Skip to main content

Advertisement

SpringerLink
Log in

State estimators for some epidemiological systems

  • Published:
Journal of Mathematical Biology Aims and scope Submit manuscript

Abstract

We consider a class of epidemiological models that includes most well-known dynamics for directly transmitted diseases, and some reduced models for indirectly transmitted diseases. We then propose a simple observer that can be applied to models in this class. The error analysis of this observer leads to a non-autonomous error equation, and a new bound for fundamental matrices is also presented. We analyse and implement this observer in two examples: the classical SIR model, and a reduced Bailey-Dietz model for vector-borne diseases. In both cases we obtain arbitrary exponential convergence of the observer. For the latter model, we also applied the observer to recover the number of susceptible using dengue infection data from a district in the city of Rio de Janeiro.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9

Similar content being viewed by others

References

  • Abdelhedi A, Boutat D, Sbita L, Tami R (2014) Extended observer to estimate the spreading of contagious disease. In 2014 European control conference (ECC), pp 1843–1848. https://doi.org/10.1109/ECC.2014.6862172

  • Abdelhedi A, Boutat D, Sbita L, Tami R, Liu D-Y (2016) Observer design for a class of nonlinear piecewise systems. Application to an epidemic model with treatment. Math Biosci 271:128–135

    Article  MathSciNet  MATH  Google Scholar 

  • Alonso-Quesada S, De la Sen M, Agarwal RP, Ibeas A (2012) An observer-based vaccination control law for an SEIR epidemic model based on feedback linearization techniques for nonlinear systems. Adv Differ Equ 161:32

    MathSciNet  MATH  Google Scholar 

  • Alvarez-Ramirez J, Meraz M, Velasco-Hernandez JX (2000) Feedback control of the chemotherapy of HIV. Int J Bifurc Chaos 10(09):2207–2219. https://doi.org/10.1142/S0218127400001377

    Article  MATH  Google Scholar 

  • Anderson RM, May RM (1991) Infectious diseases of humans: dynamics and control. Oxford Science Publications, Oxford

    Google Scholar 

  • Arino J, van den Driessche P (2003) A multi-city epidemic model. Math Popul Stud 10(3):175–193. https://doi.org/10.1080/08898480306720

    Article  MathSciNet  MATH  Google Scholar 

  • Aronna MS, Bliman P-A (2018) Interval observer for uncertain time-varying SIR-SI model of vector-borne disease. In Proceedings of European control conference. URL https://hal.inria.fr/hal-01493078

  • Audoly S, D’Angio L, Saccomani MP, Cobelli C (1998) Global identifiability of linear compartmental models-a computer algebra algorithm. IEEE Trans Biomed Eng 45(1):36–47

    Article  MATH  Google Scholar 

  • Bailey N (1975) The mathematical theory of infectious diseases and its applications. Griffin, London

    MATH  Google Scholar 

  • Becker NG (1989) Analysis of infectious disease data, vol 33. CRC Press, Boca Raton

    Google Scholar 

  • Besançon G (Ed) (2007) Nonlinear observers and applications. In Papers based on the presentations at the 28th Grenoble international summer school on control, Grenoble, France, September 2007. Lecture notes in control and information sciences 363. Berlin: Springer. xiii, 224 p

  • Bichara D, Cozic N, Iggidr A (2014) On the estimation of sequestered infected erythrocytes in Plasmodium falciparum malaria patients. Math Biosci Eng 11(4):741–759. ISSN 1547-1063. https://doi.org/10.3934/mbe.2014.11.741

  • Bjørnstad ON, Finkenstädt BF, Grenfell BT (2002) Dynamics of measles epidemics: estimating scaling of transmission rates using a time series sir model. Ecol Monogr 72(2):169–184

    Article  Google Scholar 

  • Bliman P-A, D’Avila Barros B (2017) Interval observers for SIR epidemic models subject to uncertain seasonality. In: Positive systems, volume 471 of Lecture Notes Control Inf Sci, pp 31–39. Springer, Cham

  • Bliman P-A, Efimov D, Ushirobira R (2018) A class of nonlinear adaptive observers for sir epidemic model. In Proceedings of European control conference

