Analysis and modelling of the transmission dynamics of tuberculosis in the presence of latent and active populations

Aliyu  Ibrahim; Mahdi Audu  Janda; Stella Nyambura  Kahianyu; Ass  Gueye; Peter Chola  Nkandu; Eugene Tettey  Ayerkain

doi:10.59400/jam1870

Analysis and modelling of the transmission dynamics of tuberculosis in the presence of latent and active populations

Aliyu Ibrahim Department of Mathematics, African Institute for Mathematical Sciences (AIMS), Mbour 23000, Senegal
Mahdi Audu Janda Department of Mathematics, African Institute for Mathematical Sciences (AIMS), Mbour 23000, Senegal
Stella Nyambura Kahianyu Department of Mathematics, African Institute for Mathematical Sciences (AIMS), Mbour 23000, Senegal
Ass Gueye Department of Mathematics, African Institute for Mathematical Sciences (AIMS), Mbour 23000, Senegal
Peter Chola Nkandu Department of Mathematics, African Institute for Mathematical Sciences (AIMS), Mbour 23000, Senegal
Eugene Tettey Ayerkain Department of Mathematics, African Institute for Mathematical Sciences (AIMS), Accra GA027, Ghana

Article ID: 1870

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DOI: https://doi.org/10.59400/jam1870

Keywords: tuberculosis; latent population; reproduction number; sensitivity; stability

Abstract

Tuberculosis, a chronic infectious disease caused by Mycobacterium tuberculosis, remains a significant global health challenge, particularly in developing countries. This project investigates the dynamic transmission of tuberculosis, focusing on the interplay between latent and active populations. We develop and analyze an (Susceptible, Latent, Infectious, Recovered) compartmental mathematical model to examine key parameters affecting TB transmission dynamics. Our study employs stability and sensitivity analyses to provide critical insights into the basic reproduction number and equilibrium points of the TB transmission model. Through numerical simulations, we explore how various intervention strategies impact the spread of tuberculosis. The model yields an approximate reproduction number of 0.3, suggesting that under the current conditions represented in the model, TB would naturally decline in the population. Key findings emphasize the importance of maintaining a low transmission rate and improving the recovery rate to expedite the elimination of tuberculosis. The model demonstrates the complex interplay between susceptible, infected, latent, and recovered populations over time, highlighting the persistent nature of TB due to factors such as latent activation and loss of immunity in recovered individuals. This project provides a robust foundation for public health strategies aimed at controlling and ultimately eliminating tuberculosis. Our results underscore the need for targeted interventions focusing on reducing transmission, managing latent infections, and enhancing treatment efficacy. These insights can inform policy decisions and resource allocation in TB control programs, contributing to the global effort to combat this persistent disease.

References

[1]Bagcchi S. WHO’s global tuberculosis report 2022. The Lancet Microbe. 2023; 4(1): e20.

[2]Ojo MM, Peter OJ, Goufo EFD, et al. Mathematical model for control of tuberculosis epidemiology. Journal of Applied Mathematics and Computing. 2023; 69(1): 69–87.

[3]Andam EA. Analysis of transmission dynamics of tuberculosis (TB) using differential equations: A case study of Amansie West District, Ghan [PhD thesis]. Kwame Nkrumah University of Science and Technology; 2013.

[4]Frith J. History of tuberculosis. Part 1-phthisis, consumption and the white plague. Journal of Military and Veterans Health. 2014; 22(2): 29–35.

[5]Otoo D, Osman S, Poku SA, Donkoh EK. Dynamics of tuberculosis (TB) with drug resistance to first-line treatment and leaky vaccination: A deterministic modelling perspective. Computational and Mathematical Methods in Medicine. 2021.

[6]Bhunu CO, Garira W. A two strain tuberculosis transmission model with therapy and quarantine. Mathematical Modelling and Analysis. 2009; 14(3): 291–3129.

[7]Kabunga SK, Goufo EFD, Tuong VH. Analysis and simulation of a mathematical model of tuberculosis transmission in democratic republic of the congo. Advances in Difference Equations. 2020; 2020(1): 642.

[8]Otoo D, Osman S, Poku SA, Donkoh EK. Dynamics of tuberculosis (TB) with drug resistance to first-line treatment and leaky vaccination: A deterministic modelling perspective. Computational and Mathematical Methods in Medicine. 2021.

