scholarly journals Mathematical Analysis and Numerical Solution of a Model of HIV with a Discrete Time Delay

Mathematics ◽  
2021 ◽  
Vol 9 (3) ◽  
pp. 257
Author(s):  
Abraham J. Arenas ◽  
Gilberto González-Parra ◽  
Jhon J. Naranjo ◽  
Myladis Cogollo ◽  
Nicolás De La Espriella
Keyword(s):  
Mathematical Model ◽  
Viral Entry ◽  
Globally Asymptotically Stable ◽  
Reproduction Ratio ◽  
Hiv Virus ◽  
Discrete Time Delay ◽  
Standard Difference ◽  
Lasalle Invariant Principle ◽  
Delay Differential ◽  
The Mathematical Model

We propose a mathematical model based on a set of delay differential equations that describe intracellular HIV infection. The model includes three different subpopulations of cells and the HIV virus. The mathematical model is formulated in such a way that takes into account the time between viral entry into a target cell and the production of new virions. We study the local stability of the infection-free and endemic equilibrium states. Moreover, by using a suitable Lyapunov functional and the LaSalle invariant principle, it is proved that if the basic reproduction ratio is less than unity, the infection-free equilibrium is globally asymptotically stable. In addition, we designed a non-standard difference scheme that preserves some relevant properties of the continuous mathematical model.

scholarly journals Nonlinear Dynamics of the Introduction of a New SARS-CoV-2 Variant with Different Infectiousness

Mathematics ◽  
2021 ◽  
Vol 9 (13) ◽  
pp. 1564
Author(s):  
Gilberto Gonzalez-Parra ◽  
Abraham J. Arenas
Keyword(s):  
Mathematical Model ◽  
Basic Reproduction Number ◽  
Reproduction Number ◽  
Transmission Rate ◽  
Potential Outcomes ◽  
Public Concern ◽  
Globally Asymptotically Stable ◽  
Lasalle Invariant Principle ◽  
The Mathematical Model ◽  
The Basic Reproduction Number

Several variants of the SARS-CoV-2 virus have been detected during the COVID-19 pandemic. Some of these new variants have been of health public concern due to their higher infectiousness. We propose a theoretical mathematical model based on differential equations to study the effect of introducing a new, more transmissible SARS-CoV-2 variant in a population. The mathematical model is formulated in such a way that it takes into account the higher transmission rate of the new SARS-CoV-2 strain and the subpopulation of asymptomatic carriers. We find the basic reproduction number R0 using the method of the next generation matrix. This threshold parameter is crucial since it indicates what parameters play an important role in the outcome of the COVID-19 pandemic. We study the local stability of the infection-free and endemic equilibrium states, which are potential outcomes of a pandemic. Moreover, by using a suitable Lyapunov functional and the LaSalle invariant principle, it is proved that if the basic reproduction number is less than unity, the infection-free equilibrium is globally asymptotically stable. Our study shows that the new more transmissible SARS-CoV-2 variant will prevail and the prevalence of the preexistent variant would decrease and eventually disappear. We perform numerical simulations to support the analytic results and to show some effects of a new more transmissible SARS-CoV-2 variant in a population.

GLOBAL DYNAMICS OF A DELAYED HIV-1 INFECTION MODEL WITH ABSORPTION AND SATURATION INFECTION

2012 ◽  
Vol 05 (03) ◽  
pp. 1260012 ◽  
Author(s):  
RUI XU
Keyword(s):  
Viral Entry ◽  
Chronic Infection ◽  
Infection Model ◽  
Global Dynamics ◽  
Asymptotically Stable ◽  
Basic Reproduction Ratio ◽  
Globally Asymptotically Stable ◽  
Reproduction Ratio ◽  
Virus Particles ◽  
Hiv 1

In this paper, an HIV-1 infection model with absorption, saturation infection and an intracellular delay accounting for the time between viral entry into a target cell and the production of new virus particles is investigated. By analyzing the characteristic equations, the local stability of an infection-free equilibrium and a chronic-infection equilibrium of the model is established. By using suitable Lyapunov functionals and LaSalle's invariance principle, it is proved that if the basic reproduction ratio is less than unity, the infection-free equilibrium is globally asymptotically stable; and if the basic reproduction ratio is greater than unity, sufficient condition is derived for the global stability of the chronic-infection equilibrium.

