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Rashid S, Hamidi SZ, Akram S, Raza MA, Elagan SK, Alsubei BMT. Theoretical and mathematical codynamics of nonlinear tuberculosis and COVID-19 model pertaining to fractional calculus and probabilistic approach. Sci Rep 2024; 14:8827. [PMID: 38632309 DOI: 10.1038/s41598-024-59261-7] [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: 01/10/2024] [Accepted: 04/08/2024] [Indexed: 04/19/2024] Open
Abstract
Severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) is a novel virus known as coronavirus 2 (SARS-CoV-2) that affects the pulmonary structure and results in the coronavirus illness 2019 (COVID-19). Tuberculosis (TB) and COVID-19 codynamics have been documented in numerous nations. Understanding the complexities of codynamics is now critically necessary as a consequence. The aim of this research is to construct a co-infection model of TB and COVID-19 in the context of fractional calculus operators, white noise and probability density functions, employing a rigorous biological investigation. By exhibiting that the system possesses non-negative and bounded global outcomes, it is shown that the approach is both mathematically and biologically practicable. The required conditions are derived, guaranteeing the eradication of the infection. Sensitivity analysis and bifurcation of the submodel are also investigated with system parameters. Furthermore, existence and uniqueness results are established, and the configuration is tested for the existence of an ergodic stationary distribution. For discovering the system's long-term behavior, a deterministic-probabilistic technique for modeling is designed and operated in MATLAB. By employing an extensive review, we hope that the previously mentioned approach improves and leads to mitigating the two diseases and their co-infections by examining a variety of behavioral trends, such as transitions to unpredictable procedures. In addition, the piecewise differential strategies are being outlined as having promising potential for scholars in a range of contexts because they empower them to include particular characteristics across multiple time frame phases. Such formulas can be strengthened via classical technique, power-law, exponential decay, generalized Mittag-Leffler kernels, probability density functions and random procedures. Furthermore, we get an accurate description of the probability density function encircling a quasi-equilibrium point if the effect of TB and COVID-19 minimizes the propagation of the codynamics. Consequently, scholars can obtain better outcomes when analyzing facts using random perturbations by implementing these strategies for challenging issues. Random perturbations in TB and COVID-19 co-infection are crucial in controlling the spread of an epidemic whenever the suggested circulation is steady and the amount of infection eliminated is closely correlated with the random perturbation level.
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Affiliation(s)
- Saima Rashid
- Department of Mathematics, Government College University, Faisalabad, 38000, Pakistan
- Department of Computer Science and Mathematics, Lebanese American University, Beirut, 11022801, Lebanon
| | - Sher Zaman Hamidi
- Department of Physics, Nangarhar University, Jalalabad City, Nangarhar, 2601, Afghanistan.
| | - Saima Akram
- Department of Mathematics, Government College Women University Faisalabad, Faisalabad, 38000, Pakistan
- Centre for Advanced Studies in Pure and Applied Mathematics, Bahauddin Zakariya, Multan, 60000, Pakistan
| | - Muhammad Aon Raza
- Department of Mathematics, Government College University, Faisalabad, 38000, Pakistan
| | - S K Elagan
- Department of Mathematics and Statistics, College of Science, Taif University, P.O. Box 11099, Taif, 21944, Saudi Arabia
| | - Beida Mohsen Tami Alsubei
- Mathematics Program, Department of Science and Technology, Ranyah University College, Taif University, P.O. Box 11099, Taif, 21944, Saudi Arabia
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Li W, Cai S, Zhai X, Ou J, Zheng K, Wei F, Mao X. Transmission dynamics of symptom-dependent HIV/AIDS models. MATHEMATICAL BIOSCIENCES AND ENGINEERING : MBE 2024; 21:1819-1843. [PMID: 38454662 DOI: 10.3934/mbe.2024079] [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: 03/09/2024]
Abstract
In this study, we proposed two, symptom-dependent, HIV/AIDS models to investigate the dynamical properties of HIV/AIDS in the Fujian Province. The basic reproduction number was obtained, and the local and global stabilities of the disease-free and endemic equilibrium points were verified to the deterministic HIV/AIDS model. Moreover, the indicators $ R_0^s $ and $ R_0^e $ were derived for the stochastic HIV/AIDS model, and the conditions for stationary distribution and stochastic extinction were investigated. By using the surveillance data from the Fujian Provincial Center for Disease Control and Prevention, some numerical simulations and future predictions on the scale of HIV/AIDS infections in the Fujian Province were conducted.
