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Khan T, Rihan FA, Al-Mdallal QM. An epidemiological model for analysing pandemic trends of novel coronavirus transmission with optimal control. JOURNAL OF BIOLOGICAL DYNAMICS 2024; 18:2299001. [PMID: 38156669 DOI: 10.1080/17513758.2023.2299001] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/09/2023] [Accepted: 12/15/2023] [Indexed: 01/03/2024]
Abstract
Symptomatic and asymptomatic individuals play a significant role in the transmission dynamics of novel Coronaviruses. By considering the dynamical behaviour of symptomatic and asymptomatic individuals, this study examines the temporal dynamics and optimal control of Coronavirus disease propagation using an epidemiological model. Biologically and mathematically, the well-posed epidemic problem is examined, as well as the threshold quantity with parameter sensitivity. Model parameters are quantified and their relative impact on the disease is evaluated. Additionally, the steady states are investigated to determine the model's stability and bifurcation. Using the dynamics and parameters sensitivity, we then introduce optimal control strategies for the elimination of the disease. Using real disease data, numerical simulations and model validation are performed to support theoretical findings and show the effects of control strategies.
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Affiliation(s)
- Tahir Khan
- Department of Mathematical Sciences, College of Science, UAE University, Al-Ain, United Arab Emirates
| | - Fathalla A Rihan
- Department of Mathematical Sciences, College of Science, UAE University, Al-Ain, United Arab Emirates
| | - Qasem M Al-Mdallal
- Department of Mathematical Sciences, College of Science, UAE University, Al-Ain, United Arab Emirates
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2
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Farhan M, Shah Z, Ling Z, Shah K, Abdeljawad T, Islam S, Garalleh HAL. Global dynamics and computational modeling for analyzing and controlling Hepatitis B: A novel epidemic approach. PLoS One 2024; 19:e0304375. [PMID: 38935766 PMCID: PMC11210885 DOI: 10.1371/journal.pone.0304375] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/23/2024] [Accepted: 05/12/2024] [Indexed: 06/29/2024] Open
Abstract
Hepatitis B virus (HBV) infection is a global public health issue. We offer a comprehensive analysis of the dynamics of HBV, which can be successfully controlled with vaccine and treatment. Hepatitis B virus (HBV) causes a significantly more severe and protracted disease compared to hepatitis A. While it initially presents as an acute disease, in approximately 5 to 10% of cases, it can develop into a chronic disease that causes permanent damage to the liver. The hepatitis B virus can remain active outside the body for at least seven days. If the virus penetrates an individual's body without immunization, it may still result in infection. Upon exposure to HBV, the symptoms often last for a duration ranging from 10 days to 6 months. In this study, we developed a new model for Hepatitis B Virus (HBV) that includes asymptomatic carriers, vaccination, and treatment classes to gain a comprehensive knowledge of HBV dynamics. The basic reproduction number [Formula: see text] is calculated to identify future recurrence. The local and global stabilities of the proposed model are evaluated for values of [Formula: see text] that are both below and above 1. The Lyapunov function is employed to ensure the global stability of the HBV model. Further, the existence and uniqueness of the proposed model are demonstrated. To look at the solution of the proposed model graphically, we used a useful numerical strategy, such as the non-standard finite difference method, to obtain more thorough numerical findings for the parameters that have a significant impact on disease elimination. In addition, the study of treatment class in the population, we may assess the effectiveness of alternative medicines to treat infected populations can be determined. Numerical simulations and graphical representations are employed to illustrate the implications of our theoretical conclusions.
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Affiliation(s)
- Muhammad Farhan
- School of Mathematical Science, Yangzhou University, Yangzhou, China
- Department of Mathematics, Abdul Wali Khan University Mardan, Mardan, Khyber Pakhtunkhwa, Pakistan
| | - Zahir Shah
- Department of Mathematical Sciences, University of Lakki Marwat, Lakki Marwat, KPK, Pakistan
| | - Zhi Ling
- School of Mathematical Science, Yangzhou University, Yangzhou, China
| | - Kamal Shah
- Department of Mathematics and Sciences, Prince Sultan University, Riyadh, Saudi Arabia
- Department of Computer Science and Mathematics Lebanese American University, Byblos, Lebanon
| | - Thabet Abdeljawad
- Department of Mathematics and Sciences, Prince Sultan University, Riyadh, Saudi Arabia
| | - Saeed Islam
- Department of Mathematics, Abdul Wali Khan University Mardan, Mardan, Khyber Pakhtunkhwa, Pakistan
| | - Hakim A. L. Garalleh
- Department of Mathematical Science, College of Engineering, University of Business and Technology, Jeddah, Saudi Arabia
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3
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Khan T, Rihan FA, Ibrahim M, Li S, Alamri AM, AlQahtani SA. Modeling different infectious phases of hepatitis B with generalized saturated incidence: An analysis and control. MATHEMATICAL BIOSCIENCES AND ENGINEERING : MBE 2024; 21:5207-5226. [PMID: 38872533 DOI: 10.3934/mbe.2024230] [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: 06/15/2024]
Abstract
Hepatitis B is one of the global health issues caused by the hepatitis B virus (HBV), producing 1.1 million deaths yearly. The acute and chronic phases of HBV are significant because worldwide, approximately 250 million people are infected by chronic hepatitis B. The chronic stage is a long-term, persistent infection that can cause liver damage and increase the risk of liver cancer. In the case of multiple phases of infection, a generalized saturated incidence rate model is more reasonable than a simply saturated incidence because it captures the complex dynamics of the different infection phases. In contrast, a simple saturated incidence rate model assumes a fixed shape for the incidence rate curve, which may not accurately reflect the dynamics of multiple infection phases. Considering HBV and its various phases, we constructed a model to present the dynamics and control strategies using the generalized saturated incidence. First, we proved that the model is well-posed. We then found the reproduction quantity and model equilibria to discuss the time dynamics of the model and investigate the conditions for stabilities. We also examined a control mechanism by introducing various controls to the model with the aim to increase the population of those recovered and minimize the infected people. We performed numerical experiments to check the biological significance and control implementation.
