Sains Malaysiana 51(7)(2022): 2197-2209

http://doi.org/10.17576/jsm-2022-5107-21

Mathematical Model of Pertussis and Pneumonia Co-Infection in Infants with Maternally Derived Immunity

(Model Matematik bagi Jangkitan Bersama Batuk Kokol dan Pneumonia pada Bayi dengan Imuniti daripada Ibu)

AISHA ALIYU YAKUBU^1,2, FARAH AINI ABDULLAH^1* & YAZARIAH MOHD YATIM¹

¹School of Mathematical Sciences, Universiti Sains Malaysia, 11800 USM, Pulau Pinang, Malaysia

²Ibrahim Badamasi Babangida University Lapai, PMB 11, Niger State Nigeria

Received: 30 July 2021/Accepted: 1 December 2021

Abstract

The transmission dynamics of a pertussis-pneumonia co-infection model is analyzed. The model takes into account temporary immunity of infected infants and includes a maternally derived immunity compartment. The basic reproduction number of the co-infected model is obtained using the next generation matrix, and stability analysis is carried out. The model exhibits four equilibria, namely, the pertussis-free equilibrium, the pneumonia-free equilibrium, the co-infection-free equilibrium and co-infection endemic equilibrium. Subsequently, the local stability of the co-infection-free equilibrium is analyzed and is shown to be locally asymptotically stable. Similarly, by constructing a suitable Lyapunov function, the co-infection endemic equilibrium is shown to be globally asymptotically stable. Numerical simulations are carried out to illustrate the validity of these results.

Keywords: Childhood respiratory diseases; common quadratic Lyapunov function; global stability; maternally derived immunity; pertussis-pneumonia co-infection

Abstrak

Analisis dinamik penularan model jangkitan bersama batuk kokol-radang paru-paru dijalankan. Model ini mengambil kira imuniti sementara bayi yang dijangkiti dan merangkumi ruang imuniti yang berasal daripada ibu. Nombor pembiakan asas model yang dijangkiti bersama diperoleh menggunakan matriks generasi berikutnya dan analisis kestabilan dilakukan. Model ini menunjukkan empat keseimbangan, iaitu, keseimbangan bebas batuk kokol, keseimbangan bebas radang paru-paru, keseimbangan bebas jangkitan dan keseimbangan endemik jangkitan bersama. Selepas itu, kestabilan tempatan keseimbangan bebas jangkitan dianalisis dan terbukti ia stabil secara asimptot. Begitu juga dengan membina fungsi Lyapunov yang sesuai, keseimbangan endemik jangkitan bersama terbukti stabil secara asimptot. Simulasi berangka dijalankan untuk menggambarkan kesahihan hasil ini.

Kata kunci: Fungsi kuadratik umum Lyapunov; imuniti terbitan ibu; jangkitan bersama batuk kokol-radang paru-paru; kestabilan global; penyakit respiratori kanak-kanak

REFERENCES

Aggarwal, R. 2020. Dynamics of HIV-TB co-infection with detection as optimal intervention strategy. International Journal of Non-Linear Mechanics 120: 103388.

Barger-Kamate, B., Deloria Knoll, M., Kagucia, E.W., Prosperi, C., Baggett, H.C., Brooks, W.A., Feikin, D.R., Hammitt, L.L., Howie, S.R., Levine, O.S. & Madhi, S.A. 2016. Pertussis-associated pneumonia in infants and children from low-and middle-income countries participating in the PERCH study. Clinical Infectious Diseases 63(4): 187-196.

Bichara, D., Iggidr, A. & Sallet, G. 2014. Global analysis of multi-strains SIS, SIR and MSIR epidemic models. Journal of Applied Mathematics and Computing 44(1-2): 273-292.

Birger, R.B., Kouyos, R.D., Cohen, T., Griffiths, E.C., Huijben, S., Mina, M.J., Volkova, V., Grenfell, B. & Metcalf, C.J.E. 2015. The potential impact of coinfection on antimicrobial chemotherapy and drug resistance. Trends in Microbiology 23(9): 537-544.

Burrell, C.J., Howard, C.R. & Murphy, F.A. 2016. Fenner and White's Medical Virology. 5th ed. Massachusetts: Academic Press. p. 604.

Chang, I.F., Lee, P.I., Lu, C.Y., Chen, J.M., Huang, L.M. & Chang, L.Y. 2019. Resurgence of pertussis in Taiwan during 2009-2015 and its impact on infants. Journal of Microbiology, Immunology and Infection 52(4): 542-548.

Cheng, Y.H., You, S.H., Lin, Y.J., Chen, S.C., Chen, W.Y., Chou, W.C., Hsieh, N.H. & Liao, C.M. 2017. Mathematical modeling of post coinfection with influenza. A virus and Streptococcus pneumoniae, with implications for pneumonia and COPD-risk assessment. International Journal of Chronic Obstructive Pulmonary Disease 12: 1973.

Cheon, M.K., Na, H., Han, S.B., Kwon, H.J., Chun, Y.H. & Kang, J.H. 2015. Pertussis accompanying recent mycoplasma infection in a 10-year-old girl. Infection & Chemotherapy 47(3): 197-201.