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Affiliation(s)
- Wenshuang Li
- School of Mathematics and Statistics, Fuzhou University, Fuzhou 350116, Fujian, China
| | - Shaojian Cai
- Fujian Provincial Center for Disease Control and Prevention, Fuzhou 350012, China
| | - Xuanpei Zhai
- School of Physical Science and Technology, ShanghaiTech University, Shanghai 201210, China
| | - Jianming Ou
- Fujian Provincial Center for Disease Control and Prevention, Fuzhou 350012, China
| | - Kuicheng Zheng
- Fujian Provincial Center for Disease Control and Prevention, Fuzhou 350012, China
| | - Fengying Wei
- School of Mathematics and Statistics, Fuzhou University, Fuzhou 350116, Fujian, China
- Center for Applied Mathematics of Fujian Province, Fuzhou University, Fuzhou 350116, Fujian, China
- Key Laboratory of Operations Research and Control of Universities in Fujian, Fuzhou University, Fuzhou 350116, Fujian, China
| | - Xuerong Mao
- Department of Mathematics and Statistics, University of Strathclyde, Glasgow G1 1XH, UK
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Adeyemo S, Sangotola A, Korosteleva O. Modeling Transmission Dynamics of Tuberculosis-HIV Co-Infection in South Africa. EPIDEMIOLOGIA 2023; 4:408-419. [PMID: 37873885 PMCID: PMC10594517 DOI: 10.3390/epidemiologia4040036] [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: 08/05/2023] [Revised: 10/07/2023] [Accepted: 10/09/2023] [Indexed: 10/25/2023] Open
Abstract
South Africa has the highest number of people living with the human immunodeficiency virus (HIV) in the world, accounting for nearly one in five people living with HIV globally. As of 2021, 8 million people in South Africa were infected with HIV, which is 13% of the country's total population. Approximately 450,000 people in the country develop tuberculosis (TB) disease every year, and 270,000 of those are HIV positive. This suggests that being HIV positive significantly increases one's susceptibility to TB, accelerating the spread of the epidemic. To better understand the disease burden at the population level, a Susceptible-Infected-Recovered-Dead (SIRD) TB-HIV co-infection epidemic model is presented. Parameter values are estimated using the method of moments. The disease-free equilibrium and basic reproduction number of the model are also obtained. Finally, numeric simulations are carried out for a 30-year period to give insights into the transmission dynamics of the co-infection.
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Affiliation(s)
- Simeon Adeyemo
- Department of Mathematics and Statistics, California State University, Long Beach, CA 90840, USA;
| | - Adekunle Sangotola
- Department of Physical Sciences, Bells University of Technology, Ota 112212, Ogun, Nigeria;
| | - Olga Korosteleva
- Department of Mathematics and Statistics, California State University, Long Beach, CA 90840, USA;
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Panchal J, Acharya F, Joshi K. A noninteger order SEITR dynamical model for TB. ADVANCES IN CONTINUOUS AND DISCRETE MODELS 2022; 2022:27. [PMID: 35450198 PMCID: PMC8959566 DOI: 10.1186/s13662-022-03700-0] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 07/15/2021] [Accepted: 03/14/2022] [Indexed: 11/30/2022]
Abstract
This research paper designs the noninteger order SEITR dynamical model in the Caputo sense for tuberculosis. The authors of the article have classified the infection compartment into four different compartments such as newly infected unrecognized individuals, diagnosed patients, highly infected patients, and patients with delays in treatment which provide better detail of the TB infection dynamic. We estimate the model parameters using the least square curve fitting and demonstrate that the proposed model provides a good fit to tuberculosis confirmed cases of India from the year 2000 to 2020. Further, we compute the basic reproduction number as \documentclass[12pt]{minimal}
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\begin{document}$\Re _{0} \approx 1.73$\end{document}ℜ0≈1.73 of the model using the next-generation matrix method and the model equilibria. The existence and uniqueness of the approximate solution for the SEITR model is validated using the generalized Adams–Bashforth–Moulton method. The graphical representation of the fractional order model is given to validate the result using the numerical simulation. We conclude that the fractional order model is more realistic than the classical integer order model and provide more detailed information about the real data of the TB disease dynamics.
