1
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Alshorman A, Al-Hosainat N, Jackson T. Analysis of HIV latent infection model with multiple infection stages and different drug classes. JOURNAL OF BIOLOGICAL DYNAMICS 2022; 16:713-732. [PMID: 36264087 DOI: 10.1080/17513758.2022.2113828] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/17/2021] [Accepted: 08/09/2022] [Indexed: 06/16/2023]
Abstract
Latently infected CD4+ T cells represent one of the major obstacles to HIV eradication even after receiving prolonged highly active anti-retroviral therapy (HAART). Long-term use of HAART causes the emergence of drug-resistant virus which is then involved in HIV transmission. In this paper, we develop mathematical HIV models with staged disease progression by incorporating entry inhibitor and latently infected cells. We find that entry inhibitor has the same effect as protease inhibitor on the model dynamics and therefore would benefit HIV patients who developed resistance to many of current anti-HIV medications. Numerical simulations illustrate the theoretical results and show that the virus and latently infected cells reach an infected steady state in the absence of treatment and are eliminated under treatment whereas the model including homeostatic proliferation of latently infected cells maintains the virus at low level during suppressive treatment. Therefore, complete cure of HIV needs complete eradication of latent reservoirs.
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Affiliation(s)
- Areej Alshorman
- Department of Mathematics, University of Michigan, Ann Arbor, MI, USA
| | | | - Trachette Jackson
- Department of Mathematics, University of Michigan, Ann Arbor, MI, USA
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2
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Al Agha AD, Elaiw AM, Azoz SA, Ramadan E. Stability analysis of within-host SARS-CoV-2/HIV coinfection model. MATHEMATICAL METHODS IN THE APPLIED SCIENCES 2022; 45:MMA8457. [PMID: 35942149 PMCID: PMC9348514 DOI: 10.1002/mma.8457] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 09/04/2021] [Revised: 04/24/2022] [Accepted: 05/21/2022] [Indexed: 06/15/2023]
Abstract
The world has been suffering from the coronavirus disease 2019 (COVID-19) since late 2019. COVID-19 is caused by a virus called the severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2). The human immunodeficiency virus (HIV) coinfection with SARS-CoV-2 has been reported in many patients around the world. This has raised the alarm for the importance of understanding the dynamics of coinfection and its impact on the lives of patients. As in other pandemics, mathematical modeling is one of the important tools that can help medical and experimental studies of COVID-19. In this paper, we develop a within-host SARS-CoV-2/HIV coinfection model. The model consists of six ordinary differential equations. It depicts the interactions between uninfected epithelial cells, infected epithelial cells, free SARS-CoV-2 particles, uninfected CD4+ T cells, infected CD4+ T cells, and free HIV particles. We confirm that the solutions of the developed model are biologically acceptable by proving their nonnegativity and boundedness. We compute all possible steady states and derive their positivity conditions. We choose suitable Lyapunov functions to prove the global asymptotic stability of all steady states. We run some numerical simulations to enhance the global stability results. Based on our model, weak CD4+ T cell immune response or low CD4+ T cell counts in SARS-CoV-2/HIV coinfected patient increase the concentrations of infected epithelial cells and SARS-CoV-2 viral load. This causes the coinfected patient to suffer from severe SARS-CoV-2 infection. This result agrees with many studies which showed that HIV patients are at greater risk of suffering from severe COVID-19 when infected. More studies are needed to understand the nature of SARS-CoV-2/HIV coinfection and the role of different immune responses during infection.
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Affiliation(s)
- Afnan D. Al Agha
- Department of Mathematical Science, College of EngineeringUniversity of Business and TechnologyJeddahSaudi Arabia
| | - Ahmed M. Elaiw
- Department of Mathematics, Faculty of ScienceKing Abdulaziz UniversityJeddahSaudi Arabia
- Department of Mathematics, Faculty of ScienceAl‐Azhar UniversityAssiutEgypt
| | - Shaimaa A. Azoz
- Department of Mathematics, Faculty of ScienceAssiut UniversityAssiutEgypt
| | - Esraa Ramadan
- Department of Mathematics, Faculty of ScienceAssiut UniversityAssiutEgypt
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3
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Uniform Persistence and Global Attractivity in a Delayed Virus Dynamic Model with Apoptosis and Both Virus-to-Cell and Cell-to-Cell Infections. MATHEMATICS 2022. [DOI: 10.3390/math10060975] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/16/2022]
Abstract
In this paper, we study the global dynamics of a delayed virus dynamics model with apoptosis and both virus-to-cell and cell-to-cell infections. When the basic reproduction number R0>1, we obtain the uniform persistence of the model, and give some explicit expressions of the ultimate upper and lower bounds of any positive solution of the model. In addition, by constructing the appropriate Lyapunov functionals, we obtain some sufficient conditions for the global attractivity of the disease-free equilibrium and the chronic infection equilibrium of the model. Our results extend existing related works.
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Elaiw AM, Al Agha AD, Azoz SA, Ramadan E. Global analysis of within-host SARS-CoV-2/HIV coinfection model with latency. EUROPEAN PHYSICAL JOURNAL PLUS 2022; 137:174. [PMID: 35106266 PMCID: PMC8793338 DOI: 10.1140/epjp/s13360-022-02387-2] [Citation(s) in RCA: 2] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/16/2021] [Accepted: 01/04/2022] [Indexed: 05/05/2023]
Abstract
The coronavirus disease 2019 (COVID-19) is a respiratory disease caused by a virus called the severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2). In this paper, we analyze a within-host SARS-CoV-2/HIV coinfection model. The model is made up of eight ordinary differential equations. These equations describe the interactions between healthy epithelial cells, latently infected epithelial cells, productively infected epithelial cells, SARS-CoV-2 particles, healthy CD 4 + T cells, latently infected CD 4 + T cells, productively infected CD 4 + T cells, and HIV particles. We confirm that the solutions of the developed model are bounded and nonnegative. We calculate the different steady states of the model and derive their existence conditions. We choose appropriate Lyapunov functions to show the global stability of all steady states. We execute some numerical simulations to assist the theoretical contributions. Based on our results, weak CD 4 + T cell immunity in SARS-CoV-2/HIV coinfected patients causes an increase in the concentrations of productively infected epithelial cells and SARS-CoV-2 particles. This may lead to severe SARS-CoV-2 infection in HIV patients. This result agrees with many studies that discussed the high risk of severe infection and death in HIV patients when they get SARS-CoV-2 infection. On the other hand, increasing the death rate of infected epithelial cells during the latency period can reduce the severity of SARS-CoV-2 infection in HIV patients. More studies are needed to understand the dynamics of SARS-CoV-2/HIV coinfection and find better ways to treat this vulnerable group of patients.
