1
|
Türkün C, Gölgeli M, Atay FM. A mathematical interpretation for outbreaks of bacterial meningitis under the effect of time-dependent transmission parameters. NONLINEAR DYNAMICS 2023; 111:1-18. [PMID: 37361004 PMCID: PMC10235855 DOI: 10.1007/s11071-023-08577-6] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 02/21/2023] [Accepted: 05/08/2023] [Indexed: 06/28/2023]
Abstract
We consider a SIR-type compartmental model divided into two age classes to explain the seasonal exacerbations of bacterial meningitis, especially among children outside of the meningitis belt. We describe the seasonal forcing through time-dependent transmission parameters that may represent the outbreak of the meningitis cases after the annual pilgrimage period (Hajj) or uncontrolled inflows of irregular immigrants. We present and analyse a mathematical model with time-dependent transmission. We consider not only periodic functions in the analysis but also general non-periodic transmission processes. We show that the long-time average values of transmission functions can be used as a stability marker of the equilibrium. Furthermore, we interpret the basic reproduction number in case of time-dependent transmission functions. Numerical simulations support and help visualize the theoretical results.
Collapse
Affiliation(s)
- Can Türkün
- Department of Mathematics, TOBB University of Economics and Technology, Ankara, Turkey
- Present Address: Department of Industrial Engineering, Altınbaş University, Istanbul, Turkey
| | - Meltem Gölgeli
- Department of Mathematics, TOBB University of Economics and Technology, Ankara, Turkey
| | | |
Collapse
|
2
|
Musa SS, Yusuf A, Bakare EA, Abdullahi ZU, Adamu L, Mustapha UT, He D. Unravelling the dynamics of Lassa fever transmission with differential infectivity: Modeling analysis and control strategies. MATHEMATICAL BIOSCIENCES AND ENGINEERING : MBE 2022; 19:13114-13136. [PMID: 36654038 DOI: 10.3934/mbe.2022613] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/17/2023]
Abstract
Epidemic models have been broadly used to comprehend the dynamic behaviour of emerging and re-emerging infectious diseases, predict future trends, and assess intervention strategies. The symptomatic and asymptomatic features and environmental factors for Lassa fever (LF) transmission illustrate the need for sophisticated epidemic models to capture more vital dynamics and forecast trends of LF outbreaks within countries or sub-regions on various geographic scales. This study proposes a dynamic model to examine the transmission of LF infection, a deadly disease transmitted mainly by rodents through environment. We extend prior LF models by including an infectious stage to mild and severe as well as incorporating environmental contributions from infected humans and rodents. For model calibration and prediction, we show that the model fits well with the LF scenario in Nigeria and yields remarkable prediction results. Rigorous mathematical computation divulges that the model comprises two equilibria. That is disease-free equilibrium, which is locally-asymptotically stable (LAS) when the basic reproduction number, $ {\mathcal{R}}_{0} $, is $ < 1 $; and endemic equilibrium, which is globally-asymptotically stable (GAS) when $ {\mathcal{R}}_{0} $ is $ > 1 $. We use time-dependent control strategy by employing Pontryagin's Maximum Principle to derive conditions for optimal LF control. Furthermore, a partial rank correlation coefficient is adopted for the sensitivity analysis to obtain the model's top rank parameters requiring precise attention for efficacious LF prevention and control.