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Mekonen KG, Obsu LL. Mathematical modeling and analysis for the co-infection of COVID-19 and tuberculosis. Heliyon 2022; 8:e11195. [PMID: 36281374 PMCID: PMC9583685 DOI: 10.1016/j.heliyon.2022.e11195] [Citation(s) in RCA: 2] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/11/2022] [Revised: 03/05/2022] [Accepted: 10/17/2022] [Indexed: 11/06/2022] Open
Abstract
We developed a TB-COVID-19 co-infection epidemic model using a non-linear dynamical system by subdividing the human population into seven compartments. The biological well-posedness of the formulated mathematical model was studied via proving properties like boundedness of solutions, no-negativity, and the solution's dependence on the initial data. We then computed the reproduction numbers separately for TB and COVID-19 sub-models. The criterion for stability conditions for stationary points was examined. The basic reproduction number of sub-models used to suggest the mitigation and persistence of the diseases. Qualitative analysis of the sub-models revealed that the disease-free stationary points are both locally and globally stable provided the respective reproduction numbers are smaller than unit. The endemic stationary points for each sub-models were globally stable if their respective basic reproduction numbers are greater than unit. In each sub-model, we performed an analysis of sensitive parameters concerning the corresponding reproduction numbers. Results from sensitivity indices of the parameters revealed that deceasing contact rate and increasing the transferring rates from the latent stage to an infected class of individuals leads to mitigating the two diseases and their co-infections. We have also studied the analytical behavior of the full co-infection model by deriving the equilibrium points and investigating the conditions of their stability. The numerical experiments of the proposed co-infection model agree with the findings in the analytical results.
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Affiliation(s)
| | - Legesse Lemecha Obsu
- Department of Applied Mathematics, Adama Science and Technology University, Adama, Ethiopia
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Allali K, Harroudi S, Torres DFM. Optimal control of an HIV model with a trilinear antibody growth function. DISCRETE & CONTINUOUS DYNAMICAL SYSTEMS - S 2022. [DOI: 10.3934/dcdss.2021148] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 02/05/2023]
Abstract
<p style='text-indent:20px;'>We propose and study a new mathematical model of the human immunodeficiency virus (HIV). The main novelty is to consider that the antibody growth depends not only on the virus and on the antibodies concentration but also on the uninfected cells concentration. The model consists of five nonlinear differential equations describing the evolution of the uninfected cells, the infected ones, the free viruses, and the adaptive immunity. The adaptive immune response is represented by the cytotoxic T-lymphocytes (CTL) cells and the antibodies with the growth function supposed to be trilinear. The model includes two kinds of treatments. The objective of the first one is to reduce the number of infected cells, while the aim of the second is to block free viruses. Firstly, the positivity and the boundedness of solutions are established. After that, the local stability of the disease free steady state and the infection steady states are characterized. Next, an optimal control problem is posed and investigated. Finally, numerical simulations are performed in order to show the behavior of solutions and the effectiveness of the two incorporated treatments via an efficient optimal control strategy.</p>
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Michel L, Silva CJ, Torres DFM. Model-free based control of a HIV/AIDS prevention model. MATHEMATICAL BIOSCIENCES AND ENGINEERING 2021; 19:759-774. [PMID: 34903011 DOI: 10.3934/mbe.2022034] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 02/05/2023]
Abstract
Controlling an epidemiological model is often performed using optimal control theory techniques for which the solution depends on the equations of the controlled system, objective functional and possible state and/or control constraints. In this paper, we propose a model-free control approach based on an algorithm that operates in 'real-time' and drives the state solution according to a direct feedback on the state solution that is aimed to be minimized, and without knowing explicitly the equations of the controlled system. We consider a concrete epidemic problem of minimizing the number of HIV infected individuals, through the preventive measure pre-exposure prophylaxis (PrEP) given to susceptible individuals. The solutions must satisfy control and mixed state-control constraints that represent the limitations on PrEP implementation. Our model-free based control algorithm allows to close the loop between the number of infected individuals with HIV and the supply of PrEP medication 'in real time', in such a manner that the number of infected individuals is asymptotically reduced and the number of individuals under PrEP medication remains below a fixed constant value. We prove the efficiency of our approach and compare the model-free control solutions with the ones obtained using a classical optimal control approach via Pontryagin maximum principle. The performed numerical simulations allow us to conclude that the model-free based control strategy highlights new and interesting performances compared with the classical optimal control approach.
