Volume 5, Issue 1, March 2020, Page: 16-38
A Mathematical Model and Analysis of an SVEIR Model for Streptococcus Pneumonia with Saturated Incidence Force of Infection
Opara Chiekezi Zephaniah, Department of Mathematics, Federal University of Technology, Owerri, Nigeria
Uche-Iwe Ruth Nwaugonma, Department of Mathematics, Federal University of Technology, Owerri, Nigeria
Inyama Simeon Chioma, Department of Mathematics, Federal University of Technology, Owerri, Nigeria
Omame Adrew, Department of Mathematics, Federal University of Technology, Owerri, Nigeria
Received: Oct. 16, 2019;       Accepted: Nov. 28, 2019;       Published: Feb. 19, 2020
DOI: 10.11648/j.mma.20200501.13      View  277      Downloads  116
In this paper, the dynamics of SVEIR model with saturated incidence force of infection and saturated vaccination function for Streptococcus pneumonia (that is, model that monitors the temporal transmission dynamics of the disease in the presence of preventive vaccine) was formulated and analyzed. The basic reproduction number that determines disease extinction and disease survival was revealed. The existing threshold conditions of all kinds of the equilibrium points are obtained and proved to be locally asymptotic stable for disease-free equilibrium using linearization method and Lyapunov functional method for Endemic equilibrium. Qualitative Analysis of the model was obtained and the positive of solution obtained. It was revealed that the model is positively –invariant and attracting. Thus the region is positively invariant. Hence, it is sufficient to consider the dynamics of the model (1) in the given region. In this region, the model can be considered as been epidemiologically and mathematically well-posed. The governing model was normalized and also Adomian Decomposition method was used to compute an approximate solution of the non-linear system of differential equations governing the model. Maple was used in carrying out the simulations (numerical solutions) of the model. Graphical results were presented and discussed to illustrate the solution of the problem. The achieved results reveal that the disease will die out within the community if the vaccination coverage is above the critical vaccination proportion. The study indicates that we should improve the efficiency and enlarge the capacity of the treatment to control the spread of disease.
Mathematical Model, SVEIR Model, Streptococcus Pneumonia, Saturated Incidence Force Ofinfection
To cite this article
Opara Chiekezi Zephaniah, Uche-Iwe Ruth Nwaugonma, Inyama Simeon Chioma, Omame Adrew, A Mathematical Model and Analysis of an SVEIR Model for Streptococcus Pneumonia with Saturated Incidence Force of Infection, Mathematical Modelling and Applications. Vol. 5, No. 1, 2020, pp. 16-38. doi: 10.11648/j.mma.20200501.13
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Acedo L, Gonzalez-Parra G, &Arenas A. (2010) An exact global solution for the classical Epidemic model; Nonlinear Analysis: Real World Applications; 11 (3): 1819–1825.
Alexander M. E, & Moghadas S. M. (2005) Bifurcation analysis of SIRS epidemic model with generalizedIncidence, SIAM Journal on Applied Mathematics, 65 (5): 1794–181.
Batt, S. L., Charalambous, B. M., Solomon, A. W., Knirsch, C., Massae, P. A., Safari, S., Sam, N. E., Everett, D., Mabey, D. C. & Gillespie, S. H. (2003) Impact of azithromycin administration for trachoma control on the carriage of antibiotic-resistant Streptococcus pneumoniae. Antimicrobial Agents and Chemotherapy 47: 2765-2769.
Blossom, D. B., Namayanja-Kaye, G., Nankya-Mutayoba, J., Mukasa, J. B., Bakka, H., Rwambuya, S., Windau, A., Bajaksouzian, S., Walker, C. J. & Joloba, M. L. (2006) Oropharyngeal colonization by Streptococcus pneumonia among HIV-infected adults in Uganda: assessing prevalence and antimicrobial susceptibility. International Journal of Infectious Diseases 10: 458-464.
Bogaert, D., De Groot, R. & Hermans, P. (2004) Streptococcus pneumonia colonization: the key to pneumococcal disease. Lancet Infectious 4: 144-154.
Centers for Disease Control and Prevention (2017) “Streptococcus pneumonia” Published on official website https://www.cdc.gov/pneumococcal/clinicians/streptococcus-pneumoniae.html.
Diftz K., Schenzle D. Proportionate mixing models for age-dependent infection transmission J. Math. Biol., 22 (1985): 117-120.