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Affiliation(s)
- Tahir Khan
- Department of Mathematical Sciences, UAE University, P.O.Box 15551, Al-Ain, United Arab Emirates
| | - Fathalla A Rihan
- Department of Mathematical Sciences, UAE University, P.O.Box 15551, Al-Ain, United Arab Emirates
| | - Muhammad Ibrahim
- Department of Mathematics, University of Malakand Chakdara, Dir (L), Pakhtunkhwa, Pakistan
| | - Shuo Li
- School of Mathematics and Data Sciences, Changji University, Changji 831100, Xinjiang, China
| | - Atif M Alamri
- Software Engineering Department, College of Computer and Information Sciences, King Saud University, Riyadh, Saudi Arabia
| | - Salman A AlQahtani
- Computer Engineering Department, College of Computer and Information Sciences, King Saud University, Riyadh, Saudi Arabia
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Teklu SW. Impacts of optimal control strategies on the HBV and COVID-19 co-epidemic spreading dynamics. Sci Rep 2024; 14:5328. [PMID: 38438440 PMCID: PMC10912759 DOI: 10.1038/s41598-024-55111-8] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/30/2023] [Accepted: 02/20/2024] [Indexed: 03/06/2024] Open
Abstract
Different cross-sectional and clinical research studies investigated that chronic HBV infected individuals' co-epidemic with COVID-19 infection will have more complicated liver infection than HBV infected individuals in the absence of COVID-19 infection. The main objective of this study is to investigate the optimal impacts of four time dependent control strategies on the HBV and COVID-19 co-epidemic transmission using compartmental modeling approach. The qualitative analyses of the model investigated the model solutions non-negativity and boundedness, calculated all the models effective reproduction numbers by applying the next generation operator approach, computed all the models disease-free equilibrium point (s) and endemic equilibrium point (s) and proved their local stability, shown the phenomenon of backward bifurcation by applying the Center Manifold criteria. By applied the Pontryagin's Maximum principle, the study re-formulated and analyzed the co-epidemic model optimal control problem by incorporating four time dependent controlling variables. The study also carried out numerical simulations to verify the model qualitative results and to investigate the optimal impacts of the proposed optimal control strategies. The main finding of the study reveals that implementation of protections, COVID-19 vaccine, and treatment strategies simultaneously is the most effective optimal control strategy to tackle the HBV and COVID-19 co-epidemic spreading in the community.
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Teklu SW. Analysis of HBV and COVID-19 Coinfection Model with Intervention Strategies. COMPUTATIONAL AND MATHEMATICAL METHODS IN MEDICINE 2023; 2023:6908757. [PMID: 37811291 PMCID: PMC10558273 DOI: 10.1155/2023/6908757] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 01/12/2023] [Revised: 02/21/2023] [Accepted: 03/20/2023] [Indexed: 10/10/2023]
Abstract
Coinfection of hepatitis B virus (HBV) and COVID-19 is a common public health problem throughout some nations in the world. In this study, a mathematical model for hepatitis B virus (HBV) and COVID-19 coinfection is constructed to investigate the effect of protection and treatment mechanisms on its spread in the community. Necessary conditions of the proposed model nonnegativity and boundedness of solutions are analyzed. We calculated the model reproduction numbers and carried out the local stabilities of disease-free equilibrium points whenever the associated reproduction number is less than unity. Using the well-known Castillo-Chavez criteria, the disease-free equilibrium points are shown to be globally asymptotically stable whenever the associated reproduction number is less than unity. Sensitivity analysis proved that the most influential parameters are transmission rates. Moreover, we carried out numerical simulation and shown results: some parameters have high spreading effect on the disease transmission, single infections have great impact on the coinfection transmission, and using protections and treatments simultaneously is the most effective strategy to minimize and also to eradicate the HBV and COVID-19 coinfection spreading in the community. It is concluded that to control the transmission of both diseases in a population, efforts must be geared towards preventing incident infection with either or both diseases.
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Anley DT, Dagnaw M, Belay DG, Tefera D, Tessema ZT, Molla A, Jemal SS, Zewde EA, Azanaw MM, Aragie G, Melsew YA. Modelling of Hepatitis B Virus vertical transmission dynamics in Ethiopia: a compartmental modelling approach. BMC Infect Dis 2023; 23:366. [PMID: 37259048 DOI: 10.1186/s12879-023-08343-4] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/31/2023] [Accepted: 05/22/2023] [Indexed: 06/02/2023] Open
Abstract
BACKGROUND Hepatitis B (HB) is a virus which causes a potentially fatal liver infection. It is a DNA virus belonging to the Hepadnaviridae virus family. Africa, after Asia, has the second highest number of chronic HBV carriers and is considered a high-endemic region. Ethiopia is classified as a country with a high prevalence of viral hepatitis and with nations that lack a systematic strategy for viral hepatitis surveillance. METHODS S-I-C-R deterministic model was developed and the numerical simulations were done in "R" statistical and programming software. Fixed population assumption was considered so as to develop a simple model which could predict the HBV vertical transmission for the next 5 decades. RESULTS The model revealed that significant number of populations will be infected and become carrier till the end the next 49 years even though it has decreasing trend. It was predicted that 271,719 people will die of HBV complications if no intervention will be made on its vertical transmission. The sensitivity analysis result showed that the force of infection has the most important parameter in the vertical transmission dynamics of hepatitis B. Provision of hepatitis B immunoglobulin (HBVIG) and vaccines at the time of delivery could decrease the force of infection by more than half and 51,892 lives will be saved if the intervention is offered for 50% of deliveries in Ethiopia. CONCLUSION Despite the fact that the incidence of HBV vertical transmission is substantial, it is expected to decline during the next five decades. However, the situation necessitates immediate attention, since it results in thousands of deaths if no action is taken. Offering HBVIG and vaccinations to the 50% of infants can save many lives and reduces the force of infection by more than a half.
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Affiliation(s)
- Denekew Tenaw Anley
- Department of Public Health, College of Health Sciences, Debre Tabor University, Debre Tabor, Ethiopia.