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Affiliation(s)
- Loïc Michel
- École Centrale de Nantes-LS2N, UMR 6004 CNRS, Nantes 44300, France.,Univ Lyon, INSA Lyon, Université Claude Bernard Lyon 1, École Centrale de Lyon, CNRS, Ampère, UMR 5005, Villeurbanne 69621, France
| | - Cristiana J Silva
- Center for Research and Development in Mathematics and Applications (CIDMA), Department of Mathematics, University of Aveiro, Aveiro 3810-193, Portugal
| | - Delfim F M Torres
- Center for Research and Development in Mathematics and Applications (CIDMA), Department of Mathematics, University of Aveiro, Aveiro 3810-193, Portugal
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Vaz S, Torres DFM. A dynamically-consistent nonstandard finite difference scheme for the SICA model. MATHEMATICAL BIOSCIENCES AND ENGINEERING 2021; 18:4552-4571. [PMID: 34198453 DOI: 10.3934/mbe.2021231] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 02/05/2023]
Abstract
In this work, we derive a nonstandard finite difference scheme for the SICA (Susceptible-Infected-Chronic-AIDS) model and analyze the dynamical properties of the discretized system. We prove that the discretized model is dynamically consistent with the continuous, maintaining the essential properties of the standard SICA model, namely, the positivity and boundedness of the solutions, equilibrium points, and their local and global stability.
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Affiliation(s)
- Sandra Vaz
- Center of Mathematics and Applications (CMA-UBI), Department of Mathematics, University of Beira Interior, Covilhã 6201-001, Portugal
| | - Delfim F M Torres
- Center for Research and Development in Mathematics and Applications (CIDMA), Department of Mathematics, University of Aveiro, Aveiro 3810-193, Portugal
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Mathematical Analysis of a Fractional COVID-19 Model Applied to Wuhan, Spain and Portugal. AXIOMS 2021. [DOI: 10.3390/axioms10030135] [Citation(s) in RCA: 8] [Impact Index Per Article: 2.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 02/05/2023]
Abstract
We propose a qualitative analysis of a recent fractional-order COVID-19 model. We start by showing that the model is mathematically and biologically well posed. Then, we give a proof on the global stability of the disease free equilibrium point. Finally, some numerical simulations are performed to ensure stability and convergence of the disease free equilibrium point.
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Aggarwal R, Kovacs T. Assessing the Effects of Holling Type-II Treatment Rate on HIV-TB Co-infection. Acta Biotheor 2021; 69:1-35. [PMID: 32556779 DOI: 10.1007/s10441-020-09385-w] [Citation(s) in RCA: 4] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/05/2019] [Accepted: 06/03/2020] [Indexed: 12/17/2022]
Abstract
In this paper, a HIV-TB co-infection model is explored which incorporates a non-linear treatment rate for TB. We begin with presenting a HIV-TB co-infection model and analyze both HIV and TB sub-models separately. The basic reproduction numbers corresponding to HIV-only, TB-only and the HIV-TB full model are computed. The disease-free equilibrium point of the HIV sub-model is shown to be locally as well as globally asymptotically stable when its corresponding reproduction number is less than unity. The HIV-only model exhibits a transcritical bifurcation. On the other hand, for the TB sub-model, the disease-free equilibrium point is locally asymptotically stable but may not be globally asymptotically stable. We have also analyzed the full HIV-TB co-infection model. Numerical simulations are performed to investigate the effect of treatment rate in the presence of resource limitation for TB infected individuals, which emphasize the fact that to reduce co-infection from the population programs to accelerate the treatment of TB should be implemented.
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Affiliation(s)
- Rajiv Aggarwal
- Deshbandhu College, University of Delhi, New Delhi, 110019, India.
| | - Tamas Kovacs
- Postdoctoral Researcher at ELTE, Institute for Theoretical Physics, Eotvos University, Budapest, Hungary
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Das S, Ghosh P, Banerjee S, Pyne S, Chattopadhyay J, Mukhopadhyay I. Determination of critical community size from an HIV/AIDS model. PLoS One 2021; 16:e0244543. [PMID: 33507898 PMCID: PMC7842972 DOI: 10.1371/journal.pone.0244543] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/03/2020] [Accepted: 12/11/2020] [Indexed: 11/19/2022] Open
Abstract
After an epidemic outbreak, the infection persists in a community long enough to engulf the entire susceptible population. Local extinction of the disease could be possible if the susceptible population gets depleted. In large communities, the tendency of eventual damp down of recurrent epidemics is balanced by random variability. But, in small communities, the infection would die out when the number of susceptible falls below a certain threshold. Critical community size (CCS) is considered to be the mentioned threshold, at which the infection is as likely as not to die out after a major epidemic for small communities unless reintroduced from outside. The determination of CCS could aid in devising systematic control strategies to eradicate the infectious disease from small communities. In this article, we have come up with a simplified computation based approach to deduce the CCS of HIV disease dynamics. We consider a deterministic HIV model proposed by Silva and Torres, and following Nåsell, introduce stochasticity in the model through time-varying population sizes of different compartments. Besides, Metcalf’s group observed that the relative risk of extinction of some infections on islands is almost double that in the mainlands i.e. infections cease to exist at a significantly higher rate in islands compared to the mainlands. They attributed this phenomenon to the greater recolonization in the mainlands. Interestingly, the application of our method on demographic facts and figures of countries in the AIDS belt of Africa led us to expect that existing control measures and isolated locations would assist in temporary eradication of HIV infection much faster. For example, our method suggests that through systematic control strategies, after 7.36 years HIV epidemics will temporarily be eradicated from different communes of island nation Madagascar, where the population size falls below its CCS value, unless the disease is reintroduced from outside.