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Affiliation(s)
- A M Elaiw
- Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah, 21589 Saudi Arabia
- Department of Mathematics, Faculty of Science, Al-Azhar University, Assiut Branch, Assiut, Egypt
| | - A D Al Agha
- Department of Mathematical Science, College of Engineering, University of Business and Technology, Jeddah, 21361 Saudi Arabia
| | - S A Azoz
- Department of Mathematics, Faculty of Science, Assiut University, Assiut, Egypt
| | - E Ramadan
- Department of Mathematics, Faculty of Science, Assiut University, Assiut, Egypt
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5
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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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Chhetri B, Bhagat VM, Muthusamy S, Ananth VS, Vamsi DKK, Sanjeevi CB. Time Optimal Control Studies on COVID-19 Incorporating Adverse Events of the Antiviral Drugs. COMPUTATIONAL AND MATHEMATICAL BIOPHYSICS 2021. [DOI: 10.1515/cmb-2020-0125] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/15/2022] Open
Abstract
Abstract
COVID -19 pandemic has resulted in more than 257 million infections and 5.15 million deaths worldwide. Several drug interventions targeting multiple stages of the pathogenesis of COVID -19 can significantly reduce induced infection and thus mortality. In this study, we first develop SIV model at within-host level by incorporating the intercellular time delay and analyzing the stability of equilibrium points. The model dynamics admits a disease-free equilibrium and an infected equilibrium with their stability based on the value of the basic reproduction number R
0. We then formulate an optimal control problem with antiviral drugs and second-line drugs as control measures and study their roles in reducing the number of infected cells and viral load. The comparative study conducted in the optimal control problem suggests that if the first-line antiviral drugs show adverse effects, considering these drugs in reduced amounts along with the second-line drugs would be very effective in reducing the number of infected cells and viral load in a COVID-19 infected patient. Later, we formulate a time-optimal control problem with the goal of driving the system from any initial state to the desired infection-free equilibrium state in finite minimal time. Using Pontryagin’s Minimum Principle, it is shown that the optimal control strategy is of the bang-bang type, with the possibility of switching between two extreme values of the optimal controls. Numerically, it is shown that the desired infection-free state is achieved in a shorter time when the higher values of the optimal controls. The results of this study may be very helpful to researchers, epidemiologists, clinicians and physicians working in this field.
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Affiliation(s)
- Bishal Chhetri
- Department of Mathematics and Computer Science , Sri Sathya Sai Institute of Higher Learning - SSSIHL , India
| | - Vijay M. Bhagat
- Central Leprosy Teaching and Research Institute - CLTRI , Chennai , India
| | - Swapna Muthusamy
- Central Leprosy Teaching and Research Institute - CLTRI , Chennai , India
| | - V S Ananth
- Department of Mathematics and Computer Science , Sri Sathya Sai Institute of Higher Learning - SSSIHL , India
| | - D. K. K. Vamsi
- Department of Mathematics and Computer Science , Sri Sathya Sai Institute of Higher Learning - SSSIHL , India
| | - Carani B Sanjeevi
- Central Leprosy Teaching and Research Institute - CLTRI , Chennai , India ; Department of Medicine , Karolinska Institute , Stockholm , Sweden , E-mail:
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7
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Bugalia S, Tripathi JP, Wang H. Mathematical modeling of intervention and low medical resource availability with delays: Applications to COVID-19 outbreaks in Spain and Italy. MATHEMATICAL BIOSCIENCES AND ENGINEERING : MBE 2021; 18:5865-5920. [PMID: 34517515 DOI: 10.3934/mbe.2021295] [Citation(s) in RCA: 9] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/13/2023]
Abstract
Infectious diseases have been one of the major causes of human mortality, and mathematical models have been playing significant roles in understanding the spread mechanism and controlling contagious diseases. In this paper, we propose a delayed SEIR epidemic model with intervention strategies and recovery under the low availability of resources. Non-delayed and delayed models both possess two equilibria: the disease-free equilibrium and the endemic equilibrium. When the basic reproduction number $ R_0 = 1 $, the non-delayed system undergoes a transcritical bifurcation. For the delayed system, we incorporate two important time delays: $ \tau_1 $ represents the latent period of the intervention strategies, and $ \tau_2 $ represents the period for curing the infected individuals. Time delays change the system dynamics via Hopf-bifurcation and oscillations. The direction and stability of delay induced Hopf-bifurcation are established using normal form theory and center manifold theorem. Furthermore, we rigorously prove that local Hopf bifurcation implies global Hopf bifurcation. Stability switching curves and crossing directions are analyzed on the two delay parameter plane, which allows both delays varying simultaneously. Numerical results demonstrate that by increasing the intervention strength, the infection level decays; by increasing the limitation of treatment, the infection level increases. Our quantitative observations can be useful for exploring the relative importance of intervention and medical resources. As a timing application, we parameterize the model for COVID-19 in Spain and Italy. With strict intervention policies, the infection numbers would have been greatly reduced in the early phase of COVID-19 in Spain and Italy. We also show that reducing the time delays in intervention and recovery would have decreased the total number of cases in the early phase of COVID-19 in Spain and Italy. Our work highlights the necessity to consider the time delays in intervention and recovery in an epidemic model.
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Affiliation(s)
- Sarita Bugalia
- Department of Mathematics, Central University of Rajasthan, Bandar Sindri, Kishangarh-305817, Ajmer, Rajasthan, India
| | - Jai Prakash Tripathi
- Department of Mathematics, Central University of Rajasthan, Bandar Sindri, Kishangarh-305817, Ajmer, Rajasthan, India
| | - Hao Wang
- Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton AB T6G 2G1, Canada
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8
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Li B, Jiao F. A delayed HIV-1 model with cell-to-cell spread and virus waning. JOURNAL OF BIOLOGICAL DYNAMICS 2020; 14:802-825. [PMID: 33084532 DOI: 10.1080/17513758.2020.1836272] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/21/2020] [Accepted: 09/29/2020] [Indexed: 06/11/2023]
Abstract
In this paper, we propose and analyse a delayed HIV-1 model with both viral and cellular transmissions and virus waning. We obtain the threshold dynamics of the proposed model, characterized by the basic reproduction number R0 . If R0<1 , the infection-free steady state is globally asymptotically stable; whereas if R0>1 , the system is uniformly persistent. When the delays are positive, we show that the intracellular delays in both viral and cellular infections may lead to stability switches of the infected steady state. Both analytical and numerical results indicate that if the effect of cell-to-cell transmission is ignored, then the risk of HIV-1 infection will be underestimated. Moreover, the viral load of model without virus waning is higher than the one of model with virus waning. These results highlight the important role of two ways of viral transmission and virus waning on HIV-1 infection.
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Affiliation(s)
- Bing Li
- School of Mathematical Science, Harbin Normal University, Harbin, People's Republic of China
| | - Feng Jiao
- Center for Applied Mathematics, Guangzhou University, Guangzhou, People's Republic of China
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9
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Naeem Jan M, Ali N, Zaman G, Ahmad I, Shah Z, Kumam P. HIV-1 infection dynamics and optimal control with Crowley-Martin function response. COMPUTER METHODS AND PROGRAMS IN BIOMEDICINE 2020; 193:105503. [PMID: 32387782 DOI: 10.1016/j.cmpb.2020.105503] [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: 03/09/2020] [Revised: 04/02/2020] [Accepted: 04/06/2020] [Indexed: 06/11/2023]
Abstract
BACKGROUND AND OBJECTIVE As we all know, mathematical models provide very important information for the study of the human immunodeficiency virus type. Mathematical model of human immunodeficiency virus type-1 (HIV-1) infection with contact rate represented by Crowley-Martin function response is taken into account. The aims of this novel study is to checkthe local and global stability of the disease and also prevent the outbreak from the community. METHODS The mathematical model as well as optimal system of nonlinear differential equations are tackled numerically by Runge-Kutta fourth-order method. For global stability we use Lyapunov-LaSalle invariance principle and for the description of optimal control, Pontryagin's maximum principle is used. RESULTS Graphical results are depicted and examined with different parameters values versus the basic reproductive number R0 and also the plots with and without control. The density of infected cells continued to increase without treatment, but the concentration of these cells decreased after treatment. The intensity of the pathogenic virus before and after the optimal treatment. This indicates a sharp drop in the rate of pathogenic viruses after treatment. It prevents the production of viruses by preventing cell infection and minimizing side effects. CONCLUSIONS We analysed the model by defining the basic reproductive number, showing the boundedness, positivity and permanence of the solution, and proving the local and global stability of the infection-free state. We show that the threshold quantity R0 < 1, the elimination of HIV-1 infection from the T cell population, is eradicated; while for the threshold quantity R0 > 1, HIV-1 infection remains in the host. When the threshold quantity R0 > 1, then it shows that the steady-state of chronic disease is globally stable. Optimal control strategies are developed with the optimal control pair for the description of optimal control. To reduce the density of infected cells and viruses as well as maximize the density of healthy cells is determined by the objective functional.
