Deterministic and Stochastic Models of the Spread of Streptococcal Disease and Its Sequel

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Nichaphat Patanarapeelert
Natsuda Yokchoo
Klot Patanarapeelert*

Abstract

The beta-hemolytic group A Streptococcus (GAS) is responsible for its sequel, acute rheumatic fever (ARF), which may lead to the more serious condition on other heart diseases. To gain a better understanding of the transmission in a population, we formulated epidemic models using a standard compartmental model and a continuous-time Markov chain. The models allow for the contribution of disease carrier and the effect of treatment. The equilibrium points and stability are analyzed in relation to the basic reproduction number based on the deterministic model. For the stochastic model, numerical simulation of sample paths is performed. The results indicate that the dynamic behavior for the two approaches depends on the epidemic threshold. Under stable endemic condition, most sample paths fluctuate around its mean and deterministic curve. On the other hand, when the basic reproduction number is less than one, the stochastic system undergoes a minor outbreak, while the deterministic curve approaches zero. The results are expected to be the first step of a deeper analysis of stochastic treatment linked to its deterministic counterpart.   


 


Keywords: group A Streptococcus; acute rheumatic fever; deterministic model; stochastic model; carriers


*Corresponding author: Tel.: (+66) 94 264 8255


                                          E-mail: klotpat@gmail.com

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References

[1] Doura, K., Melendez-Morales, J.D, Meyer, G.G. and Perez, L.E.,1999. An SIS Model of Streptococcal Disease with a Class of Beta-Hemolytic Carriers. Biometrics Unit Technical Reports, Number BU-1524-M.
[2] Armstrong, C., 2010. AHA guidelines on prevention of rheumatic fever and diagnosis and treatment of acute streptococcal pharyngitis. American Family Physician, 81(3), 346-359.
[3] Fauci, A.S., Braunwald, E., Kasper, D.L., Hauser, S.L., Longo, D.L., Jameson, J.L. and Loscalzo, J., 2008. Harrison's Principles of Internal Medicine. Vol II. 17th ed. New York: McGrow-Hill.
[4] Mayo Clinic, 2017. Rheumatic Fever. [online] Available at: https://www.mayoclinic.org/ diseases-conditions/rheumatic-fever/symptoms-causes/syc-20354588
[5] Choby, C., 2009. Diagnosis and treatment of streptococcal pharyngitis.American Family Physician, 79(5), 383-390.
[6] National Heart Foundation of New Zealand, 2019. Group A Streptococcal Sore Throat Management Guideline. Auckland: National Heart Foundation of New Zealand.
[7] Wessels, M.R., 2011. Streptococcal pharyngitis. The New England Journal of Medicine, 364 (7), 648-655.
[8] Chauhan, S., Kashyap, N., Kanga, A., Thakur, K., Sood, A. and Chandel, L., 2016. Genetic diversity among group a streptococcus isolated from throats of healthy and symptomatic children. Journal of tropical pediatrics, 62(2), 152–157.
[9] Weber, R., 2014. Pharyngitis. Primary Care: Clinics in Office Practice, 41(1), 91-98.
[10] Department of Health, 2011. Streptococcal Infections (Invasive Group A Strep, GAS). [online] Available at: https://www.health.ny.gov/diseases/communicable/streptococcal/group_a/ fact_ sheet.htm
[11] Martin, J.M., Green, M., Barbadora, K.A. and Wald, E.R., 2004. Group A streptococci among school-aged children: clinical characteristics and the carrier state. Pediatrics, 114(5), 1212–1219.
[12] McGuinness, M., Sibanda, N. and Welsh, M., 2015. Modeling acute rheumatic fever. Mathematical Population Studies, 22(4), 187-208.
[13] Barreto, M. L., Teixeira, M.G., and Carmo, E.H., 2006. Infectious diseases epidemiology. Journal of Epidemiology and Community Health, 60(3), 192-195.
[14] Centers for Disease Control and Prevention, 2012. Principles of Epidemiology in Public Health Practice, Third Edition an Introduction to Applied Epidemiology and Biostatistics. [online] Available at: https://www.cdc.gov/ophss/csels/dsepd/ss1978/lesson1/section10.html