| | - Mequanente Dagnaw
- Department of Epidemiology and Biostatistics, Institute of Public Health, College of Medicine and Health Sciences, University of Gondar, Gondar, Ethiopia
| | - Daniel Gashaneh Belay
- Department of Epidemiology and Biostatistics, Institute of Public Health, College of Medicine and Health Sciences, University of Gondar, Gondar, Ethiopia
| | - Dawit Tefera
- Department of Epidemiology and Biostatistics, Institute of Public Health, College of Medicine and Health Sciences, University of Gondar, Gondar, Ethiopia
| | - Zemenu Tadesse Tessema
- Department of Epidemiology and Biostatistics, Institute of Public Health, College of Medicine and Health Sciences, University of Gondar, Gondar, Ethiopia
| | - Ayenew Molla
- Department of Epidemiology and Biostatistics, Institute of Public Health, College of Medicine and Health Sciences, University of Gondar, Gondar, Ethiopia
| | - Sebwedin Surur Jemal
- Department of Statistics, College of Natural and Computational Science, Mizan Tepi University, Mizan Tepi, Ethiopia
| | - Edget Abebe Zewde
- Department of Biomedical Sciences, College of Health Sciences, Debre Tabor University, Debre Tabor, Ethiopia
| | - Melkalem Mamuye Azanaw
- Department of Public Health, College of Health Sciences, Debre Tabor University, Debre Tabor, Ethiopia
| | - Getachew Aragie
- Department of Padiatrics and Child Health Nursing, Debre Tabor University, Debre Tabor, Ethiopia
| | - Yayehirad Alemu Melsew
- Department of Epidemiology and Preventive Medicine, School of Public Health and Preventive Medicine, Monash University, 553 St Kilda Road, Melbourne, VIC, 3004, Australia
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Xue T, Zhang L, Fan X. Dynamic modeling and analysis of Hepatitis B epidemic with general incidence. MATHEMATICAL BIOSCIENCES AND ENGINEERING : MBE 2023; 20:10883-10908. [PMID: 37322965 DOI: 10.3934/mbe.2023483] [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
New stochastic and deterministic Hepatitis B epidemic models with general incidence are established to study the dynamics of Hepatitis B virus (HBV) epidemic transmission. Optimal control strategies are developed to control the spread of HBV in the population. In this regard, we first calculate the basic reproduction number and the equilibrium points of the deterministic Hepatitis B model. And then the local asymptotic stability at the equilibrium point is studied. Secondly, the basic reproduction number of the stochastic Hepatitis B model is calculated. Appropriate Lyapunov functions are constructed, and the unique global positive solution of the stochastic model is verified by Itô formula. By applying a series of stochastic inequalities and strong number theorems, the moment exponential stability, the extinction and persistence of HBV at the equilibrium point are obtained. Finally, using the optimal control theory, the optimal control strategy to eliminate the spread of HBV is developed. To reduce Hepatitis B infection rates and to promote vaccination rates, three control variables are used, for instance, isolation of patients, treatment of patients, and vaccine inoculation. For the purpose of verifying the rationality of our main theoretical conclusions, the Runge-Kutta method is applied to numerical simulation.
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Affiliation(s)
- Tingting Xue
- School of Mathematics and Physics, Xinjiang Institute of Engineering, Urumqi, Xinjiang, China
| | - Long Zhang
- School of Mathematics and Physics, Xinjiang Institute of Engineering, Urumqi, Xinjiang, China
| | - Xiaolin Fan
- School of Mathematics and Physics, Xinjiang Institute of Engineering, Urumqi, Xinjiang, China
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Mathematical Model of Hepatitis B Disease with Optimal Control and Cost-Effectiveness Analysis. COMPUTATIONAL AND MATHEMATICAL METHODS IN MEDICINE 2023. [DOI: 10.1155/2023/5215494] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 02/11/2023]
Abstract
In this paper, a mathematical model of hepatitis B disease with a two-dose vaccine series has been formulated and analyzed. We demonstrated that the model’s disease-free equilibrium is globally asymptotically stable when the basic reproduction number
is less than one, whereas the model’s endemic equilibrium is locally asymptotically stable when
is greater than one. Sensitivity analysis is performed, and based on its results, the model is extended to an optimal control problem by incorporating two control interventions, namely, prevention and enhanced newborn vaccination. Finally, simulation analyses of the model are conducted to illustrate the theoretical findings and effectiveness of each strategy, which indicates that the use of prevention efforts is the most cost-saving strategy.
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Agbomola JO, Loyinmi AC. Modelling the impact of some control strategies on the transmission dynamics of Ebola virus in human-bat population: An optimal control analysis. Heliyon 2022; 8:e12121. [PMID: 36561665 PMCID: PMC9763758 DOI: 10.1016/j.heliyon.2022.e12121] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/22/2022] [Revised: 10/09/2022] [Accepted: 11/26/2022] [Indexed: 12/12/2022] Open
Abstract
Qualitative research and comprehensive public awareness to nip the transmission of Ebola virus in the bud before it becomes a global threat is fast becoming imperative especially now that the Gambia Ebola virus is mutated. It is therefore necessary to consider and investigate a vector-host transmission model for possible control strategy of this deadly disease. Hence, in this study, we presented a novel and feasible human-bat (host-vector) S h E h I h R h i s R h n i - S b E b I b model which foretells the spread and severity of the Ebola virus from bats to humans to investigate the combined effects of three control strategies viz: (1) allowing specialized and designated agencies to bury deceased from Ebola infection without relatives touching or curdling the remains as usually practiced in most part of Africa as last respect for their departed love ones ( k 1 ), (2) systematic and deliberate depopulation of bats in the metropolis (through persecution with pesticide exposure, pre capturing, chemical timber treatment for roosts destruction) to discourage hunting them for food by virtue of their proximity ( k 2 ) and (3) immediate treatment of infected individuals in isolation ( k 3 ). We established, among others, the endemic equilibrium, disease-free equilibrium, global and local stability, non-negativity, and boundedness of the model to prove the epidemiological feasibility of the model. The reality of the presence of optimal control remarkably influences the dynamics of transmission of the virus and simulated results also confirm the great effect of the combination of the control strategies k 1 , k 2 and k 3 in flattening the curve of Ebola transmission (fig 1 - fig 8). Health workers and policy makers are better informed with fundamental precautions that could help eradicate Ebola from the populace.
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Affiliation(s)
| | - Adedapo Chris Loyinmi
- Department of Mathematics, Tai Solarin University of Education, Ijagun, Ogun State, Nigeria
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Chhetri B, Bhagat VM, Vamsi DKK, Ananth VS, Prakash B, Muthusamy S, Deshmukh P, Sanjeevi CB. Optimal Drug Regimen and Combined Drug Therapy and Its Efficacy in the Treatment of COVID-19: A Within-Host Modeling Study. Acta Biotheor 2022; 70:16. [PMID: 35588019 PMCID: PMC9118007 DOI: 10.1007/s10441-022-09440-8] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/03/2021] [Accepted: 05/11/2022] [Indexed: 11/29/2022]
Abstract
The COVID-19 pandemic has resulted in more than 524 million cases and 6 million deaths worldwide. Various drug interventions targeting multiple stages of COVID-19 pathogenesis can significantly reduce infection-related mortality. The current within-host mathematical modeling study addresses the optimal drug regimen and efficacy of combination therapies in the treatment of COVID-19. The drugs/interventions considered include Arbidol, Remdesivir, Interferon (INF) and Lopinavir/Ritonavir. It is concluded that these drugs, when administered singly or in combination, reduce the number of infected cells and viral load. Four scenarios dealing with the administration of a single drug, two drugs, three drugs and all four are discussed. In all these scenarios, the optimal drug regimen is proposed based on two methods. In the first method, these medical interventions are modeled as control interventions and a corresponding objective function and optimal control problem are formulated. In this framework, the optimal drug regimen is derived. Later, using the comparative effectiveness method, the optimal drug regimen is derived based on the basic reproduction number and viral load. The average number of infected cells and viral load decreased the most when all four drugs were used together. On the other hand, the average number of susceptible cells decreased the most when Arbidol was administered alone. The basic reproduction number and viral load decreased the most when all four interventions were used together, confirming the previously obtained finding of the optimal control problem. The results of this study can help physicians make decisions about the treatment of the life-threatening COVID-19 infection.
