourn, D. Manheim, J. Panovska-Griffiths, “Testing, tracing and isolation in compartmental models”, PLOS Comput. Biol., 17:3 (2021), e1008633 | DOI

[41] K.M. Thompson, D.A. Kalkowska, “Review of poliovirus modeling performed from 2000 to 2019 to support global polio eradication”, Expert Rev. Vaccines, 19:7 (2020), 661–686 | DOI

[42] L. Zhou, Y. Wang, Y. Xiao, M.Y. Li, “Global dynamics of a discrete age-structured SIR epidemic model with applications to measles vaccination strategies”, Math. Biosci., 308 (2019), 27–37 | DOI | MR | Zbl

[43] A. Zumla, G. Ippolito, B. McCloskey, M. Bates, R. Ansumana, D. Heymann, R. Kock, F. Ntoumi, “Enhancing preparedness for tackling new epidemic threats”, Lancet Respir. Med., 5:8 (2017), 606–608 | DOI

[44] D. Wilson, J. Taaffe, “Tailoring the local HIV/AIDS response to local HIV/AIDS epidemics”, Major infectious diseases, 3rd ed., eds. K.K. Holmes et. al., The International Bank for Reconstruction and Development / The World Bank, 2017, 157–178

[45] M. Kretzschmar, L. Wiessing, “New challenges for mathematical and statistical modeling of HIV and hepatitis C virus in injecting drug users”, AIDS, 22:13 (2008), 1527–1537 | DOI

[46] S.I. Kabanikhin, O.I. Krivorot'ko, “Mathematical modeling of the Wuhan COVID-2019 epidemic and inverse problems”, Comput. Math. Math. Phys., 60:11 (2020), 1889–1899 | DOI | MR | Zbl

[47] O.I. Krivorot'ko, S.I. Kabanikhin, N.Yu. Zyat'kov, A.Yu. Prikhod'ko, N.M. Prokhoshin, M.A. Shishlenin, “Mathematical modeling and forecasting of COVID-19 in Moscow and Novosibirsk region”, Sib. Zh. Vychisl. Mat., 23:4 (2020), 395–414 | MR | Zbl

[48] W.O. Kermack, A.G. McKendrick, “A contribution to the mathematical theory of epidemics”, Proc. R. Soc. Lond., Ser. A, 115 (1927), 700–721 | DOI | Zbl

[49] A.L. Lloyd, “Estimating variability in models for recurrent epidemics: assessing the use of moment closure techniques”, Theor. Popul. Biol., 65:1 (2004), 49–65 | DOI | Zbl

[50] S.L. Currie, et al, “A prospective study to examine persistent HCV reinfection in injection drug users who have previously cleared the virus”, Drug Alcohol Depend., 93:1–2 (2008), 148–154 | DOI

[51] A.A. Gershon, M.D. Gershon, “Pathogenesis and current approaches to control of varicella-zoster virus infections”, Clin. Microbiol. Rev., 26:4 (2013), 728–743 | DOI

[52] O. Puhach, et al, “Infectious viral load in unvaccinated and vaccinated individuals infected with ancestral, Delta or Omicron SARS-CoV-2”, Nat. Med., 28:7 (2022), 1491–1500 | DOI

[53] F.A. Inthamoussou, F. Valenciaga, G. Núñez, “Extended SEIR model for health policies assessment against the COVID-19 pandemic: the case of argentina”, J. Healthc. Informa. Res., 6:1 (2022), 91–111 | DOI

[54] I. Cherednik, “Modeling the waves of covid-19”, Acta Biotheoretica, 70:8 (2021)

[55] A. Singanayagam, et al, “Duration of infectiousness and correlation with RT-PCR cycle threshold values in cases of COVID-19, England, January to May 2020”, Eurosurveillance, 25:32 (2020), 2001483 | DOI

[56] D. Greenhalgh, “Threshold and stability results for an epidemic model with an age-structured meeting rate”, IMA J. Math. Appl. Med. Biol., 5:2 (1988), 81–100 | DOI | MR

[57] R.M. Anderson, J.A. Crombie, B.T. Grenfell, “The epidemiology of mumps in the UK: A preliminary study of virus transmission, herd immunity and the potential impact of immunization”, Epidemiol. Infect., 99:1 (1987), 65–84 | DOI

[58] H. Zhong, W. Wang, “Mathematical modelling for scarlet fever with direct and indirect infections”, J. Biol. Dyn., 14:1 (2020), 767–787 | DOI | MR