BIFURCATION ANALYSIS ON AN HIV-1 MODEL WITH CONSTANT INJECTION OF RECOMBINANT

2012 ◽  
Vol 22 (03) ◽  
pp. 1250062 ◽  
Author(s):  
PEI YU ◽  
XINGFU ZOU
Keyword(s):  
Hopf Bifurcation ◽  
Equilibrium Solution ◽  
Injection Rate ◽  
Asymptotically Stable ◽  
Equilibrium Solutions ◽  
Globally Asymptotically Stable ◽  
Reproduction Ratio ◽  
New Model ◽  
Hiv Virus ◽  
Hiv 1

This paper is a continuation of our previous work on an HIV-1 therapy model of fighting a virus with another virus [Jiang et al., 2009]. The work in [Jiang et al., 2009] investigated cascading bifurcations between equilibrium solutions, as well as Hopf bifurcation from a double-infected equilibrium solution. In this paper, we propose a modification of the model in [Revilla & Garcia-Ramos, 2003; Jiang et al., 2009] by adding a constant η to the recombinant virus equation, which accounts for the treatment of constant injection of recombinants. We study the dynamics of the new model and find that η plays an important role in the therapy. Unlike the previous model without injection of recombinant, which has three equilibrium solutions, this new model can only allow two biologically meaningful equilibrium solutions. It is shown that there is [Formula: see text] depending on η, such that the HIV free equilibrium solution [Formula: see text] is globally asymptotically stable when the basic reproduction ratio, [Formula: see text]; [Formula: see text] becomes unstable when [Formula: see text]. In the latter case, there occurs the double-infection equilibrium solution, [Formula: see text], which is stable when [Formula: see text] for some [Formula: see text] larger than [Formula: see text], and loses its stability when [Formula: see text] passes the critical value [Formula: see text] and bifurcates into a family of limit cycles through Hopf bifurcation. Our results show that appropriate injection rate can help eliminate the HIV virus in the sense that the HIV free equilibrium can be made globally asymptotically stable by choosing η > 0 sufficiently large. This is in contrast to the conclusion for the case with η = 0 in which, the recombinants do not help eliminate the HIV virus but only help reduce the HIV load in the long term sense.

scholarly journals PEMODELAN MATEMATIKA DENGAN METODE RUNGE KUTTA UNTUK PENYAKIT CAMPAK MENGGUNAKAN MATLAB R2010a

2018 ◽  
Vol 4 (2) ◽  
Author(s):  
Ketut Queena Fredlina ◽  
Komang Tri Werthi
Keyword(s):  
Mathematical Model ◽  
Mathematical Models ◽  
Measles Virus ◽  
Immunization Coverage ◽  
Living Environment ◽  
Basic Reproduction Ratio ◽  
Reproduction Ratio ◽  
Runge Kutta Method ◽  
Runge Kutta ◽  
The Mathematical Model