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Affiliation(s)
- Bishal Chhetri
- Department of Mathematics and Computer Science, Sri Sathya Sai Institute of Higher Learning - SSSIHL, Anantapur, India
| | - Vijay M. Bhagat
- Central Leprosy Teaching and Research Institute - CLTRI, Chennai, India
| | - D. K. K. Vamsi
- Department of Mathematics and Computer Science, Sri Sathya Sai Institute of Higher Learning - SSSIHL, Anantapur, India
| | - V. S. Ananth
- Department of Mathematics and Computer Science, Sri Sathya Sai Institute of Higher Learning - SSSIHL, Anantapur, India
| | - Bhanu Prakash
- Department of Mathematics and Computer Science, Sri Sathya Sai Institute of Higher Learning - SSSIHL, Anantapur, India
| | - Swapna Muthusamy
- Central Leprosy Teaching and Research Institute - CLTRI, Chennai, India
| | - Pradeep Deshmukh
- Department of Community Medicine, All India Institute of Medical Sciences - AIIMS, Nagpur, India
| | - Carani B. Sanjeevi
- Sri Sathya Sai Institute of Higher Learning - SSSIHL, Anantapur, India
- Department of Medicine, Karolinska Institute, Stockholm, Sweden
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Jan MN, Zaman G, Ali N, Ahmad I, Shah Z. Optimal control application to the epidemiology of HBV and HCV co-infection. INT J BIOMATH 2021. [DOI: 10.1142/s1793524521501011] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/18/2022]
Abstract
It is very important to note that a mathematical model plays a key role in different infectious diseases. Here, we study the dynamical behaviors of both hepatitis B virus (HBV) and hepatitis C virus (HCV) with their co-infection. Actually, the purpose of this work is to show how the bi-therapy is effective and include an inhibitor for HCV infection with some treatments, which are frequently used against HBV. Local stability, global stability and its prevention from the community are studied. Mathematical models and optimality system of nonlinear DE are solved numerically by RK4. We use linearization, Lyapunov function and Pontryagin’s maximum principle for local stability, global stability and optimal control, respectively. Stability curves and basic reproductive number are plotted with and without control versus different values of parameters. This study shows that the infection will spread without control and can cover with treatment. The intensity of HBV/HCV co-infection is studied before and after optimal treatment. This represents a short drop after treatment. First, we formulate the model then find its equilibrium points for both. The models possess four distinct equilibria: HBV and HCV free, and endemic. For the proposed problem dynamics, we show the local as well as the global stability of the HBV and HCV. With the help of optimal control theory, we increase uninfected individuals and decrease the infected individuals. Three time-dependent variables are also used, namely, vaccination, treatment and isolation. Finally, optimal control is classified into optimality system, which we can solve with Runge–Kutta-order four method for different values of parameters. Finally, we will conclude the results for implementation to minimize the infected individuals.
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Affiliation(s)
- Muhammad Naeem Jan
- Department of Mathematics, University of Malakand, Chakdara, Dir(Lower), Khyber Pakhtunkhwa, Pakistan
| | - Gul Zaman
- Department of Mathematics, University of Malakand, Chakdara, Dir(Lower), Khyber Pakhtunkhwa, Pakistan
| | - Nigar Ali
- Department of Mathematics, University of Malakand, Chakdara, Dir(Lower), Khyber Pakhtunkhwa, Pakistan
| | - Imtiaz Ahmad
- Department of Mathematics, University of Malakand, Chakdara, Dir(Lower), Khyber Pakhtunkhwa, Pakistan
| | - Zahir Shah
- Department of Mathematical Sciences, University of Lakki Marwat, Lakki Marwat 28420, Khyber Pakhtunkhwa, Pakistan
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Optimal Voluntary Vaccination of Adults and Adolescents Can Help Eradicate Hepatitis B in China. GAMES 2021. [DOI: 10.3390/g12040082] [Citation(s) in RCA: 3] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 12/26/2022]
Abstract
Hepatitis B (HBV) is one of the most common infectious diseases, with a worldwide annual incidence of over 250 million people. About one-third of the cases are in China. While China made significant efforts to implement a nationwide HBV vaccination program for newborns, a significant number of susceptible adults and teens remain. In this paper, we analyze a game-theoretical model of HBV dynamics that incorporates government-provided vaccination at birth coupled with voluntary vaccinations of susceptible adults and teens. We show that the optimal voluntary vaccination brings the disease incidence to very low levels. This result is robust and, in particular, due to a high HBV treatment cost, essentially independent from the vaccine cost.
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Tilahun GT, Woldegerima WA, Mohammed N. A fractional order model for the transmission dynamics of hepatitis B virus with two-age structure in the presence of vaccination. ARAB JOURNAL OF BASIC AND APPLIED SCIENCES 2021. [DOI: 10.1080/25765299.2021.1896423] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/21/2022] Open
Affiliation(s)
| | - Woldegebriel Assefa Woldegerima
- Department of Mathematics and Applied Mathematics, University of Pretoria, Pretoria, South Africa
- Department of Mathematics, CNCS, Mekelle University, Mekelle, Ethiopia
| | - Nesredin Mohammed
- Department of Mathematics, CNCS, Haramaya University, Haramaya, Ethiopia
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14
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Din A, Li Y, Shah MA. The Complex Dynamics of Hepatitis B Infected Individuals with Optimal Control. JOURNAL OF SYSTEMS SCIENCE AND COMPLEXITY 2021; 34:1301-1323. [PMID: 33564220 PMCID: PMC7861011 DOI: 10.1007/s11424-021-0053-0] [Citation(s) in RCA: 10] [Impact Index Per Article: 3.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/19/2020] [Revised: 06/12/2020] [Indexed: 05/08/2023]
Abstract
This paper proposes various stages of the hepatitis B virus (HBV) besides its transmissibility and nonlinear incidence rate to develop an epidemic model. The authors plan the model, and then prove some basic results for the well-posedness in term of boundedness and positivity. Moreover, the authors find the threshold parameter R 0, called the basic/effective reproductive number and carry out local sensitive analysis. Furthermore, the authors examine stability and hence condition for stability in terms of R 0. By using sensitivity analysis, the authors formulate a control problem in order to eradicate HBV from the population and proved that the control problem actually exists. The complete characterization of the optimum system was achieved by using the 4th-order Runge-Kutta procedure.