[15] Bargh, G.J.M., Hoch, J.S., Hwang, S,W., Speechley, M., Willey, B.M. and McGeer, A., 2007. Group A streptococcal carriage among residents of an urban homeless shelter. Infectious Diseases and Medical Microbiology, 18(5), 316-317.
[16] PRINTO, 2016. Information on Pediatric Rheumatic Diseases. [online] Available at: https://www.printo.it/pediatric-rheumatology/TH/info/10/
[17] Sika-Paotonu, D., Beaton, A., Raghu, A., Steer, A. and Carapetis, J., 2017. Acute Rheumatic Fever and Rheumatic Heart Disease. In: J.J. Ferretti, D.L. Stevens and V.A. Fischetti, eds. Streptococcus pyogenes : Basic Biology to Clinical Manifestations [Internet]. Oklahoma City: : University of Oklahoma Health Sciences Center.
[18] RHD Australia, 2020. What is Acute Rheumatic Fever? [online] Available at: https://www. rhdaustralia.org.au/what-acute-rheumatic-fever
[19] Bagnall, E.M., Ho, M.J. and McCormick, I.A., 2015. A 39-year-old man with recurrent rheumatic fever. CMAJ, 187(1), 50-54.
[20] Verkaeren, E., Epelboin, L., Epelboin, S., Boddaert, N., Brossier, F. and Caumes, E. 2014. Recurrent Streptococcus pyogenes genital infection in a woman: test and treat the partner! International Journal of Infectious Diseases, 29, 37-39.
[21] Passali, D., Lauriello, M., Passali, G.C., Passali, F.M. and Bellussi, L., 2017. Group A Streptococcus and its antibiotic resistance. Acta Otorhinolaryngologica Italica, 27(1), 27-32.
[22] Sela, S. and Barzilai, A., 1999. Why do we fail with penicillin in the treatment of group A streptococcus infections? Annals of Medicine, 31(5), 303-307.
[23] Olivier, C., 2000. Rheumatic fever-is it still a problem? Journal of Antimicrobial Chemotherapy, 45(suppl.1), 13-21.
[24] Yokchoo, N., Patanarapeelert, N. and Patanarapeelert, K., 2019. The effect of group A streptococcal carrier on the epidemic model of acute rheumatic fever. Theoretical Biology and Medical Modelling, 16(1), 1-9.
[25] Martin, J., J.J. Ferretti, D.L. Stevens and V.A. Fischetti, ed., 2016. The Streptococcus pyogenes carrier state. In Streptococcus pyogenes:Basic Biology to Clinical Manifestations [Internet]. Oklahoma: University of Oklahoma Health Sciences Center.
[26] DeMuri, G.P. and Wald, E.R., 2014. The group A streptococcal carrier state reviewed: still an enigma. Journal of the Pediatric Infectious Diseases Society, 3(4), 336-342.
[27] Gerber, M.A., Baltimore, R.S., Eaton, C.B., Gewitz, M., Rowley, A.H., Shulman, S.T. and Taubert, K.A., 2009. Prevention of rheumatic fever and diagnosis and treatment of acute Streptococcal pharyngitis: a scientific statement from the American Heart Association Rheumatic Fever, Endocarditis, and Kawasaki Disease Committee of the Council on Cardiovascular Disease in the Young, the Interdisciplinary Council on Functional Genomics and Translational Biology, and the Interdisciplinary Council on Quality of Care and Outcomes Research: endorsed by the American Academy of Pediatrics. Circulationaha, 119(11), 1541- 1551.
[28] Krause, R.M., Rammelkamp, C.H., Denny, F.W. and Wannamaker, L.W., 1962. Studies of the carrier state following infection with group A streptococci. I. Effect of climate. The Journal of Clinical Investigation, 41(3), 568-574.
[29] Gillespie, D.T., 1997. Exact stochastic simulation of coupled chemical reactions. The Journal of Physical Chemistry, 81(25), 2340-2361.
[30] Ayoub, E.M., 1992. Resurgence of rheumatic fever in the United States: The changing picture of preventable disease. Postgraduate Medicine, 92(3), 133-142.
[31] Spencer, S., 2008. Stochastic Epidemic Models for Emerging Diseases. Ph.D. University of Nottingham.
[32] van den Driessche, P. and Watmough, J., 2002. Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Mathematical Biosciences, 180(1-2), 29-48.
[33] Suli, E., 2014. Numerical Solution of Ordinary Differential Equations. [online] Available at: https://people.maths.ox.ac.uk/suli/nsodes.pdf