Stability Analysis of a Mathematical Model for Glioma-Immune Interaction under Optimal Therapy

2019 ◽  
Vol 20 (3-4) ◽  
pp. 269-285 ◽  
Author(s):  
Subhas Khajanchi
Keyword(s):  
Mathematical Model ◽  
Optimal Control ◽  
Stability Analysis ◽  
Model Simulation ◽  
Equilibrium Points ◽  
Glioma Cells ◽  
Adjoint System ◽  
Cellular Immunotherapy ◽  
Theory Of Optimal Control ◽  
Malignant Glioma Cells

AbstractWe investigate a mathematical model using a system of coupled ordinary differential equations, which describes the interplay of malignant glioma cells, macrophages, glioma specific CD8+T cells and the immunotherapeutic drug Adoptive Cellular Immunotherapy (ACI). To better understand under what circumstances the glioma cells can be eliminated, we employ the theory of optimal control. We investigate the dynamics of the system by observing biologically feasible equilibrium points and their stability analysis before administration of the external therapy ACI. We solve an optimal control problem with an objective functional which minimizes the glioma cell burden as well as the side effects of the treatment. We characterize our optimal control in terms of the solutions to the optimality system, in which the state system coupled with the adjoint system. Our model simulation demonstrates that the strength of treatment $u_{1}(t)$ plays an important role to eliminate the glioma cells. Finally, we derive an optimal treatment strategy and then solve it numerically.

scholarly journals A Mathematical Model to Study the Effectiveness of Some of the Strategies Adopted in Curtailing the Spread of COVID-19

2020 ◽  
Vol 2020 ◽  
pp. 1-6 ◽  
Author(s):  
Isa Abdullahi Baba ◽  
Bashir Abdullahi Baba ◽  
Parvaneh Esmaili
Keyword(s):  
Mathematical Model ◽  
Stability Analysis ◽  
Numerical Simulations ◽  
Government Policy ◽  
Local Stability ◽  
Equilibrium Points ◽  
Compartmental Models ◽  
Effective Strategy ◽  
Local Stability Analysis ◽  
The Government

In this paper, we developed a model that suggests the use of robots in identifying COVID-19-positive patients and which studied the effectiveness of the government policy of prohibiting migration of individuals into their countries especially from those countries that were known to have COVID-19 epidemic. Two compartmental models consisting of two equations each were constructed. The models studied the use of robots for the identification of COVID-19-positive patients. The effect of migration ban strategy was also studied. Four biologically meaningful equilibrium points were found. Their local stability analysis was also carried out. Numerical simulations were carried out, and the most effective strategy to curtail the spread of the disease was shown.

scholarly journals DYNAMICS OF ENVIRONMENTAL POLLUTIONTHROUGH VEHICLES

2020 ◽  
Vol 5 (4) ◽  
pp. 38-47
Author(s):  
Nita Shah ◽  
Shreya Patel ◽  
Moksha Satia ◽  
Foram Thakkar
Keyword(s):  
Numerical Simulation ◽  
Mathematical Model ◽  
Air Pollution ◽  
Stability Analysis ◽  
Reproduction Number ◽  
Equilibrium Points ◽  
System Stability ◽  
System Stability Analysis ◽  
Almost All ◽  
Day By Day

In today’s time as air pollution is increasing day by day the use of non-polluted has to be increased in almost all nooks and corner of the countries. In this paper a mathematical model is developed to analyse environmental pollution through polluted and non-polluted vehicles. Basic reproduction number has been calculated which will the decide the behavior of the system. Stability analysis has been carried out at equilibrium points. Numerical simulation is done to analyse the result for various compartments.