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Affiliation(s)
- Anwarud Din
- Department of Mathematics, Sun Yat-sen University, Guangzhou, 510275 China
| | - Yongjin Li
- Department of Mathematics, Sun Yat-sen University, Guangzhou, 510275 China
| | - Murad Ali Shah
- School of Mathematical Sciences, Beijing Normal University, Beijing, 100875 China
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15
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Sahoo S, Jhunjhunwala S, Jolly MK. The Good, The Bad and The Ugly: A Mathematical Model Investigates the Differing Outcomes Among CoVID-19 Patients. J Indian Inst Sci 2020; 100:673-681. [PMID: 33041543 PMCID: PMC7533167 DOI: 10.1007/s41745-020-00205-1] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/22/2020] [Accepted: 09/14/2020] [Indexed: 12/14/2022]
Abstract
The disease caused by SARS-CoV-2—CoVID-19—is a global pandemic that has brought severe changes worldwide. Approximately 80% of the infected patients are largely asymptomatic or have mild symptoms such as fever or cough, while rest of the patients display varying degrees of severity of symptoms, with an average mortality rate of 3–4%. Severe symptoms such as pneumonia and acute respiratory distress syndrome may be caused by tissue damage, which is mostly due to aggravated and unresolved innate and adaptive immune response, often resulting from a cytokine storm.Cytokine storm: A sudden acute increase in circulating levels of different inflammation causing cytokines including IL-6, IL-1, etc. Here, we discuss how an intricate interplay among infected cells and cells of innate and adaptive immune system can lead to such diverse clinicopathological outcomes. Particularly, we discuss how the emergent nonlinear dynamics of interaction among the components of adaptive and immune system components and virally infected cells can drive different disease severity. Such minimalistic yet rigorous mathematical modeling approaches are helpful in explaining how various co-morbidity risk factors, such as age and obesity, can aggravate the severity of CoVID-19 in patients. Furthermore, such approaches can elucidate how a fine-tuned balance of infected cell killing and resolution of inflammation can lead to infection clearance, while disruptions can drive different severe phenotypes. These results can help further in a rational selection of drug combinations that can effectively balance viral clearance and minimize tissue damage.
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Affiliation(s)
- Sarthak Sahoo
- Centre for BioSystems Science and Engineering, Indian Institute of Science, Bangalore, 560012 India
| | - Siddharth Jhunjhunwala
- Centre for BioSystems Science and Engineering, Indian Institute of Science, Bangalore, 560012 India
| | - Mohit Kumar Jolly
- Centre for BioSystems Science and Engineering, Indian Institute of Science, Bangalore, 560012 India
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16
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Madubueze CE, Dachollom S, Onwubuya IO. Controlling the Spread of COVID-19: Optimal Control Analysis. COMPUTATIONAL AND MATHEMATICAL METHODS IN MEDICINE 2020; 2020:6862516. [PMID: 32963585 PMCID: PMC7499329 DOI: 10.1155/2020/6862516] [Citation(s) in RCA: 23] [Impact Index Per Article: 5.8] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 05/06/2020] [Revised: 08/25/2020] [Accepted: 09/05/2020] [Indexed: 12/31/2022]
Abstract
Coronavirus disease 2019 (COVID-19) is a disease caused by severe acute respiratory syndrome coronavirus 2 (SARS CoV-2). It was declared on March 11, 2020, by the World Health Organization as pandemic disease. The disease has neither approved medicine nor vaccine and has made governments and scholars search for drastic measures in combating the pandemic. Regrettably, the spread of the virus and mortality due to COVID-19 has continued to increase daily. Hence, it is imperative to control the spread of the disease particularly using nonpharmacological strategies such as quarantine, isolation, and public health education. This work studied the effect of these different control strategies as time-dependent interventions using mathematical modeling and optimal control approach to ascertain their contributions in the dynamic transmission of COVID-19. The model was proven to have an invariant region and was well-posed. The basic reproduction number and effective reproduction numbers were computed with and without interventions, respectively, and were used to carry out the sensitivity analysis that identified the critical parameters contributing to the spread of COVID-19. The optimal control analysis was carried out using the Pontryagin's maximum principle to figure out the optimal strategy necessary to curtail the disease. The findings of the optimal control analysis and numerical simulations revealed that time-dependent interventions reduced the number of exposed and infected individuals compared to time-independent interventions. These interventions were time-bound and best implemented within the first 100 days of the outbreak. Again, the combined implementation of only two of these interventions produced a good result in reducing infection in the population. While, the combined implementation of all three interventions performed better, even though zero infection was not achieved in the population. This implied that multiple interventions need to be deployed early in order to reduce the virus to the barest minimum.
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Affiliation(s)
- Chinwendu E. Madubueze
- Department of Mathematics/Statistics/Computer Science, University of Agriculture Makurdi, P.M.B. 2373, Markurdi, Nigeria
| | - Sambo Dachollom
- Department of Mathematics/Statistics, Akanu Ibiam Federal Polytechnic, Unwana, P.M.B. 1007 Afikpo, Ebonyi State, Nigeria
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17
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Kouenkam JPII, Mbang J, Emvudu Y. Global dynamics of a model of hepatitis B virus infection in a sub-Saharan African rural area. INT J BIOMATH 2020. [DOI: 10.1142/s1793524520500540] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/13/2022]
Abstract
We formulate and systematically study a deterministic compartmental model of Hepatitis B. This model has some important and novel features compared with the well-known basic model in the literature. Specifically, it takes into account the differential susceptibility that follows the vaccine formulation employing three-doses schedule. It points up the HbeAg status of carriers, their levels of viral replication, the fact that treatment being not curative is recommended only to a small proportion of chronic carriers, and finally the fact that only inactive carriers are able to recover from disease. The model has simple dynamical behavior which has a globally asymptotically stable disease-free equilibrium when the basic reproduction number [Formula: see text] and an endemic equilibrium when [Formula: see text]. By the use of Lyapunov functions, when it exists, we prove the global asymptotic stability of the endemic equilibrium under some conditions. Using data from Tokombere, a rural area in Cameroon, numerical simulations are performed. These numerical simulations first confirm analytical results, second they suggest that a policy based on treatment could not significantly impact the course of the infection. Third, they show as it is well known that vaccination is a very effective measure to control the infection. Furthermore, they show that neonatal vaccination influences more the course of infection than mass vaccination strategy. Nevertheless, they picture how much loss between consecutive doses of vaccine could be harmful. Finally, it is suggested that for a Sub-saharan African rural area, two-thirds of expected incidence of Hepatitis B virus infection and one third of expected prevalence of chronic carriers could be averted by 2030 if the birth dose vaccination becomes systematic and if mass vaccination rate increases to up 10%.