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Affiliation(s)
- Muhammad Naeem Jan
- 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.
| | - Gul Zaman
- 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
- Center of Excellence in Theoretical and Computational Science (TaCS-CoE), Science Laboratory Building, Faculty of Science, King Mongkut's University of Technology Thonburi (KMUTT), 126 Pracha-Uthit Road, Bang Mod, Thrung Khru, Bangkok 10140, Thailand.
| | - Poom Kumam
- KMUTT-Fixed Point Theory and Applications Research Group, Theoretical and, Computational Science Center (TaCS), Science Laboratory Building, Faculty of Science, King, Mongkut's University of Technology Thonburi (KMUTT), 126 Pracha-Uthit Road, Bang Mod, Thrung Khru, Bangkok 10140, Thailand; Department of Medical Research, China Medical University Hospital, China Medical, University, Taichung, Taiwan.
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10
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Viral Infection Dynamics Model Based on a Markov Process with Time Delay between Cell Infection and Progeny Production. MATHEMATICS 2020. [DOI: 10.3390/math8081207] [Citation(s) in RCA: 4] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 12/22/2022]
Abstract
Many human virus infections including those with the human immunodeficiency virus type 1 (HIV) are initiated by low numbers of founder viruses. Therefore, random effects have a strong influence on the initial infection dynamics, e.g., extinction versus spread. In this study, we considered the simplest (so-called, ‘consensus’) virus dynamics model and incorporated a delay between infection of a cell and virus progeny release from the infected cell. We then developed an equivalent stochastic virus dynamics model that accounts for this delay in the description of the random interactions between the model components. The new model is used to study the statistical characteristics of virus and target cell populations. It predicts the probability of infection spread as a function of the number of transmitted viruses. A hybrid algorithm is suggested to compute efficiently the system dynamics in state space domain characterized by the mix of small and large species densities.
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11
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Geng Y, Xu J. Stability and bifurcation analysis for a delayed viral infection model with full logistic proliferation. INT J BIOMATH 2020. [DOI: 10.1142/s1793524520500333] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/18/2022]
Abstract
In this paper, we study a delayed viral infection model with cellular infection and full logistic proliferations for both healthy and infected cells. The global asymptotic stabilities of the equilibria are studied by constructing Lyapunov functionals. Moreover, we investigated the existence of Hopf bifurcation at the infected equilibrium by regarding the possible combination of the two delays as bifurcation parameters. The results show that time delays may destabilize the infected equilibrium and lead to Hopf bifurcation. Finally, numerical simulations are carried out to illustrate the main results and explore the dynamics including Hopf bifurcation and stability switches.
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Affiliation(s)
- Yan Geng
- School of Science, Xi’an Polytechnic University, Xi’an 710048, Shaanxi, P. R. China
| | - Jinhu Xu
- School of Sciences, Xi’an University of Technology, Xi’an 710049, Shaanxi, P. R. China
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12
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Mathematical Model Describing HIV Infection with Time-Delayed CD4 T-Cell Activation. Processes (Basel) 2020. [DOI: 10.3390/pr8070782] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/22/2023] Open
Abstract
A mathematical model composed of two non-linear differential equations that describe the population dynamics of CD4 T-cells in the human immune system, as well as viral HIV viral load, is proposed. The invariance region is determined, classical equilibrium stability analysis is performed by using the basic reproduction number, and numerical simulations are carried out to illustrate stability results. Thereafter, the model is modified with a delay term, describing the time required for CD4 T-cell immunological activation. This generates a two-dimensional integro-differential system, which is transformed into a system with three ordinary differential equations. For the new model, equilibriums are determined, their local stability is examined, and results are studied by way of numerical simulation.
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13
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Yang J, Bi S. Stability and Hopf bifurcation of a delayed virus infection model with latently infected cells and Beddington–DeAngelis incidence. INT J BIOMATH 2020. [DOI: 10.1142/s179352452050045x] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/18/2022]
Abstract
In this paper, the dynamical behaviors for a five-dimensional virus infection model with Latently Infected Cells and Beddington–DeAngelis incidence are investigated. In the model, four delays which denote the latently infected delay, the intracellular delay, virus production period and CTL response delay are considered. We define the basic reproductive number and the CTL immune reproductive number. By using Lyapunov functionals, LaSalle’s invariance principle and linearization method, the threshold conditions on the stability of each equilibrium are established. It is proved that when the basic reproductive number is less than or equal to unity, the infection-free equilibrium is globally asymptotically stable; when the CTL immune reproductive number is less than or equal to unity and the basic reproductive number is greater than unity, the immune-free infection equilibrium is globally asymptotically stable; when the CTL immune reproductive number is greater than unity and immune response delay is equal to zero, the immune infection equilibrium is globally asymptotically stable. The results show that immune response delay may destabilize the steady state of infection and lead to Hopf bifurcation. The existence of the Hopf bifurcation is discussed by using immune response delay as a bifurcation parameter. Numerical simulations are carried out to justify the analytical results.
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Affiliation(s)
- Junxian Yang
- School of Science, Anhui Agricultural University, Hefei 230036, P. R. China
| | - Shoudong Bi
- School of Science, Anhui Agricultural University, Hefei 230036, P. R. China
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14
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Fatehi F, Kyrychko YN, Blyuss KB. Stochastic dynamics in a time-delayed model for autoimmunity. Math Biosci 2020; 322:108323. [PMID: 32092469 DOI: 10.1016/j.mbs.2020.108323] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/27/2019] [Revised: 01/21/2020] [Accepted: 02/10/2020] [Indexed: 12/18/2022]
Abstract
In this paper we study interactions between stochasticity and time delays in the dynamics of immune response to viral infections, with particular interest in the onset and development of autoimmune response. Starting with a deterministic time-delayed model of immune response to infection, which includes cytokines and T cells with different activation thresholds, we derive an exact delayed chemical master equation for the probability density. We use system size expansion and linear noise approximation to explore how variance and coherence of stochastic oscillations depend on parameters, and to show that stochastic oscillations become more regular when regulatory T cells become more effective at clearing autoreactive T cells. Reformulating the model as an Itô stochastic delay differential equation, we perform numerical simulations to illustrate the dynamics of the model and associated probability distributions in different parameter regimes. The results suggest that even in cases where the deterministic model has stable steady states, in individual stochastic realisations, the model can exhibit sustained stochastic oscillations, whose variance increases as one gets closer to the deterministic stability boundary. Furthermore, in the regime of bi-stability, whereas deterministically the system would approach one of the steady states (or periodic solutions) depending on the initial conditions, due to the presence of stochasticity, it is now possible for the system to reach both of those dynamical states with certain probability. Biological significance of this result lies in highlighting the fact that since normally in a laboratory or clinical setting one would observe a single individual realisation of the course of the disease, even for all parameters characterising the immune system and the strength of infection being the same, there is a proportion of cases where a spontaneous recovery can be observed, and similarly, where a disease can develop in a situation that otherwise would result in a normal disease clearance.
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Affiliation(s)
- Farzad Fatehi
- Department of Mathematics, University of York, York YO10 5DD, UK.
| | - Yuliya N Kyrychko
- Department of Mathematics, University of Sussex, Falmer, Brighton BN1 9QH, UK.
| | - Konstantin B Blyuss
- Department of Mathematics, University of Sussex, Falmer, Brighton BN1 9QH, UK.