  • Capasso V, Capasso V (1993) Mathematical structures of epidemic systems, vol 88. Springer, Berlin

    Book  MATH  Google Scholar 

  • Cauchemez S, Ferguson NM (2008) Likelihood-based estimation of continuous-time epidemic models from time-series data: application to measles transmission in london. J R Soc Interface 5(25):885–897

    Article  Google Scholar 

  • Cauchemez S, Carrat F, Viboud C, Valleron A, Boelle P (2004) A bayesian mcmc approach to study transmission of influenza: application to household longitudinal data. Stat Med 23(22):3469–3487

    Article  Google Scholar 

  • Coppel W (1978) Dichotomies in Stability Theory, volume 629 of Lecture Notes in Mathematics. Springer-Verlag, Berlin

  • De la Sen M, Ibeas A, Alonso-Quesada S (2011) Observer-based vaccination strategy for a true mass action SEIR epidemic model with potential estimation of all the populations. Discr Dyn Nat Soc, pages Art. ID 743067, 19

  • Diaby M, Iggidr A, Sy M (2015) Observer design for a schistosomiasis model. Mathematical biosciences, 269:17–29. ISSN 0025-5564. http://www.sciencedirect.com/science/article/pii/S0025556415001625

  • Diekmann O, Heesterbeek J (2000) Mathematical epidemiology of infectious diseases: model building, analysis and interpretation. Wiley series in mathematical and computational biology. Wiley, Chichester

    Google Scholar 

  • Dietz K (1975) Transmission and control of arbovirus diseases. In: Ludwig E, Cooke KL (eds) Epidemiology. SIAM, New York, pp 104–121

    Google Scholar 

  • Drakeley C, Corran P, Coleman P, Tongren J, McDonald S, Carneiro I, Malima R, Lusingu J, Manjurano A, Nkya W et al (2005) Estimating medium-and long-term trends in malaria transmission by using serological markers of malaria exposure. Proc Nat Acad Sci USA 102(14):5108–5113

    Article  Google Scholar 

  • Esteva L, Vargas C (1998) Analysis of a dengue disease transmission model. Math Biosci 150(2):131–151

    Article  MATH  Google Scholar 

  • Finkenstädt BF, Grenfell BT (2000) Time series modelling of childhood diseases: a dynamical systems approach. J Roy Stat Soc: Ser C (Appl Stat) 49(2):187–205

    Article  MathSciNet  MATH  Google Scholar 

  • Fisman D, Khoo E, Tuite A (2014) Early epidemic dynamics of the west african 2014 ebola outbreak:estimates derived with a simple two-parameter model. PLOS Curr. https://doi.org/10.1371/currents.outbreaks.89c0d3783f36958d96ebbae97348d571

  • Gauthier J, Hammouri H, Othman S (1992) A simple observer for nonlinear systems applications to bioreactors. IEEE Trans Autom Control 37(6):875–880

    Article  MathSciNet  MATH  Google Scholar 

  • Gérardin P, Guernier V, Perrau J, Fianu A, Le Roux K, Grivard P, Michault A, De Lamballerie X, Flahault A, Favier F (2008) Estimating chikungunya prevalence in la reunion island outbreak by serosurveys: two methods for two critical times of the epidemic. BMC Infect Dis 8(1):1

    Article  Google Scholar 

  • Guiro A, Iggidr A, Ngom D, Touré H (2009) On the stock estimation for some fishery systems. Rev Fish Biol Fish 19(3):313–327

    Article  Google Scholar 

  • Guo H, Li MY, Shuai Z (2006) Global stability of the endemic equilibrium of multigroup SIR epidemic models. Can Appl Math Q 14(3):259–284

    MathSciNet  MATH  Google Scholar 

  • Guo H, Li MY, Shuai Z (2008) A graph-theoretic approach to the method of global Lyapunov functions. In: Proceedings of the American Mathematical Society, 136(8):2793–2802, 08. https://doi.org/10.2307/20535481. http://www.jstor.org/stable/20535481