[9]Osman S, Makinde OD. A mathematical model for coinfection of listeriosis and anthrax diseases. International Journal of Mathematics and Mathematical Sciences. 2018; 2018: 1725671. doi: 10.1155/2018/1725671

[10]Otoo D, Abeasi IO, Osman S, Donkoh EK. Mathematical modeling and analysis of the dynamics of hepatitis B with optimal control. Communications in Mathematical Biology and Neuroscience. 2021; 2021: 43.

[11]Osman S, Tilahun GT, Alemu SD, Onsongo WM. Analysis of the dynamics of rabies in North Shewa, Ethiopia. Italian Journal of Pure and Applied Mathematics. 2022; 48: 877–902.

[12]Otoo D, Abeasi IO, Osman S, Donkoh EK. Stability analysis and modeling the dynamics of hepatitis B with vaccination compartment. Italian Journal of Pure and Applied Mathematics. 2022; 48: 903–927.

[13]Okosun KO, Mukamuri M, Makinde DO. Global stability analysis and control of leptospirosis. Open Mathematics. 2016; 14(1): 567–585.

[14]Kayange HL, Massawe ES, Makinde DO, Immanuel LS. Modelling and Optimal Control of Ebola Virus Disease in the Presence of Treatment and Quarantine of Infectives. International Journal of Systems Science and Applied Mathematics. 2020; 5: 43.

[15]Osman S, Togbenon HA, Otoo D. Modelling the dynamics of campylobacteriosis using nonstandard finite difference approach with optimal control. Computational and Mathematical Methods in Medicine. 2020; 2020: 8843299. doi: 10.1155/2020/8843299

[16]Onsongo WM, Mwini ED, Nyanaro BN, Osman S. The dynamics of psittacosis in human and poultry populations: A mathematical modelling perspective. Journal of Mathematical and Computational Science. 2021; 11(6): 8472–8505.

[17]Osman S, Makinde OD, Theuri DM. Mathematical modelling of listeriosis epidemics in animal and human population with optimal control. Tamkang Journal of Mathematics. 2020; 51(4): 261–287.

[18]Otoo D, Tilahun GT, Osman S, Wole GA. Modeling the dynamics of tuberculosis with drug resistance in North Shoa Zone, Oromiya Regional State, Ethiopia. Communications in Mathematical Biology and Neuroscience. 2021; 2021: 12.

[19]Eegunjobi AS, Makinde OD. Mathematical Analysis of Two Strains Covid-19 Disease Using SEIR Model. Journal of Mathematical & Fundamental Sciences. 2022; 54(2): 211–232.

[20]Yano TK, Makinde OD, Malonza DM. Modelling childhood disease outbreak in a community with inflow of susceptible and vaccinated new-born. Global Journal of Pure and Applied Mathematics. 2016; 12(5): 3895–3916.

[21]Osman S, Otoo D, Sebil C, Makinde OD. Bifurcation, sensitivity and optimal control analysis of modelling Anthrax-Listeriosis co-dynamics. Communications in Mathematical Biology and Neuroscience. 2020; 2020: 98.

[22]Osman S. Otoo D, Sebil C, Makinde OD. Bifurcation, sensitivity and optimal control analysis of modelling anthrax-listeriosis co-dynamics. Communications in Mathematical Biology and Neuroscience. 2020; 2020: 98.

[23]Osman S, Makinde OD, Theuri DM. Stability analysis and modelling of listeriosis dynamics in human and animal populations. Global Journal of Pure and Applied Mathematics. 2018; 14(1): 115–137.

[24]Hugo A, Makinde OD, Kumar S. An eco-epidemiological model for Newcastle disease in central zone Tanzania. International Journal of Computing Science and Mathematics. 2019; 10(3): 215–235.

[25]Massawe LN, Makinde OD. Parameter Estimation and Sensitivity Analysis of Bus Rapid Transit Frequency in Tanzania. International Journal of Transportation Engineering and Technology. 2023; 9(4): 79–85.

[26]Otoo D, Tilahun GT, Osman S, Wole GA. Modeling the dynamics of tuberculosis with drug resistance in North Shoa Zone, Oromiya Regional State, Ethiopia. Communications in Mathematical Biology and Neuroscience. 2021; 2021: 12.

[27]Osman S, Otoo D, Makinde OD, et al. Modeling anthrax with optimal control and cost effectiveness analysis. Applied Mathematics. 2020; 11(03): 255.

[28]Osman S, Musyoki EM, Ndung'u RM. A mathematical model for the transmission of measles with passive immunity. International Journal of Research in Mathematical and Statistical Sciences. 2019; 6: 1–8.