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15
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Li M, Zu J. The review of differential equation models of HBV infection dynamics. J Virol Methods 2019; 266:103-113. [PMID: 30716348 DOI: 10.1016/j.jviromet.2019.01.014] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/06/2018] [Revised: 12/30/2018] [Accepted: 01/24/2019] [Indexed: 12/21/2022]
Abstract
Understanding the infection and pathogenesis mechanism of hepatitis B virus (HBV) is very important for the prevention and treatment of hepatitis B. Mathematical models contribute to illuminate the dynamic process of HBV replication in vivo. Therefore, in this paper we review the viral dynamics in HBV infection, which may help us further understand the dynamic mechanism of HBV infection and efficacy of antiviral treatment. Firstly, we introduce a family of deterministic models by considering different biological mechanisms, such as, antiviral therapy, CTL immune response, multi-types of infected hepatocytes, time delay and spatial diffusion. Particularly, we briefly describe the stochastic models of HBV infection. Secondly, we introduce the commonly used parameter estimation methods for HBV viral dynamic models and briefly discuss how to use these methods to estimate unknown parameters (such as drug efficacy) through two specific examples. We also discuss the idea and method of model identification and use a specific example to illustrate its application. Finally, we propose three new research programs, namely, considering HBV drug-resistant strain, coupling within-host and between-host dynamics in HBV infection and linking population dynamics with evolutionary dynamics of HBV diversity.
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Affiliation(s)
- Miaolei Li
- School of Mathematics and Statistics, Xi'an Jiaotong University, Xi'an, Shaanxi, 710049, PR China
| | - Jian Zu
- School of Mathematics and Statistics, Xi'an Jiaotong University, Xi'an, Shaanxi, 710049, PR China.
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16
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Chamekh M, Elzaki TM, Brik N. Semi-analytical solution for some proportional delay differential equations. SN APPLIED SCIENCES 2019. [DOI: 10.1007/s42452-018-0130-8] [Citation(s) in RCA: 9] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/28/2022] Open
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17
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Miao H, Teng Z, Abdurahman X. Stability and Hopf bifurcation for a five-dimensional virus infection model with Beddington-DeAngelis incidence and three delays. JOURNAL OF BIOLOGICAL DYNAMICS 2018; 12:146-170. [PMID: 29198164 DOI: 10.1080/17513758.2017.1408861] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/06/2016] [Accepted: 11/17/2017] [Indexed: 06/07/2023]
Abstract
In this paper, the dynamical behaviours for a five-dimensional virus infection model with three delays which describes the interactions of antibody, cytotoxic T-lymphocyte (CTL) immune responses and Beddington-DeAngelis incidence are investigated. The reproduction numbers for virus infection, antibody immune response, CTL immune response, CTL immune competition and antibody immune competition, respectively, are calculated. By using the Lyapunov functionals and linearization method, the threshold conditions on the local and global stability of the equilibria for infection-free, immune-free, antibody response, CTL response and interior, respectively, are established. The existence of Hopf bifurcation with immune delay as a bifurcation parameter is investigated by using the bifurcation theory. Numerical simulations are presented to justify the analytical results.
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Affiliation(s)
- Hui Miao
- a School of Applied Mathematics , Shanxi University of Finance and Economics , Taiyuan , People's Republic of China
| | - Zhidong Teng
- b College of Mathematics and System Sciences , Xinjiang University , Urumqi , People's Republic of China
| | - Xamxinur Abdurahman
- b College of Mathematics and System Sciences , Xinjiang University , Urumqi , People's Republic of China
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18
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Sahani SK. A delayed HIV infection model with apoptosis and viral loss. JOURNAL OF BIOLOGICAL DYNAMICS 2018; 12:1012-1034. [PMID: 30462570 DOI: 10.1080/17513758.2018.1547427] [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: 12/11/2017] [Accepted: 11/06/2018] [Indexed: 06/09/2023]
Abstract
In this paper, a delayed human immunodeficiency virus (HIV) model with apoptosis of cells has been studied. Both immunological and intracellular delay have been incorporated to make the model more relevant. Firstly, the model has been investigated using local stability analysis. Next, the global stability analysis of steady states has been performed. The stability switch criteria taking the delay as the bifurcating parameter, leading to Hopf bifurcation has been studied. The transition of the system from order to chaos has been explored, and the analytical results have been verified by numerical simulations. The results thus can be used to describe the extensive dynamics exhibited by the model introduced in this article. The effects of apoptosis on viral load has been studied in the model numerically.
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Affiliation(s)
- Saroj Kumar Sahani
- a Faculty of Mathematics & Computer Science , South Asian University , New Delhi , India
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19
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Abstract
In this paper, an HIV infection model with eclipse phase, humoral immune response and immunological delay has been discussed. By studying the characteristic equations of the model, the local stability analysis of various equilibrium points has been explored. Treating the delay as the bifurcation parameter, it has been shown that the delay can destabilize the stability of the infected steady-state leading to Hopf bifurcation and periodic solutions. By using Lyapunov functionals and LaSalle’s invariance principle, the global stability analysis of the boundary equilibrium points has also been explored. It has also been shown numerically that the inclusion of the drug therapy in the model generates sporadic outbursts of virus called viral blips. In the end, numerical simulations have been employed to justify the analytical results proved in the paper. Biologically, the proposed model can explain the presence of viral blips in the system, during the introduction of HAART, as observed in the HIV-infected patient. These blips could mark the viral advancement in the system, thus resulting in complete immunological failure. One of the reasons for these viral blips can be the presence of delay in the activation of immunological response. But the development of drug-resistive virus could also be the reason for this sudden rise in viral loads. Moreover, the incorporation of the delay in the model generates oscillations and periodicity in the model, thus validating the long latency period seen in most HIV-infected patients. Also, a longer delay in the activation of immune response marks a viral advancement in the viral timeline resulting in viral blips, which signify the evolution of the virus.
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Affiliation(s)
- SAROJ KUMAR SAHANI
- Faculty of Mathematics and Computer Science, South Asian University, Akbar Bhawan, Chanakyapuri, New Delhi 110021, India
| | - YASHI
- Faculty of Mathematics and Computer Science, South Asian University, Akbar Bhawan, Chanakyapuri, New Delhi 110021, India
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20
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Sahani SK, Yashi. Effects of delay in immunological response of HIV infection. INT J BIOMATH 2018. [DOI: 10.1142/s1793524518500766] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/18/2022]
Abstract
In this paper, a human immunodeficiency virus (HIV) infection model with both the types of immune responses, the antibody and the killer cell immune responses has been introduced. The model has been made more logical by including two delays in the activation of both the immune responses, along with the combination drug therapy. The inclusion of both the delayed immune responses provides a greater understanding of long-term dynamics of the disease. The dependence of the stability of the steady states of the model on the reproduction number [Formula: see text] has been explored through stability theory. Moreover, the global stability analysis of the infection-free steady state and the infected steady state has been proved with respect to [Formula: see text]. The bifurcation analysis of the infected steady state with respect to both delays has been performed. Numerical simulations have been carried out to justify the results proved. This model is capable of explaining the long-term dynamics of HIV infection to a greater extent than that of the existing model as it captures some basic parameters involved in the system such as immunological delay and immune response. Similarly, the model also explains the basic understanding of the disease dynamics as a result of activation of the immune response toward the virus.