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Affiliation(s)
- Sarmistha Das
- Human Genetics Unit, Indian Statistical Institute, Kolkata, West Bengal, India
| | - Pramit Ghosh
- Deben Mahata Government Medical College & Hospital, Purulia, West Bengal, India
| | - Sandip Banerjee
- Department of Mathematics, Indian Institute of Technology, Roorkee, Uttarakhand, India
| | - Saumyadipta Pyne
- Public Health Dynamics Lab and Department of Biostatistics, Graduate School of Public Health, University of Pittsburgh, Pittsburgh, Pennsylvania, United States of America
- Health Analytics Network, Pittsburgh, Pennsylvania, United States of America
| | - Joydev Chattopadhyay
- Agricultural and Ecological Research Unit, Indian Statistical Institute, Kolkata, West Bengal, India
| | - Indranil Mukhopadhyay
- Human Genetics Unit, Indian Statistical Institute, Kolkata, West Bengal, India
- * E-mail:
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A T, Aggarwal R, Raj YA. A fractional order HIV-TB co-infection model in the presence of exogenous reinfection and recurrent TB. NONLINEAR DYNAMICS 2021; 104:4701-4725. [PMID: 34075277 PMCID: PMC8159726 DOI: 10.1007/s11071-021-06518-9] [Citation(s) in RCA: 6] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/04/2020] [Accepted: 05/04/2021] [Indexed: 05/13/2023]
Abstract
In this article, a novel fractional order model has been introduced in Caputo sense for HIV-TB co-infection in the presence of exogenous reinfection and recurrent TB along with the treatment for both HIV and TB. The main aim of considering the fractional order model is to incorporate the memory effect of both diseases. We have analyzed both sub-models separately with fractional order. The basic reproduction number, which measures the contagiousness of the disease, is determined. The HIV sub-model is shown to have a locally asymptotically stable disease-free equilibrium point when the corresponding reproduction number, R H , is less than unity, whereas, for R H > 1 , the endemic equilibrium point comes into existence. For the TB sub-model, the disease-free equilibrium point has been proved to be locally asymptotically stable for R T < 1 . The existence of TB endemic equilibrium points in the presence of reinfection and recurrent TB for R T < 1 justifies the existence of backward bifurcation under certain restrictions on the parameters. Further, we numerically simulate the fractional order model to verify the analytical results and highlight the role of fractional order in co-infection modeling. The fractional order derivative is shown to have a crucial role in determining the transmission dynamics of HIV-TB co-infection. It is concluded that the memory effect plays a significant role in reducing the infection prevalence of HIV-TB co-infection. An increment in the number of recovered individuals can also be observed when the memory effect is taken into consideration by introducing fractional order model.
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Affiliation(s)
- Tanvi A
- Department of Mathematics, Deshbandhu College, University of Delhi, New Delhi, 110019 India
| | - Rajiv Aggarwal
- Department of Mathematics, Deshbandhu College, University of Delhi, New Delhi, 110019 India
| | - Yashi A. Raj
- Department of Mathematics, Shaheed Rajguru College of Applied Sciences for Women, University of Delhi, New Delhi, 110096 India
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Tanvi, Aggarwal R. Estimating the impact of antiretroviral therapy on HIV-TB co-infection: Optimal strategy prediction. INT J BIOMATH 2020. [DOI: 10.1142/s1793524521500042] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/18/2022]
Abstract
In this paper, a nonlinear population model for HIV-TB co-infection has been proposed. The model is incorporated with the effect of early and late initiation of HIV treatment in co-infectives already on TB treatment, on the occurrence of Immune Reconstitution Inflammatory syndrome (IRIS). A 15-dimensional (15D) mathematical model has been developed in this study. We begin with considering constant treatment rates and thereafter, proceed to time-dependent treatment rates for co-infectives as control parameters. The basic reproduction number, a threshold quantity, corresponding to each HIV and TB sub-model has been computed in case of constant controls. With constant values of control parameters, mathematical analysis shows the existence and local stability of the disease-free equilibrium point and the endemic equilibrium point for the model. Together with time-dependent parameters, an optimal control problem is introduced and solved using Pontryagin’s maximum principle with an objective to minimize the number of infectives and disease induced deaths along with the cost of treatment. Numerical simulations are performed to examine the effect of reproduction numbers on control profiles and to identify, the ideal combination of treatment strategies which provides minimum burden on a society. Numerical results imply that if both HIV and TB are endemic in the population, then in order to bring in minimum burden from the co-infection, optimal control efforts must be enforced rather than constant treatment rate.
