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

Diserahkan: 30 Julai 2021/Diterima: 1 Disember 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

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^*Pengarang untuk surat-menyurat; email: farahaini@usm.my