Controlling Asthma Due to Air Pollution

2020 ◽  
pp. 1-22
Author(s):  
Ankush H. Suthar ◽  
Purvi M. Pandya
Keyword(s):  
Mathematical Model ◽  
Optimal Control ◽  
Air Pollution ◽  
Control Theory ◽  
Global Stability ◽  
Optimal Control Theory ◽  
Positive Impact ◽  
Equilibrium Points ◽  
Local And Global Stability ◽  
Number Of Individuals

The health of our respiratory systems is directly affected by the atmosphere. Nowadays, eruption of respiratory disease and malfunctioning of lung due to the presence of harmful particles in the air is one of the most sever challenge. In this chapter, association between air pollution-related respiratory diseases, namely dyspnea, cough, and asthma, is analysed by constructing a mathematical model. Local and global stability of the equilibrium points is proved. Optimal control theory is applied in the model to optimize stability of the model. Applied optimal control theory contains four control variables, among which first control helps to reduce number of individuals who are exposed to air pollutants and the remaining three controls help to reduce the spread and exacerbation of asthma. The positive impact of controls on the model and intensity of asthma under the influence of dyspnea and cough is observed graphically by simulating the model.

Stability Analysis and Optimization of By-Pass Controlled Heat Exchanger With Boiling

1976 ◽  
Vol 98 (2) ◽  
pp. 161-166 ◽  
Author(s):  
J. S. Ansari
Keyword(s):  
Mathematical Model ◽  
Optimal Control ◽  
Stability Analysis ◽  
Heat Exchanger ◽  
Flow Rate ◽  
Distributed Parameter ◽  
Final Temperature ◽  
Razumikhin Theorem ◽  
Sustained Oscillations ◽  
Intermediate Variables

A heat exchanger with boiling is considered. The final temperature of steam is controlled with the help of a controller which regulates the flow rate of by-pass water mixing with the outcoming steam. The simplest known mathematical model retaining the nonlinear and distributed parameter nature of the process is adopted. A known method of analysis, namely, Liapunov-Razumikhin theorem, is used to derive results on stability. An interesting feature of the system is that a positive feedback is required for stability. If the control is designed on the basis of minimization of the error in the final temperature alone, then the optimal control, requiring a negative feeedback, leads to sustained oscillations in the intermediate variables, even when the output is steady. The analysis, therefore suggests that meaningful optimization must take into account fluctuations in intermediate variables in addition to the error. A derivative control is shown to improve the transient response.

scholarly journals Fundamentals of Synthesized Optimal Control

Mathematics ◽  
2020 ◽  
Vol 9 (1) ◽  
pp. 21
Author(s):  
Askhat Diveev ◽  
Elizaveta Shmalko ◽  
Vladimir Serebrenny ◽  
Peter Zentay
Keyword(s):  
Mathematical Model ◽  
Optimal Control ◽  
Control Method ◽  
Direct Method ◽  
Equilibrium Points ◽  
Control Object ◽  
Optimal Control Method ◽  
Optimal Value ◽  
New Formulation ◽  
The Mathematical Model

This paper presents a new formulation of the optimal control problem with uncertainty, in which an additive bounded function is considered as uncertainty. The purpose of the control is to ensure the achievement of terminal conditions with the optimal value of the quality functional, while the uncertainty has a limited impact on the change in the value of the functional. The article introduces the concept of feasibility of the mathematical model of the object, which is associated with the contraction property of mappings if we consider the model of the object as a one-parameter mapping. It is shown that this property is sufficient for the development of stable practical systems. To find a solution to the stated problem, which would ensure the feasibility of the system, the synthesized optimal control method is proposed. This article formulates the theoretical foundations of the synthesized optimal control. The method consists in making the control object stable relative to some point in the state space and to control the object by changing the position of the equilibrium points. The article provides evidence that this approach is insensitive to the uncertainties of the mathematical model of the object. An example of the application of the method for optimal control of a group of robots is given. A comparison of the synthesized optimal control method with the direct method on the model without disturbances and with them is presented.

scholarly journals Stability Analysis of Mathematical Model on HIV Infection with the Effects of Antiretroviral therapy

