Eitorial Mathematical Modeling in Virology by Differential...
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Editorial Mathematical Modeling in Virology by Differential Equations Khalid Hattaf , 1 Ahmed M. Elaiw , 2 Abid A. Lashari, 3 and Noura Yousfi 4 1 Centre R´ egional des M´ etiers de l’Education et de la Formation (CRMEF), 20340 Derb Ghalef, Casablanca, Morocco 2 Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah, Saudi Arabia 3 Department of Mathematics, Stockholm University, 106 91 Stockholm, Sweden 4 Laboratory of Analysis, Modeling and Simulation (LAMS), Faculty of Sciences Ben M’sik, Hassan II University, P.O. Box 7955 Sidi Othman, Casablanca, Morocco Correspondence should be addressed to Khalid Hattaf; [email protected] Received 27 March 2018; Accepted 28 March 2018; Published 3 May 2018 Copyright © 2018 Khalid Hattaf et al. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Viruses are microscopic organisms that need to penetrate inside a cell of their host to multiply and replicate. Various viruses infect directly the human body such as the human immunodeficiency virus (HIV), hepatitis B virus (HBV), and hepatitis C virus (HCV). ere are also viruses that affect the human body through a living vector such as West Nile virus which is a virus most commonly spread to people by mosquito bites. Infectious diseases caused by these viruses represent a major global health problem by causing mortality of millions of people and expenditure of enormous amount of money in health care and disease control. In this special issue, many mathematical models, ordinary differential equations (ODEs), delay differential equations (DDEs), partial differ- ential equations (PDEs), and fractional differential equations (FDEs), have been proposed and developed to better describe the dynamics of these infectious diseases and establish con- trol strategies to limit their evolution and spread. One paper of this special issue proposes a mathematical model with ODEs and PDEs to describe the dynamics of vec- tor-borne diseases such as West Nile virus, malaria, and dengue. e proposed model incorporates the waning of vaccine-induced immunity. Furthermore, the global stability of the equilibria of the model is established. More precisely, it is proved that when the basic reproduction number is less than one, the disease-free equilibrium is globally asymptoti- cally stable, which implies that the disease dies out. However, when the basic reproduction number is larger than one, the endemic equilibrium is globally asymptotically stable, which means that the disease persists in the population. Another paper presents a delayed model formulated by DDEs in order to study the early stage of HBV infection and impact of the delay in the infection process on the adaptive immune response, which includes cytotoxic T lymphocytes (CTL) cells and antibodies. e stability analysis of equilibria and the optimal treatment strategies are investigated. Another paper focuses on the qualitative analysis of a generalized virus dynamics model with distributed delays and two modes of transmission, one by virus-to-cell infection and the other by cell-to-cell transmission. e infection transmission process is modeled by general incidence func- tions for both modes of transmission. Moreover, many known viral infection models with discrete and distributed delays are extended and improved. Another paper deals with a family of periodic SEIRS epidemic models. e global dynamics of these models is fully determined by a threshold parameter, namely, the basic reproduction number. Numerical simulations are carried out to support the theoretical results. Another paper of this special issue proposes a fractional order model of HIV infection with specific functional response and cure rate. is functional response covers the most functional responses used by several authors such as the saturated incidence rate, the Beddington-DeAngelis functional response, and the Crowley-Martin functional Hindawi International Journal of Differential Equations Volume 2018, Article ID 2721481, 2 pages https://doi.org/10.1155/2018/2721481
Transcript of Eitorial Mathematical Modeling in Virology by Differential...
EditorialMathematical Modeling in Virology by Differential Equations
Khalid Hattaf ,1 AhmedM. Elaiw ,2 Abid A. Lashari,3 and Noura Yousfi 4
1Centre Regional des Metiers de l’Education et de la Formation (CRMEF), 20340 Derb Ghalef, Casablanca, Morocco2Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah, Saudi Arabia3Department of Mathematics, Stockholm University, 106 91 Stockholm, Sweden4Laboratory of Analysis, Modeling and Simulation (LAMS), Faculty of Sciences Ben M’sik, Hassan II University,P.O. Box 7955 Sidi Othman, Casablanca, Morocco
Correspondence should be addressed to Khalid Hattaf; [email protected]
Received 27 March 2018; Accepted 28 March 2018; Published 3 May 2018
Copyright © 2018 Khalid Hattaf et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Viruses are microscopic organisms that need to penetrateinside a cell of their host to multiply and replicate. Variousviruses infect directly the human body such as the humanimmunodeficiency virus (HIV), hepatitis B virus (HBV),and hepatitis C virus (HCV). There are also viruses thataffect the human body through a living vector such as WestNile virus which is a virus most commonly spread to peopleby mosquito bites. Infectious diseases caused by these virusesrepresent a major global health problem by causing mortalityofmillions of people and expenditure of enormous amount ofmoney in health care and disease control. In this special issue,many mathematical models, ordinary differential equations(ODEs), delay differential equations (DDEs), partial differ-ential equations (PDEs), and fractional differential equations(FDEs), have been proposed and developed to better describethe dynamics of these infectious diseases and establish con-trol strategies to limit their evolution and spread.
One paper of this special issue proposes a mathematicalmodel with ODEs and PDEs to describe the dynamics of vec-tor-borne diseases such as West Nile virus, malaria, anddengue. The proposed model incorporates the waning ofvaccine-induced immunity. Furthermore, the global stabilityof the equilibria of the model is established. More precisely,it is proved that when the basic reproduction number is lessthan one, the disease-free equilibrium is globally asymptoti-cally stable, which implies that the disease dies out. However,when the basic reproduction number is larger than one, the
endemic equilibrium is globally asymptotically stable, whichmeans that the disease persists in the population.
Another paper presents a delayed model formulated byDDEs in order to study the early stage of HBV infection andimpact of the delay in the infection process on the adaptiveimmune response, which includes cytotoxic T lymphocytes(CTL) cells and antibodies.The stability analysis of equilibriaand the optimal treatment strategies are investigated.
Another paper focuses on the qualitative analysis of ageneralized virus dynamics model with distributed delaysand twomodes of transmission, one by virus-to-cell infectionand the other by cell-to-cell transmission. The infectiontransmission process is modeled by general incidence func-tions for bothmodes of transmission.Moreover,many knownviral infectionmodels with discrete and distributed delays areextended and improved.
Another paper deals with a family of periodic SEIRSepidemic models. The global dynamics of these models isfully determined by a threshold parameter, namely, the basicreproduction number. Numerical simulations are carried outto support the theoretical results.
Another paper of this special issue proposes a fractionalorder model of HIV infection with specific functionalresponse and cure rate. This functional response covers themost functional responses used by several authors such as thesaturated incidence rate, the Beddington-DeAngelisfunctional response, and the Crowley-Martin functional
HindawiInternational Journal of Differential EquationsVolume 2018, Article ID 2721481, 2 pageshttps://doi.org/10.1155/2018/2721481
2 International Journal of Differential Equations
response. It is shown that the model is mathematically andbiologically well-posed. In addition, the local and globalstabilities of the equilibria are investigated.
Khalid HattafAhmed M. ElaiwAbid A. Lashari
Noura Yousfi
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