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Affiliation(s)
- Saroj Kumar Sahani
- Department of Mathematics, South Asian University, Akbar Bhawan, Chankyapuri, New Delhi 110021, India
| | - Yashi
- Department of Mathematics, South Asian University, Akbar Bhawan, Chankyapuri, New Delhi 110021, India
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21
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Jiang C, Wang K, Song L. Global dynamics of a delay virus model with recruitment and saturation effects of immune responses. MATHEMATICAL BIOSCIENCES AND ENGINEERING : MBE 2018; 14:1233-1246. [PMID: 29161858 DOI: 10.3934/mbe.2017063] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/07/2023]
Abstract
In this paper, we formulate a virus dynamics model with the recruitment of immune responses, saturation effects and an intracellular time delay. With the help of uniform persistence theory and Lyapunov method, we show that the global stability of the model is totally determined by the basic reproductive number R0. Furthermore, we analyze the effects of the recruitment of immune responses on virus infection by numerical simulation. The results show ignoring the recruitment of immune responses will result in overestimation of the basic reproductive number and the severity of viral infection.
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Affiliation(s)
- Cuicui Jiang
- Department of Mathematics, School of Biomedical Engineering, Third Military Medical University, Chongqing 400038, China
| | - Kaifa Wang
- Department of Mathematics, School of Biomedical Engineering, Third Military Medical University, Chongqing 400038, China
| | - Lijuan Song
- Department of Mathematics, School of Biomedical Engineering, Third Military Medical University, Chongqing 400038, China
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22
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Jenner A, Yun CO, Yoon A, Kim PS, Coster ACF. Modelling heterogeneity in viral-tumour dynamics: The effects of gene-attenuation on viral characteristics. J Theor Biol 2018; 454:41-52. [PMID: 29857083 DOI: 10.1016/j.jtbi.2018.05.030] [Citation(s) in RCA: 8] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/18/2017] [Revised: 05/25/2018] [Accepted: 05/28/2018] [Indexed: 01/16/2023]
Abstract
The use of viruses as a cancer treatment is becoming increasingly more robust; however, there is still a long way to go before a completely successful treatment is formulated. One major challenge in the field is to select which virus, out of a burgeoning number of oncolytic viruses and engineered derivatives, can maximise both treatment spread and anticancer cytotoxicity. To assist in solving this problem, an in-depth understanding of the virus-tumour interaction is crucial. In this article, we present a novel integro-differential system with distributed delays embodying the dynamics of an oncolytic adenovirus with a fixed population of tumour cells in vitro, allowing for heterogeneity to exist in the virus and cell populations. The parameters of the model are optimised in a hierarchical manner, the purpose of which is not to obtain a perfect representation of the data. Instead, we place our parameter values in the correct region of the parameter space. Due to the sparse nature of the data it is not possible to obtain the parameter values with any certainty, but rather we demonstrate the suitability of the model. Using our model we quantify how modifications to the viral genome alter the viral characteristics, specifically how the attenuation of the E1B 19 and E1B 55 gene affect the system performance, and identify the dominant processes altered by the mutations. From our analysis, we conclude that the deletion of the E1B 55 gene significantly reduces the replication rate of the virus in comparison to the deletion of the E1B 19 gene. We also found that the deletion of both the E1B 19 and E1B 55 genes resulted in a long delay in the average replication start time of the virus. This leads us to propose the use of E1B 19 gene-attenuated adenovirus for cancer therapy, as opposed to E1B 55 gene-attenuated adenoviruses.
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Affiliation(s)
- Adrianne Jenner
- School of Mathematics and Statistics, University of Sydney, Sydney, NSW, Australia
| | - Chae-Ok Yun
- Department of Bioengineering, College of Engineering, Hanyang University, Seoul, South Korea
| | - Arum Yoon
- Department of Bioengineering, College of Engineering, Hanyang University, Seoul, South Korea
| | - Peter S Kim
- School of Mathematics and Statistics, University of Sydney, Sydney, NSW, Australia
| | - Adelle C F Coster
- School of Mathematics and Statistics, University of New South Wales, Sydney, NSW, Australia.
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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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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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25
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Miao H, Teng Z, Abdurahman X, Li Z. Global stability of a diffusive and delayed virus infection model with general incidence function and adaptive immune response. COMPUTATIONAL AND APPLIED MATHEMATICS 2018. [PMCID: PMC7149116 DOI: 10.1007/s40314-017-0543-9] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/13/2023]
Abstract
In this paper, the dynamical behaviors for a five-dimensional virus infection model with diffusion and two delays which describes the interactions of antibody, cytotoxic T-lymphocyte (CTL) immune responses and a general incidence function are investigated. The reproduction numbers for virus infection, antibody immune response, CTL immune response, CTL immune competition and antibody immune competition, respectively, are calculated. By using the Lyapunov functionals and linearization methods, the threshold conditions on the global stability of the equilibria for infection-free, immune-free, antibody response, CTL response and antibody and CTL responses, respectively, are established if the space is assumed as homogeneous. When the space is inhomogeneous, the effects of diffusion, intracellular delay and production delay are obtained by the numerical simulations.
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Affiliation(s)
- Hui Miao
- School of Applied Mathematics, Shanxi University of Finance and Economics, Taiyuan, 030006 People’s Republic of China
| | - Zhidong Teng
- College of Mathematics and System Sciences, Xinjiang University, Urumqi, 830046 People’s Republic of China
| | - Xamxinur Abdurahman
- College of Mathematics and System Sciences, Xinjiang University, Urumqi, 830046 People’s Republic of China
| | - Zhiming Li
- College of Mathematics and System Sciences, Xinjiang University, Urumqi, 830046 People’s Republic of China
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26
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Wen B, Wang J, Teng Z. A discrete-time analog for coupled within-host and between-host dynamics in environmentally driven infectious disease. ADVANCES IN DIFFERENCE EQUATIONS 2018; 2018:69. [PMID: 32226450 PMCID: PMC7100524 DOI: 10.1186/s13662-018-1522-1] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/26/2017] [Accepted: 02/09/2018] [Indexed: 05/03/2023]
Abstract
In this paper, we establish a discrete-time analog for coupled within-host and between-host systems for an environmentally driven infectious disease with fast and slow two time scales by using the non-standard finite difference scheme. The system is divided into a fast time system and a slow time system by using the idea of limit equations. For the fast system, the positivity and boundedness of the solutions, the basic reproduction number and the existence for infection-free and unique virus infectious equilibria are obtained, and the threshold conditions on the local stability of equilibria are established. In the slow system, except for the positivity and boundedness of the solutions, the existence for disease-free, unique endemic and two endemic equilibria are obtained, and the sufficient conditions on the local stability for disease-free and unique endemic equilibria are established. To return to the coupling system, the local stability for the virus- and disease-free equilibrium, and virus infectious but disease-free equilibrium is established. The numerical examples show that an endemic equilibrium is locally asymptotically stable and the other one is unstable when there are two endemic equilibria.
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Affiliation(s)
- Buyu Wen
- College of Mathematics and Systems Science, Xinjiang University, Urumqi, People’s Republic of China
| | - Jianpeng Wang
- College of Mathematics and Systems Science, Xinjiang University, Urumqi, People’s Republic of China
| | - Zhidong Teng
- College of Mathematics and Systems Science, Xinjiang University, Urumqi, People’s Republic of China
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27
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Alshorman A, Wang X, Joseph Meyer M, Rong L. Analysis of HIV models with two time delays. JOURNAL OF BIOLOGICAL DYNAMICS 2017; 11:40-64. [PMID: 26889761 DOI: 10.1080/17513758.2016.1148202] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/05/2023]
Abstract
Time delays can affect the dynamics of HIV infection predicted by mathematical models. In this paper, we studied two mathematical models each with two time delays. In the first model with HIV latency, one delay is the time between viral entry into a cell and the establishment of HIV latency, and the other delay is the time between cell infection and viral production. We defined the basic reproductive number and showed the local and global stability of the steady states. Numerical simulations were performed to evaluate the influence of time delays on the dynamics. In the second model with HIV immune response, one delay is the time between cell infection and viral production, and the other delay is the time needed for the adaptive immune response to emerge to control viral replication. With two positive delays, we obtained the stability crossing curves for the model, which were shown to be a series of open-ended curves.