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Affiliation(s)
- Jean Pierre II Kouenkam
- Department of Mathematics, Faculty of Science, University of Yaounde I, P. O. Box 812 Yaounde, Cameroon
| | - Joseph Mbang
- Department of Mathematics, Faculty of Science, University of Yaounde I, P. O. Box 812 Yaounde, Cameroon
- UMI 209 IRD/UPMC UMMISCO, Bondy, Projet MASAIE INRIA Grand Est, France and Projet GRIMCAPE, LIRIMA, Cameroun
| | - Yves Emvudu
- Department of Mathematics, Faculty of Science, University of Yaounde I, P. O. Box 812 Yaounde, Cameroon
- UMI 209 IRD/UPMC UMMISCO, Bondy, Projet MASAIE INRIA Grand Est, France and Projet GRIMCAPE, LIRIMA, Cameroun
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18
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KHAN MUHAMMADALTAF, SHAH SYEDAZHARALI, ULLAH SAIF, OKOSUN KAZEEMOARE, FAROOQ MUHAMMAD. OPTIMAL CONTROL ANALYSIS OF THE EFFECT OF TREATMENT, ISOLATION AND VACCINATION ON HEPATITIS B VIRUS. J BIOL SYST 2020. [DOI: 10.1142/s0218339020400057] [Citation(s) in RCA: 12] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/18/2022]
Abstract
Hepatitis B infection is a serious health issue and a major cause of deaths worldwide. This infection can be overcome by adopting proper treatment and control strategies. In this paper, we develop and use a mathematical model to explore the effect of treatment on the dynamics of hepatitis B infection. First, we formulate and use a model without control variables to calculate the basic reproduction number and to investigate basic properties of the model such as the existence and stability of equilibria. In the absence of control measures, we prove that the disease free equilibrium is locally asymptotically stable when the basic reproduction number is less than unity. Also, using persistent theorem, it is shown that the infection is uniformly persistent, whenever the basic reproduction number is greater than unity. Using optimal control theory, we incorporate into the model three time-dependent control variables and investigate the conditions required to curtail the spread of the disease. Finally, to illustrate the effectiveness of each of the control strategies on disease control and eradication, we perform numerical simulations. Based on the numerical results, we found that the first two strategies (treatment and isolation strategy) and (vaccination and isolation strategy) are not very effective as a long term control or eradication strategy for HBV. Hence, we recommend that in order to effectively control the disease, all the control measures (isolation, vaccination and treatment) must be implemented at the same time.
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Affiliation(s)
- MUHAMMAD ALTAF KHAN
- Informetrics Research Group, Ton Duc Thang University, Ho Chi Minh City, Vietnam
- Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam
| | - SYED AZHAR ALI SHAH
- Department of Mathematics, University of Peshawar, Khyber Pakhtunkhwa, Pakistan
| | - SAIF ULLAH
- Department of Mathematics, University of Peshawar, Khyber Pakhtunkhwa, Pakistan
| | | | - MUHAMMAD FAROOQ
- Department of Mathematics, University of Peshawar, Khyber Pakhtunkhwa, Pakistan
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19
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Chouhan A, Maiwand S, Ngo M, Putalapattu V, Rychtář J, Taylor D. Game-Theoretical Model of Retroactive Hepatitis B Vaccination in China. Bull Math Biol 2020; 82:80. [PMID: 32542575 DOI: 10.1007/s11538-020-00748-5] [Citation(s) in RCA: 8] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/01/2020] [Accepted: 05/14/2020] [Indexed: 02/08/2023]
Abstract
Hepatitis B (HepB) is one of the most common infectious diseases affecting over two billion people worldwide. About one third of all HepB cases are in China. In recent years, China made significant efforts to implement a nationwide HepB vaccination program and reduced the number of unvaccinated infants from 30 to 10%. However, many individuals still remain unprotected, particularly those born before 2003. Consequently, a catch-up retroactive vaccination is an important and potentially cost-effective way to reduce HepB prevalence. In this paper, we analyze a game theoretical model of HepB dynamics that incorporates government-provided vaccination at birth coupled with voluntary retroactive vaccinations. Given the uncertainty about the long-term efficacy of the HepB vaccinations, we study several scenarios. When the waning rate is relatively high, we show that this retroactive vaccination should be a necessary component of any HepB eradication effort. When the vaccine offers long-lasting protection, the voluntary retroactive vaccination brings the disease incidence to sufficiently low levels. Also, we find that the optimal vaccination rates are almost independent of the vaccination coverage at birth. Moreover, it is in an individual's self-interest to vaccinate (and potentially re-vaccinate) at a rate just slightly above the vaccine waning rate.
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Affiliation(s)
- Ali Chouhan
- Department of Biology, Virginia Commonwealth University, Richmond, VA, 23284-2012, USA
| | - Sohail Maiwand
- Department of Mathematics and Applied Mathematics, Virginia Commonwealth University, Richmond, VA, 23284-2014, USA
| | - Matthew Ngo
- Department of Biology, Virginia Commonwealth University, Richmond, VA, 23284-2012, USA
| | - Vooha Putalapattu
- Department of Psychology, Virginia Commonwealth University, Richmond, VA, 23284-2018, USA
| | - Jan Rychtář
- Department of Mathematics and Applied Mathematics, Virginia Commonwealth University, Richmond, VA, 23284-2014, USA.
| | - Dewey Taylor
- Department of Mathematics and Applied Mathematics, Virginia Commonwealth University, Richmond, VA, 23284-2014, USA
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20
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Aniji M, Kavitha N, Balamuralitharan S. Approximate solutions for HBV infection with stability analysis using LHAM during antiviral therapy. BOUNDARY VALUE PROBLEMS 2020; 2020:80. [PMID: 33719357 PMCID: PMC7223922 DOI: 10.1186/s13661-020-01373-w] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 03/05/2020] [Accepted: 04/02/2020] [Indexed: 06/12/2023]
Abstract
Hepatitis B virus (HBV) is a life-threatening virus that causes very serious liver-related diseases from the family of Hepadnaviridae having very rare qualities resembling retroviruses. In this paper, we analyze the effect of antiviral therapy through mathematical modeling by using Liao's homotopy analysis method (LHAM) that defines the connection between the target liver cells and the HBV. We also examine the basic nonlinear differential equation by LHAM to get a semi-analytical solution. This can be a very straight and direct method which provides the appropriate solution. Moreover, the local and global stability analysis of disease-free and endemic equilibrium is done using Lyapunov function. Mathematica 12 software is used to find out the solutions and graphical representations. We also discuss the numerical simulations up to sixth-order approximation and error analysis using the same software.