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Affiliation(s)
- Tanvi
- Department of Mathematics, Shaheed Rajguru College of Applied Sciences for Women, University of Delhi, New Delhi-110096, India
| | - Rajiv Aggarwal
- Department of Mathematics, Deshbandhu College, University of Delhi, New Delhi-110019, India
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Abstract
New compartmental mathematical model for the spread of the COVID-19 disease. COVID-19 pandemic in Portugal till the end of the three states of emergency. Simultaneous description of active infected and hospitalized individuals. Reproduction number in agreement with the one of Portuguese Health authorities. The possibility of a second wave of COVID-19 in Portugal is not ruled out.
We propose a compartmental mathematical model for the spread of the COVID-19 disease, showing its usefulness with respect to the pandemic in Portugal, from the first recorded case in the country till the end of the three states of emergency. New results include the compartmental model, described by a system of seven ordinary differential equations; proof of positivity and boundedness of solutions; investigation of equilibrium points and their stability analysis; computation of the basic reproduction number; and numerical simulations with official real data from the Portuguese health authorities. Besides completely new, the proposed model allows to describe quite well the spread of COVID-19 in Portugal, fitting simultaneously not only the number of active infected individuals but also the number of hospitalized individuals, respectively with a L2 error of 9.2152e−04 and 1.6136e−04 with respect to the initial population. Such results are very important, from a practical point of view, and far from trivial from a mathematical perspective. Moreover, the obtained value for the basic reproduction number is in agreement with the one given by the Portuguese authorities at the end of the three emergency states.
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Analysis and optimal control of an HIV model based on CD4 count. J Math Biol 2020; 81:209-241. [PMID: 32601724 DOI: 10.1007/s00285-020-01508-8] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/17/2019] [Revised: 12/12/2019] [Indexed: 10/24/2022]
Abstract
A non-linear mechanistic model for the transmission dynamics of HIV/AIDS is developed and analyzed. The model classified the infected individuals based on their CD4 count level. Furthermore, education campaign, voluntary testing and counseling and treatment are considered as intervention strategies for controlling the disease. The analysis of the model reveals that imperfect public enlightenment campaign can induce backward bifurcation. It has been shown that when public enlightenment campaign is [Formula: see text] effective, the disease free equilibrium is globally asymptotically stable for [Formula: see text], whereas for [Formula: see text] the global stability of the endemic equilibrium is proved only in a special case. Time dependent controls of the intervention strategies mentioned above are incorporated into the model and the optimal control strategies with minimal implementation cost are identified. In addition, cost effectiveness analysis in the form of incremental cost effectiveness ratio is carried-out to identify the most cost effective strategies. The results suggest that out of the three non dominated strategies, the strategy of educating the newly entrants only or combination of newly entrants and susceptible individuals is very cost effective using per capita GDP of Nigeria as at 2018. However, the choice of which strategy to implement depends on budgetary allocation and resource availability.
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Silva CJ, Torres DFM. On SICA Models for HIV Transmission. STUDIES IN SYSTEMS, DECISION AND CONTROL 2020. [DOI: 10.1007/978-3-030-49896-2_6] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 12/14/2022]
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17
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A stochastic analysis of the impact of fluctuations in the environment on pre-exposure prophylaxis for HIV infection. Soft comput 2019. [DOI: 10.1007/s00500-019-04611-1] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/25/2022]
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Tovar M, Arregui S, Marinova D, Martín C, Sanz J, Moreno Y. Bridging the gap between efficacy trials and model-based impact evaluation for new tuberculosis vaccines. Nat Commun 2019; 10:5457. [PMID: 31784512 PMCID: PMC6884451 DOI: 10.1038/s41467-019-13387-9] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/11/2019] [Accepted: 11/01/2019] [Indexed: 11/08/2022] Open
Abstract
In Tuberculosis (TB), given the complexity of its transmission dynamics, observations of reduced epidemiological risk associated with preventive interventions can be difficult to translate into mechanistic interpretations. Specifically, in clinical trials of vaccine efficacy, a readout of protection against TB disease can be mapped to multiple dynamical mechanisms, an issue that has been overlooked so far. Here, we describe this limitation and its effect on model-based evaluations of vaccine impact. Furthermore, we propose a methodology to analyze efficacy trials that circumvents it, leveraging a combination of compartmental models and stochastic simulations. Using our approach, we can disentangle the different possible mechanisms of action underlying vaccine protection effects against TB, conditioned to trial design, size, and duration. Our results unlock a deeper interpretation of the data emanating from efficacy trials of TB vaccines, which renders them more interpretable in terms of transmission models and translates into explicit recommendations for vaccine developers.