  • Hethcote HW (2000) The mathematics of infectious diseases. SIAM Rev 42(4):599–653

    Article  MathSciNet  MATH  Google Scholar 

  • Ibeas A, de la Sen M, Alonso-Quesada S, Zamani I (2015) Stability analysis and observer design for discrete-time SEIR epidemic models. Adv Differ Equ 122:21

    MathSciNet  MATH  Google Scholar 

  • Iggidr A (2004) Encyclopedia of Life Support Systems (EOLSS), chapter Controllability, Observability and Stability of Mathematical Models. UNESCO, Eolss Publishers, Oxford

    Google Scholar 

  • Iggidr A, Sallet G, Tsanou B (2012) Global stability analysis of a metapopulation SIS epidemic model. Math Popul Stud 19(3):115–129. https://doi.org/10.1080/08898480.2012.693844

    Article  MathSciNet  MATH  Google Scholar 

  • Iggidr A, Sallet G, Souza MO (2016) On the dynamics of a class of multi-group models for vector-borne diseases. J Math Anal Appl 441(2):723–743. ISSN 0022-247X. https://doi.org/10.1016/j.jmaa.2016.04.003. http://www.sciencedirect.com/science/article/pii/S0022247X16300324

  • Inaba H (1990) Threshold and stability results for an age-structured epidemic model. J Math Biol 28(4):411–434

    Article  MathSciNet  MATH  Google Scholar 

  • Jacquez JA, Greif P (1985) Numerical parameter identifiability and estimability: integrating identifiability, estimability, and optimal sampling design. Math Biosci 77(1):201–227

    Article  MATH  Google Scholar 

  • Jacquez JA, et al. (1985) Compartmental analysis in biology and medicine. JSTOR

  • Joseph L, Gyorkos TW, Coupal L (1995) Bayesian estimation of disease prevalence and the parameters of diagnostic tests in the absence of a gold standard. Am J Epidemiol 141(3):263–272

    Article  Google Scholar 

  • Kågström B (1977) Bounds and perturbation bounds for the matrix exponential. BIT Numer Math 17(1):39–57. https://doi.org/10.1007/BF01932398

    Article  MathSciNet  MATH  Google Scholar 

  • Keeling MJ, Rohani P (2008) Modeling infectious diseases in humans and animals. Princeton University Press, Princeton

    MATH  Google Scholar 

  • Kermack WO, McKendrick AG (1927) A contribution to the mathematical theory of epidemics. In Proceedings of the Royal Society of London a: mathematical, physical and engineering sciences, volume 115, pp 700–721

  • Kuniya T (2011) Global stability analysis with a discretization approach for an age-structured multigroup SIR epidemic model. Nonlinear Anal Real World Appl 12(5):2640–2655

    Article  MathSciNet  MATH  Google Scholar 

  • Lajmanovich A, Yorke J (1976) A deterministic model for gonorrhea in a nonhomogeneous population. Math Biosci 28:221–236

    Article  MathSciNet  MATH  Google Scholar 

  • Lloyd AL (2001) The dependence of viral parameter estimates on the assumed viral life cycle: limitations of studies of viral load data. Proc Roy Soc Lond B Biol Sci 268(1469):847–854

    Article  Google Scholar 

  • Luenberger DG (1971) An introduction to observers. IEEE Trans Autom Control AC 16(6):596–602. http://search.ebscohost.com/login.aspx?direct=true&db=inh&AN=334765&lang=fr&site=ehost-live

  • Nishiura H (2006) Mathematical and statistical analyses of the spread of dengue. Dengue Bull 30:51–67

    Google Scholar 

  • Pongsumpun P, Tang I (2003) Transmission of Dengue hemorrhagic fever in an age structured population. Math Comput Model 37(9–10):949–961