[29]Berhe HW, Makinde OD. Computational modelling and optimal control of measles epidemic in human population. Biosystems. 2020; 190: 104102.

[30]Berhe HW, Makinde OD, Theuri DM. Modelling the dynamics of direct and pathogens-induced dysentery diarrhoea epidemic with controls. Journal of biological dynamics. 2019; 13(1): 192–217.

[31]Muia DW, Osman S, Wainaina M. Modelling and analysis of trypanosomiasis transmission mechanism. Global Journal of Pure and Applied Mathematics. 2018; 14(10): 1311–1331.

[32]Osman S, Otoo D, Sebi C. Analysis of listeriosis transmission dynamics with optimal control. Applied Mathematics. 2020; 11(7): 712–737.

[33]Osman S, Makinde OD, Theuri DM. Stability analysis and modelling of listeriosis dynamics in human and animal populations. Global Journal of Pure and Applied Mathematics. 2018; 14(1): 115–137.

[34]Keno TD, Dano LB, Makinde OD. Modeling and optimal control analysis for malaria transmission with role of climate variability. Computational and Mathematical Methods. 2022; 2022(1): 9667396.

[35]Osman S, Makinde OD, Theuri DM. Mathematical modelling of transmission dynamics of anthrax in human and animal population. Mathematical Theory and Modelling. 2018.

[36]Eustace KA, Osman S, Wainaina M. Mathematical modelling and analysis of the dynamics of cholera. Global Journal of Pure and Applied Mathematics. 2018; 14(9): 1259–1275.

[37]Kanyaa JK, Osman S, Wainaina M. Mathematical modelling of substance abuse by commercial drivers. Global journal of pure and applied mathematics. 2018; 14(9): 1149–1165.

[38]Karunditu JW, Kimathi G, Osman S. Mathematical modeling of typhoid fever disease incorporating unprotected humans in the spread dynamics. Journal of Advances in Mathematics and Computer Science. 2019; 32(3): 1–11.

[39]Tessema H, Haruna I, Osman S, Kassa E. A mathematical model analysis of marriage divorce. Communications in Mathematical Biology and Neuroscience. 2022.

[40]Eyaran WE, Osman S, Wainaina M. Modelling and analysis of seir with delay differential equation. Global Journal of Pure and Applied Mathematics. 2019; 15(4): 365–382.

[41]Onsongo WM, Mwini ED, Nyanaro BN, Osman S. The dynamics of psittacosis in human and poultry populations: A mathematical modelling perspective. Journal of Mathematical and Computational Science. 2021; 11(6): 8472–8505.

[42]Karanja TW, Osman S, Wainaina M. Analysis and modelling of ringworm infections in an environment. Global Journal of Pure and Applied Mathematics. 2019; 15(5): 649–665.

[43]World Health Organization. Global tuberculosis report 2022. World Health Organization; 2022.

[44]Central Intelligence Agency. The world factbook. Available online: https://www.cia.gov/the-world-factbook/field/death-rate/ (accessed on 10 October 2024).

[45]Okosun KO, Mukamuri M, Makinde OD. Co-dynamics of trypanosomiasis and cryptosporidiosis. Applied Mathematics & Information Science. 2016; 10(6): 2137–2161.

[46]Otoo D, Edusei H, Gyan A, et al. Optimal prevention of HIV-AIDS with emphasis on unprotected and unnatural canal activities: A deterministic modelling perspective. Communications in Mathematical Biology and Neuroscience. 2023.

[47]Onsongo WM, Mwini ED, Nyanaro BN, Osman S. Stability analysis and modelling the dynamics of psittacosis in human and poultry populations. Communications in Mathematical Biology and Neuroscience. 2022.

[48]Diallo B, Okelo JA, Osman S, et al. A study of fractional bovine tuberculosis model with vaccination on human population. Communications in Mathematical Biology and Neuroscience. 2023.

[49]Konlan M, Abassawah Danquah B, Okyere E, et al. Global stability analysis and modelling onchocerciasis transmission dynamics with control measures. Infection Ecology & Epidemiology. 2024; 14(1): 2347941.

Published

2024-11-18

How to Cite

Ibrahim, A., Janda, M. A., Kahianyu, S. N., Gueye, A., Nkandu, P. C., & Ayerkain, E. T. (2024). Analysis and modelling of the transmission dynamics of tuberculosis in the presence of latent and active populations. Journal of AppliedMath, 2(6), 1870. https://doi.org/10.59400/jam1870

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