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Affiliation(s)
- Mario Tovar
- Institute for Bio-computation and Physics of Complex Systems (BIFI), University of Zaragoza, Zaragoza, Spain
- Department of Theoretical Physics, Faculty of Sciences, University of Zaragoza, Zaragoza, Spain
| | - Sergio Arregui
- Institute for Bio-computation and Physics of Complex Systems (BIFI), University of Zaragoza, Zaragoza, Spain
- Department of Theoretical Physics, Faculty of Sciences, University of Zaragoza, Zaragoza, Spain
| | - Dessislava Marinova
- Microbiology Department, Faculty of Medicine, University of Zaragoza, Zaragoza, Spain
- Networked Biomedical Research Center on Respiratory Disease CIBERES, Madrid, Spain
| | - Carlos Martín
- Microbiology Department, Faculty of Medicine, University of Zaragoza, Zaragoza, Spain
- Networked Biomedical Research Center on Respiratory Disease CIBERES, Madrid, Spain
- Service of Microbiology, Miguel Servet Hospital, IIS Aragon, Zaragoza, Spain
| | - Joaquín Sanz
- Institute for Bio-computation and Physics of Complex Systems (BIFI), University of Zaragoza, Zaragoza, Spain.
- Department of Theoretical Physics, Faculty of Sciences, University of Zaragoza, Zaragoza, Spain.
- Department of Medicine, Genetics Section, University of Chicago, Chicago, IL, USA.
| | - Yamir Moreno
- Institute for Bio-computation and Physics of Complex Systems (BIFI), University of Zaragoza, Zaragoza, Spain.
- Department of Theoretical Physics, Faculty of Sciences, University of Zaragoza, Zaragoza, Spain.
- ISI Foundation, Turin, Italy.
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Silva CJ, Torres DF. Stability of a fractional HIV/AIDS model. MATHEMATICS AND COMPUTERS IN SIMULATION 2019. [DOI: 10.1016/j.matcom.2019.03.016] [Citation(s) in RCA: 8] [Impact Index Per Article: 1.6] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 02/08/2023]
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20
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Gromov D, Bulla I, Romero-Severson EO. Systematic evaluation of the population-level effects of alternative treatment strategies on the basic reproduction number. J Theor Biol 2018; 462:381-390. [PMID: 30500598 DOI: 10.1016/j.jtbi.2018.11.029] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/05/2018] [Revised: 10/25/2018] [Accepted: 11/26/2018] [Indexed: 11/28/2022]
Abstract
An approach to estimate the influence of the treatment-type controls on the basic reproduction number, R0, is proposed and elaborated. The presented approach allows one to estimate the effect of a given treatment strategy or to compare a number of different treatment strategies on the basic reproduction number. All our results are valid for sufficiently small values of the control. However, in many cases it is possible to extend this analysis to larger values of the control as was illustrated by examples.
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Affiliation(s)
- Dmitry Gromov
- Faculty of Applied Mathematics and Control Processes, Saint Petersburg State University, St. Petersburg, Russia.
| | - Ingo Bulla
- Institut für Mathematik und Informatik, Universität Greifswald, Walther-Rathenau-Straße 47, Greifswald 17487, Germany; Université Perpignan Via Domitia, IHPE UMR 5244, CNRS, Perpignan F-66860, France.
| | - Ethan O Romero-Severson
- Theoretical Biology and Biophysics Group, Los Alamos National Laboratory, Los Alamos, New Mexico, USA.