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Affiliation(s)
- Areej Alshorman
- a Department of Mathematics and Statistics , Oakland University , Rochester , MI , USA
| | - Xia Wang
- b College of Mathematics and Information Science , Xinyang Normal University , Xinyang , People's Republic of China
| | - M Joseph Meyer
- a Department of Mathematics and Statistics , Oakland University , Rochester , MI , USA
| | - Libin Rong
- a Department of Mathematics and Statistics , Oakland University , Rochester , MI , USA
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28
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Nikin-Beers R, Ciupe SM. Modelling original antigenic sin in dengue viral infection. MATHEMATICAL MEDICINE AND BIOLOGY-A JOURNAL OF THE IMA 2017; 35:257-272. [DOI: 10.1093/imammb/dqx002] [Citation(s) in RCA: 17] [Impact Index Per Article: 2.4] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/21/2016] [Accepted: 01/23/2017] [Indexed: 01/04/2023]
Affiliation(s)
| | - Stanca M Ciupe
- Department of Mathematics, Virginia Tech, Blacksburg, VA, USA
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29
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Global Stability of Delayed Viral Infection Models with Nonlinear Antibody and CTL Immune Responses and General Incidence Rate. COMPUTATIONAL AND MATHEMATICAL METHODS IN MEDICINE 2017; 2016:3903726. [PMID: 28115980 PMCID: PMC5198507 DOI: 10.1155/2016/3903726] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 08/12/2016] [Accepted: 11/22/2016] [Indexed: 11/18/2022]
Abstract
The dynamical behaviors for a five-dimensional viral infection model with three delays which describes the interactions of antibody, cytotoxic T-lymphocyte (CTL) immune responses, and nonlinear incidence rate are investigated. The threshold values for viral infection, antibody response, CTL immune response, CTL immune competition, and antibody competition, respectively, are established. Under certain assumptions, the threshold value conditions on the global stability of the infection-free, immune-free, antibody response, CTL immune response, and interior equilibria are proved by using the Lyapunov functionals method, respectively. Immune delay as a bifurcation parameter is further investigated. The numerical simulations are performed in order to illustrate the dynamical behavior of the model.
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30
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HIV Dynamics With Immune Responses: Perspectives From Mathematical Modeling. CURRENT CLINICAL MICROBIOLOGY REPORTS 2016. [DOI: 10.1007/s40588-016-0049-z] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/20/2022]
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31
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Wang J, Teng Z, Miao H. Global dynamics for discrete-time analog of viral infection model with nonlinear incidence and CTL immune response. ADVANCES IN DIFFERENCE EQUATIONS 2016; 2016:143. [PMID: 32226448 PMCID: PMC7099752 DOI: 10.1186/s13662-016-0862-y] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 01/25/2016] [Accepted: 05/10/2016] [Indexed: 06/10/2023]
Abstract
In this paper, a discrete-time analog of a viral infection model with nonlinear incidence and CTL immune response is established by using the Micken non-standard finite difference scheme. The two basic reproduction numbers R 0 and R 1 are defined. The basic properties on the positivity and boundedness of solutions and the existence of the virus-free, the no-immune, and the infected equilibria are established. By using the Lyapunov functions and linearization methods, the global stability of the equilibria for the model is established. That is, when R 0 ≤ 1 then the virus-free equilibrium is globally asymptotically stable, and under the additional assumption ( A 4 ) when R 0 > 1 and R 1 ≤ 1 then the no-immune equilibrium is globally asymptotically stable and when R 0 > 1 and R 1 > 1 then the infected equilibrium is globally asymptotically stable. Furthermore, the numerical simulations show that even if assumption ( A 4 ) does not hold, the no-immune equilibrium and the infected equilibrium also may be globally asymptotically stable.
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Affiliation(s)
- Jianpeng Wang
- College of Mathematics and Systems Science, Xinjiang University, Urumqi, 830046 People’s Republic of China
| | - Zhidong Teng
- College of Mathematics and Systems Science, Xinjiang University, Urumqi, 830046 People’s Republic of China
| | - Hui Miao
- College of Mathematics and Systems Science, Xinjiang University, Urumqi, 830046 People’s Republic of China
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32
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PAWELEK KASIAA, LIU SHENGQIANG, LOLLA MADHURIU. MODELING THE SPREAD OF HOOKWORM DISEASE AND ASSESSING CHEMOTHERAPY PROGRAMS: MATHEMATICAL ANALYSIS AND COMPARISON WITH SURVEILLANCE DATA. J BIOL SYST 2016. [DOI: 10.1142/s0218339016500091] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/18/2022]
Abstract
Soil-transmitted helminthes including hookworm infections result in a major disease burden worldwide. Periodic chemotherapy treatments with anthelminthic drugs remain major intervention strategy in highly endemic areas and currently there is no approved human vaccine. We developed a model to study the spread of the hookworm infection in the population level having a variety of parasite development stages. We investigated long-term effectiveness of selective and mass chemotherapy, scenarios in which the therapy becomes less effective, and is interrupted. We analyzed the mathematical model, which includes determining the basic reproductive number and steady states, defining model initial conditions, proving positiveness and boundedness of solutions, local and global stability of the disease-free and endemic steady states, and uniform persistence of the system. We fitted our model using field data from hookworm control programs in endemic areas in China and Zimbabwe. The model provided accurate predictions of disease prevalence for all considered locations for multiple years of utilizing deworming programs. We demonstrated that administration of the deworming therapy is an effective method in terms of controlling the hookworm infection. Nevertheless, the chemotherapy programs do not eradicate the disease in the villages and the infection occurrence promptly increases once the program is interrupted, thus ultimately leading to the pre-treatment levels. Our modeling results suggest that the deworming programs should be maintained at least once per year to control the infection.
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Affiliation(s)
- KASIA A. PAWELEK
- Department of Mathematics and Computational Science, University of South Carolina Beaufort, 1 University Boulevard, Bluffton, SC 29909, USA
| | - SHENGQIANG LIU
- The Academy of Fundamental and Interdisciplinary Science, Harbin Institute of Technology, 92 Xidazhi Street, Harbin 150080, P. R. China
| | - MADHURI U. LOLLA
- Department of Mathematics and Statistics, University of North Carolina Charlotte, 9201 University City Boulevard, Charlotte, NC 28223, USA
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33
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Song H, Jiang W, Liu S. Global dynamics of two heterogeneous SIR models with nonlinear incidence and delays. INT J BIOMATH 2016. [DOI: 10.1142/s1793524516500467] [Citation(s) in RCA: 9] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/18/2022]
Abstract
To investigate the effect of heterogeneity on the global dynamics of two SIR epidemic models with general nonlinear incidence rate and infection delays, we formulate a multi-group model corresponding to the heterogeneity in the host population and a multi-stage model corresponding to heterogeneous stages of infection. Under biologically motivated considerations, we establish that the global dynamics for each of the two models is determined completely by the corresponding basic reproduction number: if the basic reproduction number is less than or equal to one, then the disease-free equilibrium is globally asymptotically stable and the disease dies out in all groups or stages; if the basic reproduction number is larger than one, then the disease will persist in all groups or stages, and there is a unique endemic equilibrium which is globally asymptotically stable. Then we conclude that the heterogeneity does not change the global dynamics of the SIR model when the incidence rate is a general nonlinear function. Our results extend a class of previous results and can be applied to the other epidemiological models. The proofs of the main results use Lyapunov functional and graph-theoretic approach.