[59] S. Basetti, J. Hodgson, T.M. Rawson, A. Majeed, “Scarlet fever: a guide for general practitioners”, London J. Prim. Care, 9:5 (2017), 77–79 | DOI

[60] H.M. Yang, A.S. Silveira, “The loss of immunity in directly transmitted infections modeling: Effects on the epidemiological parameters”, Bull. Math. Biol., 60:2 (1998), 355–372 | DOI | Zbl

[61] W.J. Edmunds, O.G. van de Heijden, M. Eerola, N.J. Gay, “Modelling rubella in Europe”, Epidemiol. Infect., 125:3 (2000), 617–634 | DOI

[62] K.C. Corey, A. Noymer, “A “post-honeymoon” measles epidemic in Burundi: mathematical model-based analysis and implications for vaccination timing”, PeerJ, 4 (2016), e2476 | DOI

[63] B.F. Finkenstädt, O.N. Bjørnstad, B.T. Grenfell, “A stochastic model for extinction and recurrence of epidemics: estimation and inference for measles outbreaks”, Biostatistics, 3:4 (2002), 493–510 | DOI | MR | Zbl

[64] G. Rozhnova, A. Nunes, “Modelling the long-term dynamics of pre-vaccination pertussis”, J. R. Soc. Interface, 9:76 (2012), 2959–2970 | DOI

[65] P. Pesco, P. Bergero, G. Fabricius, D. Hozbor, “Modelling the effect of changes in vaccine effectiveness and transmission contact rates on pertussis epidemiology”, Epidemics, 7 (2014), 13–21 | DOI

[66] H. Allen, et al, “Evidence for re-infection and persistent carriage of {Shigella} species in adult males reporting domestically acquired infection in England”, Clin. Microbiol. Infecti., 27:1 (2021), 126.e7–126.e13 | DOI | MR

[67] A.A. Shad, W.A. Shad, “Shigella sonnei: virulence and antibiotic resistance”, Arch. Microbiol., 203:1 (2021), 45–58 | DOI | MR

[68] M.W. Russell, S.D. Gray-Owen, A.E. Jerse, “Editorial: Immunity to Neisseria gonorrhoeae”, Front. Immunol., 11 (2020), 1375 | DOI

[69] H.W. Hethcote, J.A. Yorke, A. Nold, “Gonorrhea modeling: a comparison of control methods”, Math. Biosci., 58:1 (1982), 93–109 | DOI | MR | Zbl

[70] R.G. Woolthuis, J. Wallinga, M. van Boven, “Variation in loss of immunity shapes influenza epidemics and the impact of vaccination”, BMC Infect. Dis., 17 (2017), 632 | DOI

[71] A Cori, A.J. Valleron, F. Carrat, G.S. Tomba, G. Thomas, P.Y. Boëlle, “Estimating influenza latency and infectious period durations using viral excretion data”, Epidemics, 4:3 (2012), 132–138 | DOI

[72] Z. Hussain, B.C. Das, S.A. Husain, N.S. Murthy, P. Kar, “Increasing trend of acute hepatitis A in north India: need for identification of high-risk population for vaccination”, J. Gastroenterol. Hepatol., 21:4 (2006), 689–693 | DOI

[73] I. Czumbel, C. Quinten, P. Lopalco, J.C. Semenza, and ECDC expert panel working group, “Management and control of communicable diseases in schools and other child care settings: systematic review on the incubation period and period of infectiousness”, BMC Infect. Dis., 18:1 (2018), 199 | DOI

[74] G. Freer, M. Pistello, “Varicella-zoster virus infection: natural history, clinical manifestations, immunity and current and future vaccination strategies”, New Microbiol., 41:2 (2018), 95–105

[75] C.K. Aitken, J. Lewis, S.L. Tracy, T. Spelman, D.S. Bowden, M. Bharadwaj, H. Drummer, M. Hellard, “High incidence of hepatitis C virus reinfection in a cohort of injecting drug users”, Hepatology, 48:6 (2008), 1746–1752 | DOI

[76] P.K. Drain, et al, “Point-of-care HIV viral load testing: an essential tool for a sustainable global HIV/AIDS response”, Clin. Microbiol. Rev., 32:3 (2008), 00097–18

[77] B.G. Williams, V. Lima, E. Gouws, “Modelling the impact of antiretroviral therapy on the epidemic of HIV”, Curr. HIV Res., 9:6 (2011), 367–382 | DOI

[78] N.K. Martin, P. Vickerman, M. Hickman, “Mathematical modelling of hepatitis C treatment for injecting drug users”, J. Theor. Biol., 274:1 (2011), 58–66 | DOI | MR | Zbl