Clay, P.A., Duffy, M.A. & Rudolf, V.H. 2020. Within-host priority effects and epidemic timing determine outbreak severity in co-infected populations. Proceedings of the Royal Society B 287(1922): 20200046.

Cox, F.E.G. 2001. Concomitant infections, parasites and immune responses. Parasitology-Cambridge 122: 23-38.

Diekmann, O. & Heesterbeek, J.A.P. 2000. Mathematical Epidemiology of Infectious Diseases: Model Building, Analysis and Interpretation. 1st ed. John Wiley and Sons. p. 303.

Diekmann, O., Heesterbeek, J. & Metz, J.A. 1990. On the definition and the computation of the basic reproduction ratio  in models for infectious diseases in heterogeneous populations, Journal of Mathematical Biology 28(4): 365-382.

Glidden, C.K., Coon, C.A., Beechler, B.R., McNulty, C., Ezenwa, V.O. & Jolles, A.E. 2021. Co‐infection best predicts respiratory viral infection in a wild host. Journal of Animal Ecology 90(3): 602-614.

Gouveia, N. & Fletcher, T. 2000. Respiratory diseases in children and outdoor air pollution in Sao Paulo, Brazil: A time series analysis. Occupational and Environmental Medicine 57(7): 477-483.

Griffiths, E.C., Pedersen, A.B., Fenton, A. & Petchey, O.L. 2011. The nature and consequences of coinfection in humans. Journal of Infection 63(3): 200-206.

Hethcote, H.W. 2000. The mathematics of infectious diseases. SIAM Review 42(4): 599-653.

Jiang, W., Wu, M., Chen, S., Li, A., Wang, K., Wang, Y., Chen, Z., Hao, C., Shao, X. & Xu, J. 2021. Virus coinfection is a predictor of radiologically confirmed pneumonia in children with Bordetella pertussis infection. Infectious Diseases and Therapy 10(1): 335-346.

Kilgore, P.E., Salim, A.M., Zervos, M.J. & Schmitt, H.J. 2016. Pertussis: Microbiology, disease, treatment, and prevention. Clinical Microbiology Reviews 29(3): 449-486.

La Salle, J.P. 1976. The Stability of Dynamical Systems. Society for Industrial and Applied Mathematics.

Mbabazi, F.K., Mugisha, J.Y.T. & Kimathi, M. 2018. Modeling the within-host co-infection of influenza A virus and pneumococcus. Applied Mathematics and Computation 339: 488-506.

Muloiwa, R., Wolter, N., Mupere, E., Tan, T., Chitkara, A.J., Forsyth, K.D., von König, C.H.W. & Hussey, G. 2018. Pertussis in Africa: Findings and recommendations of the Global Pertussis Initiative (GPI). Vaccine 36(18): 2385-2393.

National Cancer Institute 2018. Respiratory Disease. https://www.cancer.gov/publications/dictionaries/cancer-terms/def/respiratory-disease Accessed on 16th April, 2019.

Nthiiri, J.K., Lavi, G.O. & Mayonge, A. 2015. Mathematical model of pneumonia and HIV/AIDS coinfection in the presence of protection. Int. J. Math Anal. 9(42): 2069-2085.

Okosun, K.O., Khan, M.A., Bonyah, E. & Okosun, O.O. 2019. Cholera‐schistosomiasis coinfection dynamics. Optimal Control Applications and Methods 40(4): 703-727.

Omame, A., Okuonghae, D., Umana, R.A. & Inyama, S.C. 2020. Analysis of a co-infection model for HPV-TB. Applied Mathematical Modelling 77: 881-901.

Pesco, P., Bergero, P., Fabricius, G. & Hozbor, D. 2015. Mathematical modeling of delayed pertussis vaccination in infants. Vaccine 33(41): 5475-5480.

Pinky, L. & Dobrovolny, H.M. 2016. Coinfections of the respiratory tract: Viral competition for resources. PLoS ONE 11(5): e0155589.

Shah, N.H., Sheoran, N. & Shah, Y. 2020. Dynamics of HIV-TB co-infection model. Axioms 9(1): 29.

Siddik, S.B.M., Abdullah, F.A. & Ismail, A.I.M. 2020. Mathematical model of dengue virus with predator-prey interactions. Sains Malaysiana 49(5): 1191-1200.

Tilahun, G.T. 2019. Modeling co-dynamics of pneumonia and meningitis diseases. Advances in Difference Equations 2019(1): 149.

Vargas-De-León, C. 2009. Constructions of Lyapunov functions for classic SIS, SIR and SIRS epidemic models with variable population size. Foro-Red-Mat: Revista electrónica de contenido matemático 26: 1-12.

Yakubu, A.A., Abdullah, F.A., Md Ismail, A.I. & Yatim, Y.M. 2020. Dynamical analysis on the transmission of pertussis with maternally derived immunity. Journal of Mathematics and Statistics 16(1): 104-112.

Zouari, A., Touati, A., Smaoui, H., Brun, D., Kasdaghli, K., Menif, K., Jaballah, N.B., Hassen, E.B., Guiso, N. & Kechrid, A. 2012. Dual infection with Bordetella pertussis and Mycoplasma pneumoniae in three infants. Infection 40(2): 213-217.

^*Corresponding author; email: farahaini@usm.my