Donnelly C. A. (2003) Epidemiological determinants of spread of causal agent of severe acute respiratory syndrome in Hong Kong, The Lancet Published online May 7, 2003.
Dulpl, E. (2012) Nasopharyngeal carrier rate of Streptococcus pneumonia in children: serotype distribution and antimicrobial resistance. Archieves of Iranian Medicine 15: 500.
Eckalbar J. C &Eckalbar W. L. (2011) Dynamics of an epidemic model with quadratic treatment. Nonlinear Analysis: Real World Applications. 2011; 12 (1): 320–332.
Esteva L & Matias M. (2001) A model for vector transmitted diseases with saturation incidence. Journal of Biological Systems. 2001; 9 (4): 235–245.
Faden, H., Duffy, L., Wasielewski, R., Wolf, J., Krystofik, D. & Tung, Y. (1997) Relationship between nasopharyngeal colonization and the development of otitis media in children. Journal of Infectious Diseases 175: 1440-1445.
Falade, A. G. & Ayede, A. I. (2011) Epidemiology, aetiology and management of childhood acute community-acquired pneumonia in developing counties - a review. African Journal of Medicine and Medical Sciences 40: 293-308.
Hethcote H. (2000) The Mathematics of Infectious Diseases; SIAM Review; 42 (4): 599–653.
Hethcote H., Zhien M, Shengbing L., (2002) “Effects of quarantine in six endemic models for infectious diseases Mathematical Biosciences, 180 (2002): 141-160, 10.1016/S0025-5564(02)00111-6.
He, J. H. (1999). Variational iteration method- a kind of non-linear analytical technique: some examples, Int. J. Nonlin. Mech., 34: 699-708.
He, J. H. & X. H. Wu (2006). Construction of solitary solutions and compacton-like solution by variational iteration method, Chos. Soltn. Frcts; 29 (1): 108-113.
Hoppensteadt F. (1974) “An Age Dependent Epidemic Model” Journal of the Franklin Institute 297 (5): 325-333.
Hoppensteadt F. (1974) Asymptotic stability in singular perturbation problems J. Diff. Eq, 15 (no. 3): 510-521.
Jacobs, M. R. (2004) Streptococcus pneumoniae: Epidemiology and patterns of resistance. American Journal of Medicine Supplements117: 3-15.
Joloba, M., Bajaksouzian, S., Plalavecino, E., Whalen, C. & Jacobs, M. (2001a) High prevalence of carriage of antibiotic-resistant Streptococcus pneumoniae in children in Kampala, Uganda. International Journal of Antimicrobial Agents 17: 395-400.
Kinabo G. D, Van der Ven A, Msuya, Shayo A. M, Schimana W, Ndaro A, Van Asten H, Dolmans W, Warris A, & Hermans P (2013) Dynamics of nasopharyngeal bacterial colonization in HIV exposed young infant in Tanzania; 18 (3): 286-295.
Korobeinikov, A. (2004) Lyapunov functions and global properties for SEIR and SEIS epidemic models; Math. Med. Biol.: J. IMA., 21: 75-83.
La Salle J. P. (1976) The Stability of Dynamical Systems, SIAM, No. 25.
Lannelli M, Martcheva M, & Li X.-Z., (2005) “Strain replacement in an epidemic model with super-infection and perfect vaccination,” Mathematical Biosciences, 195 (1): 23–46, 2005. View at Publisher View at Google Scholar View at MathSciNet • View at Scopus.
Li M. Y, &Muldowney J. S. (1996) A geometric approach to global-stability problems. SIAM Journal onMathematical Analysis; 27 (4): 1070–1083.
Li, J., and Brauer, F. (2008). “Continuous-time age-structured models in population dynamics and epidemiology,” in Math Epidemiol, Lecture Notes in Mathematics, eds. F. Brauer, P. van den Driessche, and J. Wu (Berlin; Heidelberg: Springer), 1945: 205–227.
Li, Y. (2004). Determination of the critical concentration of neutrophils required to block bacterial growth in tissues. J. Exp. Med. 200: 613–622. doi: 10.1084/jem.2004072.
Liu, X. B, &Yang, L. J. (2012) Stability analysis of an SEIQV epidemic model with saturated incidence rate. Nonlinear Analysis: Real World Applications; 13 (6): 2671–2679.