Collapse
Affiliation(s)
- Salihu S Musa
- Department of Applied Mathematics, Hong Kong Polytechnic University, Hong Kong, China
- Department of Mathematics, Kano University of Science and Technology, Wudil, Kano, Nigeria
| | - Abdullahi Yusuf
- Department of Computer Engineering, Biruni University, Istanbul, Turkey
| | - Emmanuel A Bakare
- Department of Mathematics, Federal University Oye Ekiti, Ekiti, Nigeria
- Biomathematics and Applied Mathematical Modelling Research Group, Federal University Oye Ekiti, Ekiti, Nigeria
| | - Zainab U Abdullahi
- Department of Biological Sciences, Federal University Dutsin-Ma, Katsina, Nigeria
| | - Lukman Adamu
- Department of Mathematical Sciences, Faculty of Science, University of Maiduguri, Nigeria
| | - Umar T Mustapha
- Department of Mathematics, Science Faculty, Federal University Dutse, Jigawa, Nigeria
| | - Daihai He
- Department of Applied Mathematics, Hong Kong Polytechnic University, Hong Kong, China
| |
Collapse
|
3
|
Alqahtani RT, Musa SS, Yusuf A. Unravelling the dynamics of the COVID-19 pandemic with the effect of vaccination, vertical transmission and hospitalization. RESULTS IN PHYSICS 2022; 39:105715. [PMID: 35720511 PMCID: PMC9192123 DOI: 10.1016/j.rinp.2022.105715] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/26/2022] [Revised: 06/02/2022] [Accepted: 06/07/2022] [Indexed: 05/12/2023]
Abstract
The coronavirus disease 2019 (COVID-19) is caused by a newly emerged virus known as severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2), transmitted through air droplets from an infected person. However, other transmission routes are reported, such as vertical transmission. Here, we propose an epidemic model that considers the combined effect of vertical transmission, vaccination and hospitalization to investigate the dynamics of the virus's dissemination. Rigorous mathematical analysis of the model reveals that two equilibria exist: the disease-free equilibrium, which is locally asymptotically stable when the basic reproduction number ( R 0 ) is less than 1 (unstable otherwise), and an endemic equilibrium, which is globally asymptotically stable when R 0 > 1 under certain conditions, implying the plausibility of the disease to spread and cause large outbreaks in a community. Moreover, we fit the model using the Saudi Arabia cases scenario, which designates the incidence cases from the in-depth surveillance data as well as displays the epidemic trends in Saudi Arabia. Through Caputo fractional-order, simulation results are provided to show dynamics behaviour on the model parameters. Together with the non-integer order variant, the proposed model is considered to explain various dynamics features of the disease. Further numerical simulations are carried out using an efficient numerical technique to offer additional insight into the model's dynamics and investigate the combined effect of vaccination, vertical transmission, and hospitalization. In addition, a sensitivity analysis is conducted on the model parameters against the R 0 and infection attack rate to pinpoint the most crucial parameters that should be emphasized in controlling the pandemic effectively. Finally, the findings suggest that adequate vaccination coupled with basic non-pharmaceutical interventions are crucial in mitigating disease incidences and deaths.
Collapse
Affiliation(s)
- Rubayyi T Alqahtani
- Department of Mathematics and Statistics, College of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh, Saudi Arabia
| | - Salihu S Musa
- Department of Applied Mathematics, Hong Kong Polytechnic University, Hong Kong, China
- Department of Mathematics, Near East University TRNC, Mersin 10, Nicosia 99138, Turkey
| | - Abdullahi Yusuf
- Department of Computer Engineering, Biruni University, Istanbul, Turkey
- Department of Mathematics, Federal University Dutse, Jigawa, Nigeria
| |
Collapse
|
4
|
Musa SS, Baba IA, Yusuf A, Sulaiman TA, Aliyu AI, Zhao S, He D. Transmission dynamics of SARS-CoV-2: A modeling analysis with high-and-moderate risk populations. RESULTS IN PHYSICS 2021; 26:104290. [PMID: 34026471 PMCID: PMC8131571 DOI: 10.1016/j.rinp.2021.104290] [Citation(s) in RCA: 13] [Impact Index Per Article: 4.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/12/2021] [Revised: 04/30/2021] [Accepted: 05/01/2021] [Indexed: 05/03/2023]
Abstract