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Affiliation(s)
- M. Aniji
- Department of Mathematics, University College of Engineering, Rajamadam, Pattukkottai, Tamilnadu, India
| | - N. Kavitha
- Department of Mathematics, University College of Engineering, Rajamadam, Pattukkottai, Tamilnadu, India
| | - S. Balamuralitharan
- Department of Mathematics, Faculty of Engineering and Technology, SRM Institute of Science and Technology, Kattankulathur, Tamilnadu, India
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21
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Khatun MS, Biswas MHA. Optimal control strategies for preventing hepatitis B infection and reducing chronic liver cirrhosis incidence. Infect Dis Model 2019; 5:91-110. [PMID: 31930183 PMCID: PMC6948267 DOI: 10.1016/j.idm.2019.12.006] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/26/2019] [Revised: 12/21/2019] [Accepted: 12/23/2019] [Indexed: 11/21/2022] Open
Abstract
Advanced liver cirrhosis has become life-threatening among non-communicable diseases nowadays. Cirrhosis, the terminal stage of liver diseases in which the liver develops scarring as a result of various long-term continuous damages. Among liver diseases, viral hepatitis is the major risk factor for chronic cirrhosis development. The present paper demonstrates a compartmental model of chronic disease liver cirrhosis describing the transmission dynamics of this disease. Applying the Pontryagin’s maximum principle, the optimal control policies such as vaccination for hepatitis B virus and treatment of other causes of cirrhosis are adopted as control measures. The target of this study is to minimize the number of infected and liver cirrhotic individuals as well as the associated cost of the control. For this purpose, the optimal control strategies are employed according to the underlying causes behind this disease. Our goal is to find the strategy of preventing hepatitis B infection which is considered one of the leading causes of cirrhosis and consequently, reduction of the chronic cirrhosis incidence. Efficiency analysis is also performed to observe the effective control among the two control strategies. The model is investigated both analytically and numerically and the numerical simulations are carried out to illustrate the analytical findings. The analysis reveals that both the vaccination and treatment could be the most fruitful way to reduce the incidence of chronic liver cirrhosis.
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22
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Optimal Impulse Vaccination Approach for an SIR Control Model with Short-Term Immunity. MATHEMATICS 2019. [DOI: 10.3390/math7050420] [Citation(s) in RCA: 10] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/17/2022]
Abstract
Vaccines are not administered on a continuous basis, but injections are practically introduced at discrete times often separated by an important number of time units, and this differs depending on the nature of the epidemic and its associated vaccine. In addition, especially when it comes to vaccination, most optimization approaches in the literature and those that have been subject to epidemic models have focused on treating problems that led to continuous vaccination schedules but their applicability remains debatable. In search of a more realistic methodology to resolve this issue, a control modeling design, where the control can be characterized analytically and then optimized, can definitely help to find an optimal regimen of pulsed vaccinations. Therefore, we propose a susceptible-infected-removed (SIR) hybrid epidemic model with impulse vaccination control and a compartment that represents the number of vaccinated individuals supposed to not acquire sufficient immunity to become permanently recovered due to the short-term effect of vaccines. A basic reproduction number, when the control is defined as a constant parameter, is calculated. Since we also need to find the optimal values of this impulse control when it is defined as a function of time, we start by stating a general form of an impulse version of Pontryagin’s maximum principle that can be adapted to our case, and then we apply it to our model. Finally, we provide our numerical simulations that are obtained via an impulse progressive-regressive iterative scheme with fixed intervals between impulse times (theoretical example of an impulse at each week), and we conclude with a discussion of our results.
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23
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Friedman A, Siewe N. Chronic hepatitis B virus and liver fibrosis: A mathematical model. PLoS One 2018; 13:e0195037. [PMID: 29634771 PMCID: PMC5892900 DOI: 10.1371/journal.pone.0195037] [Citation(s) in RCA: 10] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/21/2017] [Accepted: 03/15/2018] [Indexed: 12/14/2022] Open
Abstract
Hepatitis B virus (HBV) infection is a liver disorder that can result in cirrhosis, liver failure and hepatocellular carcinoma. HBV infection remains a major global health problem, as it affects more 350 million people chronically and kills roughly 600,000 people annually. Drugs currently used against HBV include IFN-α that decreases viremia, inflammation and the growth of liver fibrosis, and adefovir that decreases the viral load. Each of these drugs can have severe side-effects. In the present paper, we consider the treatment of chronic HBV by a combination of IFN-α and adefovir, and raise the following question: What should be the optimal ratio between IFN-α and adefovir in order to achieve the best 'efficacy' under constraints on the total amount of the drugs; here the efficacy is measured by the reduction of the levels of inflammation and of fibrosis? We develop a mathematical model of HBV pathogenesis by a system of partial differential equations (PDEs) and use the model to simulate a 'synergy map' which addresses the above question.
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Affiliation(s)
- Avner Friedman
- Mathematical Biosciences Institute & Department of Mathematics, The Ohio State University, Columbus, Ohio, United States of America
| | - Nourridine Siewe
- National Institute for Mathematical and Biological Synthesis, University of Tennessee, Knoxville, Tennessee, United States of America
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24
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KHAN MUHAMMADALTAF, ISLAM SAEED, VALVERDE JOSEC, KHAN SHERAFZAL. CONTROL STRATEGIES of HEPATITIS B WITH THREE CONTROL VARIABLES. J BIOL SYST 2018. [DOI: 10.1142/s0218339018500018] [Citation(s) in RCA: 16] [Impact Index Per Article: 2.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/18/2022]
Abstract
In this paper, we present a compartmental mathematical model of hepatitis B virus with optimal control strategies. First, we formulate the model applying the optimal control techniques which use control variables in the form of isolation, educational campaign and vaccination. We derive the conditions under which it is optimal to eradicate the disease and examine the impact of possible vaccination treatment strategies on disease transmission. When such an elimination is impossible, we use the techniques of Pontryagin’s Maximum Principle to derive the necessary conditions for the optimal control problem. The numerical results show that some effective vaccination and control can reduce the disease spread in the community.
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Affiliation(s)
- MUHAMMAD ALTAF KHAN
- Department of Mathematics, City University of Science and Information Technology, Peshawar, KP 25000, Pakistan
| | - SAEED ISLAM
- Department of Mathematics, Abdul Wali Khan, University, Mardan, KP 23200, Pakistan
| | - JOSE C. VALVERDE
- Department of Mathematics, University of Castilla-La Mancha, Spain
| | - SHER AFZAL KHAN
- Faculty of Computing and Information Technology in Rabigh, King Abdul Aziz University, Jeddah, Saudi Arabia
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25
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Liang P, Zu J, Zhuang G. A Literature Review of Mathematical Models of Hepatitis B Virus Transmission Applied to Immunization Strategies From 1994 to 2015. J Epidemiol 2017; 28:221-229. [PMID: 29276213 PMCID: PMC5911672 DOI: 10.2188/jea.je20160203] [Citation(s) in RCA: 13] [Impact Index Per Article: 1.9] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/12/2022] Open
Abstract
A mathematical model of the transmission dynamics of infectious disease is an important theoretical epidemiology method, which has been used to simulate the prevalence of hepatitis B and evaluate different immunization strategies. However, differences lie in the mathematical processes of modeling HBV transmission in published studies, not only in the model structure, but also in the estimation of certain parameters. This review reveals that the dynamics model of HBV transmission only simulates the spread of HBV in the population from the macroscopic point of view and highlights several main shortcomings in the model structure and parameter estimation. First, age-dependence is the most important characteristic in the transmission of HBV, but an age-structure model and related age-dependent parameters were not adopted in some of the compartmental models describing HBV transmission. In addition, the numerical estimation of the force of HBV infection did not give sufficient weight to the age and time factors and is not suitable using the incidence data. Lastly, the current mathematical models did not well reflect the details of the factors of HBV transmission, such as migration from high or intermediate HBV endemic areas to low endemic areas and the kind of HBV genotype. All of these shortcomings may lead to unreliable results. When the mathematical model closely reflects the fact of hepatitis B spread, the results of the model fit will provide valuable information for controlling the transmission of hepatitis B.