Mochan, E., Swigon, D., Ermentrout, B., Lukens, S., & Clermont, G. (2014). A mathematical model of intrahost pneumococcal pneumonia infection dynamics in murine strains; J. Theoritical Biol. 353: 44–54. doi: 10.1016/j.jtbi.2014.02.021.
McCullers, J. A., English, B. K., & Novak, R. (2000). Isolation and characterization of vancomycin-tolerant Streptococcus pneumoniae from the cerebrospinal fluid of a patient who developed recrudescent meningitis; J. Infect. Dis. 181: 369–373. Doi: 10.1086/315216.
MurrayJ. D. (2001) Mathematical Biology (2nd, corrected Edition), Springer-Verlag, pp. 385-396.
Nantanda, D. M., Hildenwall, H., Peterson, S., Kaddu-Mulindwa, D., Kalyesubula, I. &Tumwine, J. K. (2008) Bacterial aetiology and outcome in children with severe pneumonia in Uganda. Annals of Tropical Paediatrics: International Child Health 28: 253-260.
Nelson, P. W., Gilchrist, M. A., Coombs, D., Hyman, J. M., & Perelson, A. S. (2004). An age-structured model of HIV infection that allows for variations in the death rate of productively infected cells; Math. Bioscience 1: 267–288. Doi: 10.3934/mbe.2004.1.267.
Nuorti, P., Butler, J. C., Crutcher, J., Guevara, R., Welch, D., Holder, P., et al. (1998). An outbreak of multidrug- resistantant pneumococcal pneumonia and bacteremia among unvaccinated nursing rome residents. N. Engl. J. Med. 338: 1861–1868. Doi: 10.1056/NEJM199806253382601.
O’Brien, K. L., Wolfson, L. J., Watt, J. P., Henkle, E., Deloria-Knoll, M., McCall, N., Lee, E., Mulholland, K., Levine, O. S. & Cherian, T. (2009) Burden of disease caused by Streptococcus pneumoniae in children younger than 5 years: global estimates. Lancet 374: 893-902.
Prina, E., Ranzani, O. T., & Torres, A. (2015b). Community-acquired pneumonia; Lancet 386: 1097–1108. doi: 10.1016/S0140-6736(15)60733-4.
Schrag, S. J., Peña, C., Fernández, J., Sánchez, J., Gómez, V., Pérez, E., et al. (2001). Effect of short-course, high-dose amoxicillin therapy on resistant pneumococcal carriage: a randomized trial. JAMA 286: 49–56. Doi: 10.1001/jama.286.1.49. Available online at: http://jamanetwork.com/journals/jama/fullarticle/193977.
Shrestha, S., Foxman, B., Dawid, S., Aiello, A. E., Davis, B. M., Berus, J., et al. (2013). Time and dose-dependent risk of pneumococcal pneumonia following influenza: a model for within-host interaction between influenza and Streptococcus pneumoniae. Interf. Focus 10: 20130233. doi: 10.1098/rsif.2013.0233.
Smith, A. M., Adler, F. R., Ribeiro, R. M., Gutenkunst, R. N., McAuley, J. L., McCullers, J. A., et al. (2013). Kinetics of coinfection with influenza A virus and Streptococcus pneumoniae. PloSPathog. 9:e1003238. Doi: 10.1371/journal.ppat.1003238.
Smith, A. M., McCullers, J. A., & Adler, F. R. (2011). Mathematical model of a three-stage innate immune response to a pneumococcal lung infection; J. Theoritical Biology 276: 106–116. doi: 10.1016/j.jtbi.2011.01.052.
Van den Driessche & Watmough J. “Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission”; Mathematics Biosciences, 180 (2): 29-48.
Wang W. & Ruan S. (2004), Simulating the SARS outbreak in Beijing with limited data, J. Theoretical Biology227: 369-379.
Xu, R. (2012) Global stability of a delayed epidemic model with latent period and vaccination strategy Appl. Math. Model, 36: 5293-5300.
Zhang X, & Liu X. N. (2008) Backward bifurcation of an epidemic model with saturated treatment function. Journal of Mathematical Analysis and Applications; 348 (1): 433–443.
Zhou L, & Fan M. (2012) Dynamics of an SIR epidemic model with limited medical resources revisited. Nonlinear Analysis: Real World Applications; 13 (1): 312–324.
Omame, A., Okuonghae, D., Umana, R. A., Inyama, S. C., (2019) Analysis of a co-infection model for HPV-TB, Applied Mathematical Modelling, https://doi.org/10.1016/j.apm.2019.08.012.
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