Nigeria is second to South Africa with the highest reported cases of COVID-19 in sub-Saharan Africa. In this paper, we employ an SEIR-based compartmental model to study and analyze the transmission dynamics of SARS-CoV-2 outbreaks in Nigeria. The model incorporates different group of populations (that is, high- and- moderate risk populations) and is use to investigate the influence on each population on the overall transmission dynamics.The model, which is fitted well to the data, is qualitatively analyzed to evaluate the impacts of different schemes for controlstrategies. Mathematical analysis reveals that the model has two equilibria; i.e., disease-free equilibrium (DFE) which is local asymptotic stability (LAS) if the basic reproduction number (R 0 ) is less than 1; and unstable forR 0 > 1 , and an endemic equilibrium (EE) which is globally asymptotic stability (LAS) wheneverR 0 > 1 . Furthermore, we find that the model undergoes a phenomenon of backward bifurcation (BB, a coexistence of stable DFE and stable EE even if theR 0 < 1 ). We employ Partial Rank Correlation coefficients (PRCCs) for sensitivity analyses to evaluate the model's parameters. Our results highlight that proper surveillance, especially movement of individuals from high risk to moderate risk population, testing, as well as imposition of other NPIs measures are vital strategies for mitigating the COVID-19 epidemic in Nigeria. Besides, in the absence of an exact solution for the proposed model, we solve the model with the well-known ODE45 numerical solver and the effective numerical schemes such as Euler (EM), Runge-Kutta of order 2 (RK-2), and Runge-Kutta of order 4 (RK-4) in order to establish approximate solutions and to show the physical features of the model. It has been shown that these numerical schemes are very effective and efficient to establish superb approximate solutions for differential equations.
Collapse
Affiliation(s)
- Salihu S Musa
- Department of Applied Mathematics, Hong Kong Polytechnic University, Hong Kong
- Department of Mathematics, Kano University of Science and Technology, Wudil, Nigeria
| | - Isa A Baba
- Department of Mathematics, Bayero University Kano, Nigeria
| | - Abdullahi Yusuf
- Department of Computer Engineering, Biruni University, Istanbul, Turkey
- Department of Mathematics, Science Faculty, Federal University Dutse, Jigawa, Nigeria
| | - Tukur A Sulaiman
- Department of Computer Engineering, Biruni University, Istanbul, Turkey
- Department of Mathematics, Science Faculty, Federal University Dutse, Jigawa, Nigeria
| | - Aliyu I Aliyu
- Department of Mathematics, Science Faculty, Federal University Dutse, Jigawa, Nigeria
| | - Shi Zhao
- JC School of Public Health and Primary Care, Chinese University of Hong Kong, Hong Kong
- CUHK Shenzhen Research Institute, Shenzhen, China
| | - Daihai He
- Department of Applied Mathematics, Hong Kong Polytechnic University, Hong Kong
| |
Collapse
|
5
|
Modeling the 2014-2015 Ebola Virus Disease Outbreaks in Sierra Leone, Guinea, and Liberia with Effect of High- and Low-risk Susceptible Individuals. Bull Math Biol 2020; 82:102. [PMID: 32734342 DOI: 10.1007/s11538-020-00779-y] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/07/2019] [Accepted: 07/14/2020] [Indexed: 10/23/2022]
Abstract
Ebola virus disease (EVD) is a rare but fatal disease of humans and other primates caused by Ebola viruses. Study shows that the 2014-2015 EVD outbreak causes more than 10,000 deaths. In this paper, we propose and analyze a deterministic model to study the transmission dynamics of EVD in Sierra Leone, Guinea, and Liberia. Our analyses show that the model has two equilibria: (1) the disease-free equilibrium (DFE) which is locally asymptotically stable when the basic reproduction number ([Formula: see text]) is less than unity and unstable if it is greater than one, and (2) an endemic equilibrium (EE) which is globally asymptotically stable when [Formula: see text] is greater than unity. Furthermore, the backward bifurcation occurs, a coexistence between a stale DFE and a stable EE even if the [Formula: see text] is less than unity, which makes the disease control more strenuous and would depend on the initial size of subpopulation. By fitting to reported Ebola cases from Sierra Leone, Guinea, and Liberia in 2014-2015, our model has captured the epidemic patterns in all three countries and shed light on future Ebola control and prevention strategies.
Collapse
|