    Article  MathSciNet  MATH  Google Scholar 

  • Reiner RC, Perkins TA, Barker CM, Niu T, Chaves LF, Ellis AM, George DB, Le Menach A, Pulliam JRC, Bisanzio D, Buckee C, Chiyaka C, Cummings DAT, Garcia AJ, Gatton ML, Gething PW, Hartley DM, Johnston G, Klein EY, Michael E, Lindsay SW, Lloyd AL, Pigott DM, Reisen WK, Ruktanonchai N, Singh BK, Tatem AJ, Kitron U, Hay SI, Scott TW, Smith DL (2013) A systematic review of mathematical models of mosquito-borne pathogen transmission: 1970–2010. J Roy Soc Interface. https://doi.org/10.1098/rsif.2012.0921

  • Rocha F, Aguiar M, Souza MO, Stollenwerk N (2013) Time-scale separation and center manifold analysis describing vector-borne disease dynamics. Int J Comput Math 90(10):2105–2125

    Article  MATH  Google Scholar 

  • Ross R (1911) The prevention of malaria. John Murray, London

    Google Scholar 

  • Smith DL, Battle KE, Hay SI, Barker CM, Scott TW, McKenzie FE (2012) Ross, macdonald, and a theory for the dynamics and control of mosquito-transmitted pathogens. PLoS Pathog. https://doi.org/10.1371%2Fjournal.ppat.1002588

  • Smith HL (1995) Monotone dynamical systems: an introduction to the theory of competitive and cooperative systems. Mathematical surveys and monographs. American Mathematical Society (AMS), Providence

    Google Scholar 

  • Souza MO (2014) Multiscale analysis for a vector-borne epidemic model. J Math Biol 68(5):1269–1293

    Article  MathSciNet  MATH  Google Scholar 

  • Tami R, Boutat D, Zheng G (2013) Extended output depending normal form. Automatica 49(7):2192–2198. https://doi.org/10.1016/j.automatica.2013.03.025. http://www.sciencedirect.com/science/article/pii/S0005109813001945

  • Tami R, Boutat D, Zheng G, Kratz F (2014) Parameters and states estimation for dengue epidemic model. In 2014 European control conference (ECC), pp 528–533. https://doi.org/10.1109/ECC.2014.6862173

  • Toni T, Welch D, Strelkowa N, Ipsen A, Stumpf MP (2009) Approximate bayesian computation scheme for parameter inference and model selection in dynamical systems. J R Soc Interface 6(31):187–202

    Article  Google Scholar 

  • Towers S, Patterson-Lomba O, Castillo-Chavez C (2014) Temporal variations in the effective reproduction number of the 2014 West Africa Ebola outbreak. PLOS Curr Outbreaks

  • Velasco-Hernández JX, García JA, Kirschner D (2002) Remarks on modeling host-viral dynamics and treatment. InCastillo-Chavez CEEA (Ed) Mathematical approaches for emerging and reemerging infectious diseases: an introduction. In Proceedings of a tutorial introduction to epidemiology and immunology, pp 287–308. Springer. IMA Vol Math Appl 125

  • Wonham W (1979) Linear multivariable control: a geometric approach, 2nd edn. Springer, New York

Download references

Author information

Authors and Affiliations

  1. Université de Lorraine, CNRS, Inria, IECL, F-57000, Metz, France

    A. Iggidr

  2. Instituto de Matemática e Estatística, Universidade Federal Fluminense, Rua Prof. Marcos Waldemar de Freitas Reis, S/N, Campus do Gragoatá, Niterói, RJ, 24210-201, Brazil

    M. O. Souza

Authors
  1. A. Iggidr
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. M. O. Souza
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to M. O. Souza.

Additional information

This work was partially funded by Inria and CAPES-Brazil in the framework of the program STIC AmSud (Project MOSTICAW). M. O. Souza was partially supported by CNPq Grants # 308113/2012-8, # 486395/2013-8 and # 309079/2015-2.

Appendix A: A glimpse at the data

Appendix A: A glimpse at the data

Table 1 Number of reported in the district of Jacarepaguá, together with the corresponding proportion—the population of this district was taken from the Brazilian census of 2000 as 540000
Full size table

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Iggidr, A., Souza, M.O. State estimators for some epidemiological systems. J. Math. Biol. 78, 225–256 (2019). https://doi.org/10.1007/s00285-018-1273-3

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00285-018-1273-3

Keywords

Mathematics Subject Classification

Search

Navigation