ABSTRACT<br />Mathematical models have important roles in various fields of science. By using several assumptions, problems that exist in the living environment can be transformed in mathematical models. From the existing mathematical model, the parameters that affect the model can then be analyzed. An epidemic is an event that can be transformed into a mathematical model. Epidemic events are the occurrence of the spread or outbreak of an illness in a region. Measles is one of the causes of death in developing countries caused by the measles virus, the Paramixovirus group. In 1982 a measles immunization program in Indonesia was conducted. Based on data from the 2015 Ministry of Health, Indonesia has a medium immunization coverage in Southeast Asia, which is 84%. In 2020 Indonesia has a target rate of measles immunization coverage of 95%. Measles is a concern of the Bali Provincial Health Office because the spread of this disease is always high. Specifically in this study we will discuss mathematical models for the incidence of measles epidemics. The problem is how to construct the model and what parameters are the most significant influences in the mathematical model of measles. In making mathematical models for the spread of measles, the population is divided into 3 parts: Susceptible, Infectious, and Recovered. Furthermore, analyze the parameters and determine the basic reproduction ratio (𝑹𝟎), then numerical simulations were carried out using the Order 4 Runge Kutta method.<br />Keywords : Mathematics , Measles, basic reproduction ratio (𝑹𝟎), Runge-Kutta Methods<br />ABSTRAK<br />Model matematika memiliki peran yang cukup penting dalam berbagai bidang ilmu. Dengan menggunakan beberapa asumsi, permasalahan yang ada dalam lingkungan kehidupan dapat ditransformasikan dalam model matematika. Dari model matematika yang ada selanjutnya dapat dianalisis parameter-parameter yang mempengaruhi model tersebut. Kejadian epidemi merupakan salah satu kejadian yang dapat ditransformasikan dalam model matematika. Kejadian epidemi adalah kejadian penyebaran atau mewabahnya suatu penyakit dalam suatu wilayah. Penyakit campak merupakan salah satu penyakit penyebab kematian penduduk di negara-negara berkembang yang disebabkan oleh virus campak golongan Paramixovirus. Pada tahun 1982 program imunisasi campak di Indonesia telah dilakukan. Berdasarkan data dari Departemen Kesehatan 2015, Indonesia memiliki cakupan imunisasi kategori sedang di Asia Tenggara yakni 84%. Pada tahun 2020 Indonesia memiliki target angka cakupan imunisasi campak sebesar 95%. Penyakit campak menjadi perhatian Dinas Kesehatan Profinsi Bali karena penyebaran penyakit ini selalu ada. Secara khusus dalam penelitian ini akan membahas model matematika untuk kejadian epidemi penyakit campak. Yang menjadi permasalahan adalah bagaimana mengontruksi model dan parameter apakah yang berpengaruh paling signifikan dalam model matematika penyakit campak. Dalam pembuatan model matematika untuk penyebaran penyakit campak, populasi manusia dibagi menjadi 3 bagian yaitu : Susceptible, Infectious, dan Recovered. Selanjutnya menganalisis parameter dan menentukan nilai basic reproduction ratio (R0), kemudian dilakukan simulasi numerik dengan metode Runge Kutta Orde 4.<br />Kata kunci : model matematika, campak, basic reproduction ratio (𝑹𝟎),metode Runge-Kutta

scholarly journals A Mathematical Model of Malaria Transmission with Structured Vector Population and Seasonality

2017 ◽  
Vol 2017 ◽  
pp. 1-15 ◽  
Author(s):  
Bakary Traoré ◽  
Boureima Sangaré ◽  
Sado Traoré
Keyword(s):  
Mathematical Model ◽  
Malaria Transmission ◽  
Climatic Factors ◽  
Positive Periodic Solution ◽  
Uniform Persistence ◽  
Basic Reproduction Ratio ◽  
Globally Asymptotically Stable ◽  
Reproduction Ratio ◽  
Vector Population ◽  
Biting Rate

In this paper, we formulate a mathematical model of nonautonomous ordinary differential equations describing the dynamics of malaria transmission with age structure for the vector population. The biting rate of mosquitoes is considered as a positive periodic function which depends on climatic factors. The basic reproduction ratio of the model is obtained and we show that it is the threshold parameter between the extinction and the persistence of the disease. Thus, by applying the theorem of comparison and the theory of uniform persistence, we prove that if the basic reproduction ratio is less than 1, then the disease-free equilibrium is globally asymptotically stable and if it is greater than 1, then there exists at least one positive periodic solution. Finally, numerical simulations are carried out to illustrate our analytical results.

RISK OF DISEASE-SELECTIVE PREDATION IN AN INFECTED PREY-PREDATOR SYSTEM

2009 ◽  
Vol 17 (01) ◽  
pp. 111-124 ◽  
Author(s):  
SHARIFUL ALAM
Keyword(s):  
Risk Factors ◽  
Mathematical Model ◽  
Time Delay ◽  
Discrete Time ◽  
Selective Predation ◽  
Discrete Time Delay ◽  
Risk Of Disease ◽  
The Mathematical Model ◽  
Predator System ◽  
Infected Prey

In this paper the mathematical model of disease-selective predation as proposed by Roy and Chattopadhyay10 is considered to identify the true risk of selective predation where the predator can recognize the infected prey and avoids those during predation. Furthermore, the model is modified by adding a discrete time delay in the term involving the gestation of prey by the predator and analyzed both numerically and analytically to review the risk factors.

scholarly journals Stability analysis of a simple mathematical model for school bullying