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Affiliation(s)
- Haitao Song
- Complex Systems Research Center, Shanxi University, Taiyuan, Shanxi 030006, P. R. China
- Department of Mathematics, Harbin Institute of Technology, Harbin Heilongjiang 150001, P. R. China
| | - Weihua Jiang
- Department of Mathematics, Harbin Institute of Technology, Harbin Heilongjiang 150001, P. R. China
| | - Shengqiang Liu
- The Academy of Fundamental and Interdisciplinary Science, Harbin Institute of Technology, 3041#, 2 Yi-Kuang Street, Harbin Heilongjiang 150080, P. R. China
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34
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Li B, Chen Y, Lu X, Liu S. A delayed HIV-1 model with virus waning term. MATHEMATICAL BIOSCIENCES AND ENGINEERING : MBE 2016; 13:135-157. [PMID: 26776264 DOI: 10.3934/mbe.2016.13.135] [Citation(s) in RCA: 8] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/05/2023]
Abstract
In this paper, we propose and analyze a delayed HIV-1 model with CTL immune response and virus waning. The two discrete delays stand for the time for infected cells to produce viruses after viral entry and for the time for CD8+ T cell immune response to emerge to control viral replication. We obtain the positiveness and boundedness of solutions and find the basic reproduction number R0. If R0 < 1, then the infection-free steady state is globally asymptotically stable and the infection is cleared from the T-cell population; whereas if R0 > 1, then the system is uniformly persistent and the viral concentration maintains at some constant level. The global dynamics when R0 > 1 is complicated. We establish the local stability of the infected steady state and show that Hopf bifurcation can occur. Both analytical and numerical results indicate that if, in the initial infection stage, the effect of delays on HIV-1 infection is ignored, then the risk of HIV-1 infection (if persists) will be underestimated. Moreover, the viral load differs from that without virus waning. These results highlight the important role of delays and virus waning on HIV-1 infection.
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Affiliation(s)
- Bing Li
- Academy of Fundamental and Interdisciplinary Science, Harbin Institute of Technology, 3041#, 2 Yi-Kuang street, Harbin, 150080, China.
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35
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Global stability of one and two discrete delay models for chronic hepatitis B infection with HBV DNA-containing capsids. ACTA ACUST UNITED AC 2015. [DOI: 10.1007/s40314-015-0242-3] [Citation(s) in RCA: 31] [Impact Index Per Article: 3.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/29/2022]
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36
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Yang Y, Zou L, Ruan S. Global dynamics of a delayed within-host viral infection model with both virus-to-cell and cell-to-cell transmissions. Math Biosci 2015; 270:183-91. [PMID: 25998145 DOI: 10.1016/j.mbs.2015.05.001] [Citation(s) in RCA: 87] [Impact Index Per Article: 9.7] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/18/2015] [Revised: 04/29/2015] [Accepted: 05/06/2015] [Indexed: 10/23/2022]
Abstract
A within-host viral infection model with both virus-to-cell and cell-to-cell transmissions and three distributed delays is investigated, in which the first distributed delay describes the intracellular latency for the virus-to-cell infection, the second delay represents the intracellular latency for the cell-to-cell infection, and the third delay describes the time period that viruses penetrated into cells and infected cells release new virions. The global stability analysis of the model is carried out in terms of the basic reproduction number R0. If R0≤1, the infection-free (semi-trivial) equilibrium is the unique equilibrium and is globally stable; if R0>1, the chronic infection (positive) equilibrium exists and is globally stable under certain assumptions. Examples and numerical simulations for several special cases are presented, including various within-host dynamics models with discrete or distributed delays that have been well-studied in the literature. It is found that the global stability of the chronic infection equilibrium might change in some special cases when the assumptions do not hold. The results show that the model can be applied to describe the within-host dynamics of HBV, HIV, or HTLV-1 infection.
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Affiliation(s)
- Yu Yang
- School of Science and Technology, Zhejiang International Studies University, Hangzhou 310012, P. R. China
| | - Lan Zou
- Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, P. R. China
| | - Shigui Ruan
- Department of Mathematics, University of Miami, Coral Gables, FL 33124-4250, USA.
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37
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Hajizadeh I, Shahrokhi M. Observer-Based Output Feedback Linearization Control with Application to HIV Dynamics. Ind Eng Chem Res 2015. [DOI: 10.1021/ie5022442] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/30/2022]
Affiliation(s)
- Iman Hajizadeh
- Department of Chemical and
Petroleum Engineering, Sharif University of Technology, P.O. Box 11155-9465 Azadi Av., Tehran, Iran
| | - Mohammad Shahrokhi
- Department of Chemical and
Petroleum Engineering, Sharif University of Technology, P.O. Box 11155-9465 Azadi Av., Tehran, Iran
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38
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Arafa AAM, Rida SZ, Khalil M. A fractional-order model of HIV infection: Numerical solution and comparisons with data of patients. INT J BIOMATH 2014. [DOI: 10.1142/s1793524514500363] [Citation(s) in RCA: 24] [Impact Index Per Article: 2.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/18/2022]
Abstract
In this paper, a fractional-order model which describes the human immunodeficiency type-1 virus (HIV-1) infection is presented. Numerical solutions are obtained using a generalized Euler method (GEM) to handle the fractional derivatives. The fractional derivatives are described in the Caputo sense. We show that the model established in this paper possesses non-negative solutions. Comparisons between the results of the fractional-order model, the results of the integer model and the measured real data obtained from 10 patients during primary HIV-1 infection are presented. These comparisons show that the results of the fractional-order model give predictions to the plasma virus load of the patients better than those of the integer model.
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Affiliation(s)
- A. A. M. Arafa
- Department of Mathematics and Computer Science, Faculty of Sciences, Port Said University, Port Said, Egypt
| | - S. Z. Rida
- Department of Mathematics, Faculty of Sciences, South Valley University, Qena, Egypt
| | - M. Khalil
- Department of Mathematics, Faculty of Engineering, October University for Modern Sciences and Arts (MSA University), 6th Oct. City, Giza, Egypt
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39
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Batmani Y, Khaloozadeh H. On the design of human immunodeficiency virus treatment based on a non‐linear time‐delay model. IET Syst Biol 2014; 8:13-21. [DOI: 10.1049/iet-syb.2013.0012] [Citation(s) in RCA: 12] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/19/2022] Open
Affiliation(s)
- Yazdan Batmani
- Department of Systems and Control, Industrial Control Center of ExcellenceK. N. Toosi University of TechnologyTehranIran
| | - Hamid Khaloozadeh
- Department of Systems and Control, Industrial Control Center of ExcellenceK. N. Toosi University of TechnologyTehranIran
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40
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LI BING, LIU SHENGQIANG. A DELAYED HIV-1 MODEL WITH MULTIPLE TARGET CELLS AND GENERAL NONLINEAR INCIDENCE RATE. J BIOL SYST 2014. [DOI: 10.1142/s0218339013400123] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/18/2022]
Abstract
We investigate a delayed HIV-1 infection model with general nonlinear incidence functions and two classes of target cells: CD4+ T-cells and macrophages. To account for the time lags between viruses' entry into the corresponding two types of target cells and the production of new virus particles, we incorporate four distributed intracellular delays into the model. We show that the basic reproduction number ℜ0 is the sum of the basic reproduction numbers of HIV-1 infection with CD4+ T-cells and that with macrophages; moreover, if ℜ0 is less than or equal to one, then the HIV-1 infection is cleared from the T-cell population and macrophages; whereas if ℜ0 is larger than one, then the viral concentration maintains at some constant level. It is shown, from both our analytic and numeric results, that ignoring the contributions of macrophages to HIV-1 infection and production will underestimate both the risk of HIV-1 infection and the viral load when persisting. This highlights the important effects of multiple target cells on HIV-1 infection.