2020 ◽  
Vol 17 (1) ◽  
pp. 109-116
Author(s):  
Lilis Dwi Sapta Aprilyani ◽  
Kasbawati Kasbawati ◽  
Syamsuddin Toaha
Keyword(s):  
Mathematical Model ◽  
Stability Analysis ◽  
Antiretroviral Therapy ◽  
Hiv Infection ◽  
Protease Inhibitor ◽  
Equilibrium Points ◽  
Genetic Material ◽  
Infection Model ◽  
Antiretroviral Therapies ◽  
Hiv Infection Model

HIV is a retrovirus, a virus which has enzymes and can convert genetic material from RNA to DNA. Antiretroviral therapies are the treatment to make the activity of the virus slow. The purpose of this article is to develop a mathematical model of HIV infection by reviewing antiretroviral therapy, analyze the equilibrium point, and determine the effectiveness of antiretroviral therapy. There are two equilibrium points in this HIV infection model, namely infection-free equilibrium and infected equilibrium. Numerical simulations are carried out based on selected parameters showed that infection free equilibrium is reached when the effectiveness of antiretroviral therapy is 0,4 for RT inhibitor and 0,3 for Protease Inhibitor. This means that antiretroviral therapy may change infected conditions to infection free conditions.

Spread of Tuberculosis Among Smokers

2020 ◽  
pp. 49-73
Author(s):  
Purvi M. Pandya ◽  
Ekta N. Jayswal ◽  
Yash Shah
Keyword(s):  
Mathematical Model ◽  
Stability Analysis ◽  
Numerical Simulations ◽  
Psychological Health ◽  
Matrix Method ◽  
Equilibrium Points ◽  
Research Work ◽  
Linear Ordinary Differential Equations ◽  
Smoking Addiction ◽  
The Mathematical Model

Smoking tobacco has some hazardous implications on an individual's physical, physiological, and psychological health; health of the passive smokers near him or her; and on the surrounding environment. From carcinomas to auto-immune disorders, smoking has a role to play. Therefore, there arises a need to frame a systemic pathway to decipher relationship between smoking and a perilous disease such as tuberculosis. This research work focuses on how drugs or medications can affect individuals who are susceptible to tuberculosis because of smoking habits and also on individuals who have already developed symptoms of tuberculosis due to their smoking addiction. The mathematical model is formulated using non-linear ordinary differential equations, and then threshold is calculated for different equilibrium points using next generation matrix method. Stability analysis along with numerical simulations are carried out to validate the data.

scholarly journals Θ-SEIHRD mathematical model of Covid19-stability analysis using fast-slow decomposition

PeerJ ◽  
2020 ◽  
Vol 8 ◽  
pp. e10019
Author(s):  
OPhir Nave ◽  
Israel Hartuv ◽  
Uziel Shemesh
Keyword(s):  
Mathematical Model ◽  
Stability Analysis ◽  
Nonlinear Differential Equations ◽  
Equilibrium Points ◽  
Asymptotic Methods ◽  
Fast Subsystem ◽  
The Stability ◽  
System Variables ◽  
The Mathematical Model ◽  
Slow Decomposition

In general, a mathematical model that contains many linear/nonlinear differential equations, describing a phenomenon, does not have an explicit hierarchy of system variables. That is, the identification of the fast variables and the slow variables of the system is not explicitly clear. The decomposition of a system into fast and slow subsystems is usually based on intuitive ideas and knowledge of the mathematical model being investigated. In this study, we apply the singular perturbed vector field (SPVF) method to the COVID-19 mathematical model of to expose the hierarchy of the model. This decomposition enables us to rewrite the model in new coordinates in the form of fast and slow subsystems and, hence, to investigate only the fast subsystem with different asymptotic methods. In addition, this decomposition enables us to investigate the stability analysis of the model, which is important in case of COVID-19. We found the stable equilibrium points of the mathematical model and compared the results of the model with those reported by the Chinese authorities and found a fit of approximately 96 percent.

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