[79] M.M Cassell, R. Wilcher, R.A. Ramautarsing, N. Phanuphak, T.D. Mastro, “Go where the virus is: An HIV micro-epidemic control approach to stop HIV transmission”, Glob. Health Sci. Pract., 8:4 (2020), 614–625 | DOI

[80] N. Scott, et al, “Hepatitis C elimination in Myanmar: Modelling the impact, cost, cost-effectiveness and economic benefits”, Lancet Reg. Health West. Pac, 10 (2021), 100129

[81] Revell, A. et al, “Computational models as predictors of HIV treatment outcomes for the Phidisa cohort in South Africa”, South. Afr. J. HIV Med., 17:1 (2016), 450 | DOI

[82] Y. Zhou, Y. Shao, Y. Ruan, J. Xu, Z. Ma, C. Mei, J. Wu, “Modeling and prediction of HIV in China: transmission rates structured by infection ages”, Math. Biosci. Eng., 5:2 (2008), 403–418 | DOI | MR | Zbl

[83] T. Brown, W. Peerapatanapokin, “The Asian Epidemic Model: a process model for exploring HIV policy and programme alternatives in Asia”, Sexually Transmitted Infections, 80, suppl 1 (2004), i19–i24 | DOI

[84] A. Blake, J.E. Smith, “Modeling hepatitis C elimination among people who inject drugs in new hampshire”, JAMA Network Open, 4:8 (2021), e2119092 | DOI

[85] P. van den Driessche, “Reproduction numbers of infectious disease models”, Infect. Dis. Model., 2:3 (2017), 288–303

[86] S. Alizon, A. Hurford, N. Mideo, M. Van Baalen, “Virulence evolution and the trade-off hypothesis: history, current state of affairs and the future”, J. Evol. Biol., 22:2 (2009), 245–259 | DOI

[87] V. Tahan, C. Karaca, B. Yildirim etc., “Sexual transmission of HCV between spouses”, Am. J. Gastroenterol, 100:4 (2005), 821–824 | DOI

[88] R. Marguta, A. Parisi, “Periodicity, synchronization and persistence in pre-vaccination measles”, J. R. Soc. Interface, 13:119 (2016), 20160258 | DOI

[89] D.P. Greenberg, C.H. von König, U. Heininger, “Health burden of pertussis in infants and children”, Pediatr. Infect. Dis. J., 24:5 Suppl. (2005), S39–43 | DOI

[90] J.D. Cherry, “Pertussis: challenges today and for the future”, PLoS Pathog., 9:7 (2013), e1003418 | DOI

[91] I.S. Fazakas-Anca, A. Modrea, S. Vlase, “Using the stochastic gradient descent optimization algorithm on estimating of reactivity ratios”, Materials (Basel), 14:16 (2021), 4764 | DOI

[92] S. Bilal, B.K. Singh, A. Prasad, E. Michael, “Effects of quasiperiodic forcing in epidemic models”, Chaos, 26:9 (2016), 093115 | DOI | MR

[93] D. Gromov, I. Bulla, O. Silvia Serea, E.O. Romero-Severson, “Numerical optimal control for HIV prevention with dynamic budget allocation”, Math. Med. Biol., 35:4 (2018), 469–491 | DOI | MR | Zbl

[94] A.A. Aliyu, “Public health ethics and the COVID-19 pandemic”, Ann. Afr. Med., 20:3 (2021), 157–163 | DOI

[95] G.V. Demidenko, “Systems of differential equations of higher dimension and delay equations”, Sib. Math. J., 53:6 (2012), 1021–1028 | DOI | MR | Zbl

[96] W.C. Miller, “Infectious disease (in) epidemiology”, Epidemiology, 21:5 (2010), 593–594 | DOI

[97] G.V. Demidenko, “Systems of differential equations of higher dimension and delay equations”, Sib. Math. J., 53:6 (2012), 1021–1028 | DOI | MR | Zbl

[98] V. Petrakova, O. Krivorotko, “Mean field game for modeling of COVID-19 spread”, J. Math. Anal. Appl., 514:1 (2022), 126271 | DOI | MR | Zbl

[99] F. Brauer, C. Castillo-Chavez, Z. Feng, Control of communicable diseases in man, Springer, 2019

[100] A.S. Benenson, Red Book 2021 Report of the Committee of Infectious Diseases, American Public Health Association, 1981

[101] F. Brauer, C. Castillo-Chavez, Z. Feng, Mathematical models in epidemiology, Springer, New York, 2019 | MR | Zbl