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Affiliation(s)
- BING LI
- Department of Mathematics, Harbin Institute of Technology, Harbin 150000, P. R. China
- School of Mathematical Science, Harbin Normal University, Harbin 150000, P. R. China
| | - SHENGQIANG LIU
- Academy of Fundamental and Interdisciplinary Sciences, Harbin Institute of Technology, 3041#, 2 Yi-Kuang Street, Harbin 150080, P. R. China
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41
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Wang X, Liu S, Rong L. Permanence and extinction of a non-autonomous HIV-1
model with time delays. ACTA ACUST UNITED AC 2014. [DOI: 10.3934/dcdsb.2014.19.1783] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/18/2022]
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42
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Jiang C, Wang W. Complete classification of global dynamics of a virus model with
immune responses. ACTA ACUST UNITED AC 2014. [DOI: 10.3934/dcdsb.2014.19.1087] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/13/2022]
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43
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BAIRAGI N, ADAK D. HOW SELF-PROLIFERATION OF CD4+T CELLS AFFECT THE HIV DYNAMICS IN AN IN-HOST TARGET-CELL LIMITED HIV MODEL WITH SATURATION INFECTION RATE: A QUASI-STEADY-STATE APPROXIMATION ANALYSIS. INT J BIOMATH 2013. [DOI: 10.1142/s1793524513500046] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/18/2022]
Abstract
In this study, we consider two target-cell limited models with saturation type infection rate and intracellular delay: one without self-proliferation and the other with self-proliferation of activated CD4+T cells. We discuss about the local and global behavior of both the systems in presence and absence of intracellular delay. It is shown that the endemic equilibrium of a target-cell limited model would be unstable in presence and absence of intracellular delay only when self-proliferation of activated CD4+T cell is considered. Otherwise, all positive solutions converge to the endemic equilibrium or disease-free equilibrium depending on whether the basic reproduction ratio is greater than or less than unity. Our study suggests that amplitude of oscillation is negatively correlated with the constant input rate of CD4+T cell when intracellular delay is absent or low. However, they are positively correlated if the delay is too high. Amplitude of oscillation, on the other hand, is always positively correlated with the proliferation rate of CD4+T cell for all delay. Our mathematical and simulation analysis also suggest that there are many potential contributors who are responsible for the variation of CD4+T cells and virus particles in the blood plasma of HIV patients.
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Affiliation(s)
- N. BAIRAGI
- Center for Mathematical Biology and Ecology, Department of Mathematics, Jadavpur University, Kolkata 700032, India
| | - D. ADAK
- Center for Mathematical Biology and Ecology, Department of Mathematics, Jadavpur University, Kolkata 700032, India
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44
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Yuan Z, Zou X. Global threshold dynamics in an HIV virus model with nonlinear infection rate and distributed invasion and production delays. MATHEMATICAL BIOSCIENCES AND ENGINEERING : MBE 2013; 10:483-498. [PMID: 23458310 DOI: 10.3934/mbe.2013.10.483] [Citation(s) in RCA: 9] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/01/2023]
Abstract
We consider a mathematical model that describes the interactions of the HIV virus, CD4 cells and CTLs within host, which is a modification of some existing models by incorporating (i) two distributed kernels reflecting the variance of time for virus to invade into cells and the variance of time for invaded virions to reproduce within cells; (ii) a nonlinear incidence function f for virus infections, and (iii) a nonlinear removal rate function h for infected cells. By constructing Lyapunov functionals and subtle estimates of the derivatives of these Lyapunov functionals, we shown that the model has the threshold dynamics: if the basic reproduction number (BRN) is less than or equal to one, then the infection free equilibrium is globally asymptotically stable, meaning that HIV virus will be cleared; whereas if the BRN is larger than one, then there exist an infected equilibrium which is globally asymptotically stable, implying that the HIV-1 infection will persist in the host and the viral concentration will approach a positive constant level. This together with the dependence/independence of the BRN on f and h reveals the effect of the adoption of these nonlinear functions.
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Affiliation(s)
- Zhaohui Yuan
- College of Mathematics and Econometrics, Hunan University, Changsha, Hunan 410082, China.
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45
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Cai LM, Martcheva M, Li XZ. Competitive exclusion in a vector-host epidemic model with distributed delay(†). JOURNAL OF BIOLOGICAL DYNAMICS 2013; 7 Suppl 1:47-67. [PMID: 23421610 PMCID: PMC4053945 DOI: 10.1080/17513758.2013.772253] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 07/14/2012] [Revised: 01/20/2013] [Indexed: 06/01/2023]
Abstract
A multi-strain model of a vector-borne disease with distributed delay in the vector and the host is investigated. It is shown that if the reproduction number of the model R0 < 1, the unique disease-free equilibrium is globally asymptotically stable.Without loss of generality, strain one is assumed to have the largest reproduction number. In this case, the dominance equilibrium of strain one is shown to be locally stable. The basic reproduction number for a strain i (R(i)(0) is written as a product of the reproduction number of the vector (R(i)(v)) and the reproduction number of the host (R(i)(h)), i.e. R(i)(0) = R(i)(h)R(i)(v). The competitive exclusion principle is derived under the somewhat stronger condition that if strain one maximizes both the reproduction number of the host R(i)(h) < R(1)(h), i ≠ 1 and the reproduction number of the vector R(i)(v) < R(1)(v ), i ≠ 1,strain one dominance equilibrium is globally asymptotically stable.
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Affiliation(s)
- Li-Ming Cai
- Department of Mathematics, Xinyang Normal University, Xinyang 464000,
People's Republic of China
| | - Maia Martcheva
- Department of Mathematics, University of Florida, 358 Little Hall, PO
Box 118105, Gainesville, FL 32611-8105, USA
| | - Xue-Zhi Li
- Department of Mathematics, Xinyang Normal University, Xinyang 464000,
People's Republic of China
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46
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Wang X, Wang W. An HIV infection model based on a vectored immunoprophylaxis experiment. J Theor Biol 2012; 313:127-35. [DOI: 10.1016/j.jtbi.2012.08.023] [Citation(s) in RCA: 18] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/21/2012] [Revised: 08/18/2012] [Accepted: 08/20/2012] [Indexed: 10/28/2022]
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47
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Wang X, Liu S. Global properties of a delayed SIR epidemic model with multiple parallel infectious stages. MATHEMATICAL BIOSCIENCES AND ENGINEERING : MBE 2012; 9:685-695. [PMID: 22881032 DOI: 10.3934/mbe.2012.9.685] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/01/2023]
Abstract
In this paper, we study the global properties of an SIR epidemic model with distributed delays, where there are several parallel infective stages, and some of the infected cells are detected and treated, which others remain undetected and untreated. The model is analyzed by determining a basic reproduction number R0, and by using Lyapunov functionals, we prove that the infection-free equilibrium E0 of system (3) is globally asymptotically attractive when R0 ≤ 1, and that the unique infected equilibrium E* of system (3) exists and it is globally asymptotically attractive when R0 > 1.
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Affiliation(s)
- Xia Wang
- Academy of Fundamental and Interdisciplinary Sciences, Harbin Institute of Technology, Harbin, 150080, China.
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48
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Kinetic model of HIV infection including hematopoietic progenitor cells. Math Biosci 2012; 236:36-43. [DOI: 10.1016/j.mbs.2012.01.003] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/20/2010] [Revised: 01/10/2012] [Accepted: 01/13/2012] [Indexed: 12/16/2022]
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