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Affiliation(s)
- Peifeng Liang
- Department of Statistics, People's Hospital of Ningxia Hui Autonomous Region
| | - Jian Zu
- Department of Applied Mathematics, School of Mathematics and Statistics, Xi'an Jiaotong University
| | - Guihua Zhuang
- Department of Epidemiology and Biostatistics, School of Public Health, Xi'an Jiaotong University Health Science Center
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26
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Sun D, Liu F. Modeling and Control of a Delayed Hepatitis B Virus Model with Incubation Period and Combination Treatment. Interdiscip Sci 2017; 10:375-389. [PMID: 29260504 DOI: 10.1007/s12539-017-0275-y] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/12/2017] [Revised: 11/16/2017] [Accepted: 11/18/2017] [Indexed: 12/21/2022]
Abstract
In this paper, a hepatitis B virus (HBV) model with an incubation period and delayed state and control variables is firstly proposed. Furthermore, the combination treatment is adopted to have a longer-lasting effect than mono-therapy. The equilibrium points and basic reproduction number are calculated, and then the local stability is analyzed on this model. We then present optimal control strategies based on the Pontryagin's minimum principle with an objective function not only to reduce the levels of exposed cells, infected cells and free viruses nearly to zero at the end of therapy, but also to minimize the drug side-effect and the cost of treatment. What's more, we develop a numerical simulation algorithm for solving our HBV model based on the combination of forward and backward difference approximations. The state dynamics of uninfected cells, exposed cells, infected cells, free viruses, CTL and ALT are simulated with or without optimal control, which show that HBV is reduced nearly to zero based on the time-varying optimal control strategies whereas the disease would break out without control. At last, by the simulations, we prove that strategy A is the best among the three kinds of strategies we adopt and further comparisons have been done between model (1) and model (2).
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Affiliation(s)
- Deshun Sun
- Control and Simulation Center, Harbin Institute of Technology, Harbin, 15008, China
| | - Fei Liu
- Control and Simulation Center, Harbin Institute of Technology, Harbin, 15008, China.
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27
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Khan T, Zaman G, Chohan MI. The transmission dynamic and optimal control of acute and chronic hepatitis B. JOURNAL OF BIOLOGICAL DYNAMICS 2017; 11:172-189. [PMID: 27852159 DOI: 10.1080/17513758.2016.1256441] [Citation(s) in RCA: 30] [Impact Index Per Article: 4.3] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/15/2023]
Abstract
In this article, we present the transmission dynamic of the acute and chronic hepatitis B epidemic problem and develop an optimal control strategy to control the spread of hepatitis B in a community. In order to do this, first we present the model formulation and find the basic reproduction number [Formula: see text]. We show that if [Formula: see text] then the disease-free equilibrium is both locally as well as globally asymptotically stable. Then, we prove that the model is locally and globally asymptotically stable, if [Formula: see text]. To control the spread of this infection, we develop a control strategy by applying three control variables such as isolation of infected and non-infected individuals, treatment and vaccination to minimize the number of acute infected, chronically infected with hepatitis B individuals and maximize the number of susceptible and recovered individuals. Finally, we present numerical simulation to illustrate the feasibility of the control strategy.
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Affiliation(s)
- Tahir Khan
- a Department of Mathematics , University of Malakand, Chakdara Dir (Lower) Khyber Pakhtunkhawa , Pakistan
| | - Gul Zaman
- a Department of Mathematics , University of Malakand, Chakdara Dir (Lower) Khyber Pakhtunkhawa , Pakistan
| | - M Ikhlaq Chohan
- b Department of Business Administration and Accounting , Buraimi University College , Al-Buraimi , Oman
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28
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Goyal A, Murray JM. Recognizing the impact of endemic hepatitis D virus on hepatitis B virus eradication. Theor Popul Biol 2016; 112:60-69. [PMID: 27594346 DOI: 10.1016/j.tpb.2016.08.004] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/04/2015] [Revised: 08/24/2016] [Accepted: 08/24/2016] [Indexed: 01/04/2023]
Abstract
BACKGROUND Hepatitis delta virus (HDV) in conjunction with hepatitis B virus (HBV) increases adult morbidity and mortality. A number of studies have performed cost-benefit analyses for HBV interventions, but they have ignored the impact of HDV on these outcomes. METHODS Using a mathematical model of HBV-HDV epidemiology, we compare health benefits and cost outcomes of four interventions: testing with HBV adult vaccination (diagnosis), diagnosis with antiviral treatment for HBV infections (mono-infections), diagnosis with antiviral treatment for HBV-HDV infections (dual-infections), and awareness programs. The relationship between optimal levels and outcomes of each of these interventions and HDV prevalence in HBV infected individuals ranging from 0 to 50% is determined. RESULTS Over a 50 year period under no intervention, HBV prevalence, per capita total cost and death toll increase by 2.25%, -$11 and 2.6-fold respectively in moderate HDV endemic regions compared to mono-infected regions; the corresponding values for high HDV endemic regions are 4.2%, -$21 and 3.9-fold. Optimal interventions can be strategized similarly in mono and dually endemic regions. Only implementation of all four interventions achieves a very low HBV prevalence of around 1.5% in a moderate HDV endemic region such as China, with 2.8 million fewer deaths compared to no intervention. Although the policy of implementation of all four interventions costs additional $382 billion compared to no intervention, it still remains cost-effective with an incremental cost-effectiveness ratio of $1400/QALY. Very high efficacy awareness programs achieve less prevalence with fewer deaths at a lower cost compared to treatment and/or vaccination programs. CONCLUSION HDV substantially affects the performance of any HBV-related intervention. Its exclusion results in over-estimation of the effectiveness of HBV interventions.
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Affiliation(s)
- Ashish Goyal
- School of Mathematics and Statistics, UNSW Australia, Sydney, NSW 2052, Australia.
| | - John M Murray
- School of Mathematics and Statistics, UNSW Australia, Sydney, NSW 2052, Australia
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29
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Differences in the availability of diagnostics and treatment modalities for chronic hepatitis B across Europe. Clin Microbiol Infect 2015; 21:1027-32. [DOI: 10.1016/j.cmi.2015.07.002] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/02/2015] [Revised: 06/30/2015] [Accepted: 07/01/2015] [Indexed: 12/26/2022]
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