2022 ◽  
Vol 7 (4) ◽  
pp. 4936-4945
Author(s):  
H. A. Ashi ◽  
Keyword(s):  
Mathematical Model ◽  
Equilibrium Points ◽  
School Bullying ◽  
Drop Out ◽  
Simple Mathematical Model ◽  
Globally Asymptotically Stable ◽  
School Drop Out ◽  
New Criterion ◽  
The Mathematical Model ◽  
Commit Suicide

<abstract><p>School bullying is a highly concerned problem due to its effect on students' academic achievement. The effect might go beyond that to develop health problems, school drop out and, in some extreme cases, commit suicide for victims. On the other hand, adolescents who continuously bully over time are at risk of becoming involved in gang membership and other types of crime. Therefore, we propose a simple mathematical model for school bullying by considering two variables: the number of victims students and the number of bullies students. The main assumption employed to develop the mathematical model is that school policy bans bullying and expels students who practice this behavior to maintain a constructive educational environment within the school. We show that the model has two equilibrium points, and that both equilibrium points are locally and globally asymptotically stable under certain conditions. Also, we define a threshold parameter with a new criterion called the bullying index. Furthermore, we show that the model exhibits the phenomena of transcritical bifurcation subject to the bullying index. All the findings are supported with numerical simulations.</p></abstract>

scholarly journals A Mathematical Model of Immune-System-Melanoma Competition

2012 ◽  
Vol 2012 ◽  
pp. 1-13 ◽  
Author(s):  
Marzio Pennisi
Keyword(s):  
Mathematical Model ◽  
Specific Immunotherapy ◽  
Melanoma Tumor ◽  
Injection Point ◽  
Combined Administration ◽  
Delay Differential ◽  
The Mathematical Model ◽  
Good Agreement

We present a mathematical model developed to reproduce the immune response entitled with the combined administration of activated OT1 cytotoxic T lymphocytes (CTLs) and Anti-CD137 monoclonal antibodies. The treatment is directed against melanoma in B16 OVA mouse models exposed to a specific immunotherapy strategy. We model two compartments: the injection point compartment where the treatment is administered and the skin compartment where melanoma tumor cells proliferate. To model the migration of OT1 CTLs and antibodies from the injection to the skin compartment, we use delay differential equations (DDEs). The outcomes of the mathematical model are in good agreement with the in vivo results. Moreover, sensitivity analysis of the mathematical model underlines the key role of OT1 CTLs and suggests that a possible reduction of the number of injected antibodies should not affect substantially the treatment efficacy.

scholarly journals A mathematical model of viral oncology as an immuno-oncology instigator

2019 ◽  
Author(s):  
Tyler Cassidy ◽  
Antony R Humphries
Keyword(s):  
Mathematical Model ◽  
Distributed Delay ◽  
Cycle Duration ◽  
Necessary And Sufficient Condition ◽  
Cell Cycle Duration ◽  
Viral Therapy ◽  
Delay Differential ◽  
Necessary And Sufficient ◽  
The Mathematical Model

Abstract We develop and analyse a mathematical model of tumour–immune interaction that explicitly incorporates heterogeneity in tumour cell cycle duration by using a distributed delay differential equation. We derive a necessary and sufficient condition for local stability of the cancer-free equilibrium in which the amount of tumour–immune interaction completely characterizes disease progression. Consistent with the immunoediting hypothesis, we show that decreasing tumour–immune interaction leads to tumour expansion. Finally, by simulating the mathematical model, we show that the strength of tumour–immune interaction determines the long-term success or failure of viral therapy.

scholarly journals A Delayed Vaccination Model for Rotavirus Infection

2020 ◽  
Vol 13 (4) ◽  
pp. 840-851
Author(s):  
Florence A. Adongo ◽  
Onyango Omondi Lawrence ◽  
Job Bonyo ◽  
G. O. Lawi ◽  
Ogada A. Elisha
Keyword(s):  
Mathematical Model ◽  
Stability Analysis ◽  
Hopf Bifurcation ◽  
Numerical Analysis ◽  
Delay Differential Equations ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Rotavirus Infection ◽  
Delay Differential ◽  
Disease Free

In this work, a mathematical model for rotavirus infection incorporating delay differential equations has been formulated. Stability analysis of the model has been performed. The result shows that the Disease Free Equilibrium is globally asymptotically stable and the Endemic Equilibrium undergoes a Hopf bifurcation. Numerical analysis has been performed to validate the analysis.

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