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21
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Gao DP, Huang NJ. Optimal control analysis of a tuberculosis model. APPLIED MATHEMATICAL MODELLING 2018; 58:47-64. [PMID: 32287942 PMCID: PMC7117058 DOI: 10.1016/j.apm.2017.12.027] [Citation(s) in RCA: 16] [Impact Index Per Article: 2.7] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/26/2017] [Revised: 10/16/2017] [Accepted: 12/19/2017] [Indexed: 05/30/2023]
Abstract
In this paper, we extend the model of Liu and Zhang (Math Comput Model 54:836-845, 2011) by incorporating three control terms and apply optimal control theory to the resulting model. Optimal control strategies are proposed to minimize both the disease burden and the intervention cost. We prove the existence and uniqueness of optimal control paths and obtain these optimal paths analytically using Pontryagin's Maximum Principle. We analyse our results numerically to compare various strategies of proposed controls. It is observed that implementation of three controls is most effective and less expensive among all the strategies. Thus, we conclude that in order to reduce tuberculosis threat all the three controls must be taken into consideration concurrently.
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Affiliation(s)
- Da-Peng Gao
- Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, PR China
- School of Mathematics and Information, China West Normal University, Nanchong, Sichuan 637009, PR China
| | - Nan-Jing Huang
- Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, PR China
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22
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Optimal Control Strategy for TB-HIV/AIDS Co-Infection Model in the Presence of Behaviour Modification. Processes (Basel) 2018. [DOI: 10.3390/pr6050048] [Citation(s) in RCA: 14] [Impact Index Per Article: 2.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/05/2023] Open
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23
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Analysis and Optimal Control of an Intracellular Delayed HIV Model with CTL Immune Response. MATHEMATICS IN COMPUTER SCIENCE 2018. [DOI: 10.1007/s11786-018-0333-9] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 02/05/2023]
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24
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Area I, NdaÏrou F, J. Nieto J, J. Silva C, F. M. Torres D. Ebola model and optimal control with vaccination constraints. ACTA ACUST UNITED AC 2018. [DOI: 10.3934/jimo.2017054] [Citation(s) in RCA: 17] [Impact Index Per Article: 2.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 02/05/2023]
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25
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Rodrigues F, J. Silva C, F. M. Torres D, Maurer H. Optimal control of a delayed HIV model. DISCRETE & CONTINUOUS DYNAMICAL SYSTEMS - B 2018. [DOI: 10.3934/dcdsb.2018030] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 02/05/2023]
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26
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J. Silva C, F. M. Torres D. Modeling and optimal control of HIV/AIDS prevention through PrEP. ACTA ACUST UNITED AC 2018. [DOI: 10.3934/dcdss.2018008] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 02/05/2023]
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27
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An epidemic model for cholera with optimal control treatment. JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS 2017. [DOI: 10.1016/j.cam.2016.11.002] [Citation(s) in RCA: 12] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 02/08/2023]
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28
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Silva CJ, Torres DF. A SICA compartmental model in epidemiology with application to HIV/AIDS in Cape Verde. ECOLOGICAL COMPLEXITY 2017. [DOI: 10.1016/j.ecocom.2016.12.001] [Citation(s) in RCA: 19] [Impact Index Per Article: 2.7] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 02/08/2023]
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Tang B, Xiao Y, Wu J. Implication of vaccination against dengue for Zika outbreak. Sci Rep 2016; 6:35623. [PMID: 27774987 PMCID: PMC5075941 DOI: 10.1038/srep35623] [Citation(s) in RCA: 25] [Impact Index Per Article: 3.1] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/11/2016] [Accepted: 10/04/2016] [Indexed: 01/23/2023] Open
Abstract
Zika virus co-circulates with dengue in tropical and sub-tropical regions. Cases of co-infection by dengue and Zika have been reported, the implication of this co-infection for an integrated intervention program for controlling both dengue and Zika must be addressed urgently. Here, we formulate a mathematical model to describe the transmission dynamics of co-infection of dengue and Zika with particular focus on the effects of Zika outbreak by vaccination against dengue among human hosts. Our analysis determines specific conditions under which vaccination against dengue can significantly increase the Zika outbreak peak, and speed up the Zika outbreak peak timing. Our results call for further study about the co-infection to direct an integrated control to balance the benefits for dengue control and the damages of Zika outbreak.
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Affiliation(s)
- Biao Tang
- School of Mathematics and Statistics, Xi'an Jiaotong University, Xi'an 710049, PR China
- Centre for Disease Modelling, York Institute for Health Research, York University, Toronto, ON, M3J 1P3, Canada
| | - Yanni Xiao
- School of Mathematics and Statistics, Xi'an Jiaotong University, Xi'an 710049, PR China
| | - Jianhong Wu
- Centre for Disease Modelling, York Institute for Health Research, York University, Toronto, ON, M3J 1P3, Canada
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31
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Stability and optimal control of a delayed HIV model. MATHEMATICAL METHODS IN THE APPLIED SCIENCES 2016. [DOI: 10.1002/mma.4207] [Citation(s) in RCA: 12] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 02/05/2023]
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