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Yang Kuang

Arizona State University, Mathematics, Faculty Member

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Arizona State University, July, 1997--Present, ProfessorAugust, 1985--July, 1988, University of Alberta, Canada, Ph.DNov., 1984--Aug., 1985, Mathematical Institute, University of Oxford, D. Phil. candidate.Aug., 1983--July, 1984, Applied Mathematical Institute of Academic Sinica, P.R.C.Aug., 1980--Nov., 1984, University of Science and Technology of China, P.R.C.
Address: Bellevue, Washington, United States

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University of Zagreb

Kansas State University

Prof. Mohamed Labib Salem

Tanta University

Heather D Baker

University of Toronto

Roshan Chitrakar

Nepal College of Information Technology

Michael J. Gonzalez

Universidad de Puerto Rico Rio

Graduate Center of the City University of New York

Mikail F Lumentut

National Taiwan University

Hossein Sadegh Lafmejani

Università degli Studi di Parma (Italy)

University of Sarajevo

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Papers by Yang Kuang

Dynamics of a mechanistically derived stoichiometric producer-grazer model

Journal of Biological Dynamics, Jul 1, 2008

One of the simplest predator-prey models that tracks the quantity and the quality of prey is the ... more

Analysis of a Model of the Glucose‐Insulin Regulatory System with Two Delays

Siam Journal on Applied Mathematics, 2007

We continue a recent attempt to better understand the glucose-insulin regulatory system via a mat... more We continue a recent attempt to better understand the glucose-insulin regulatory system via a mathematical model of delay differential equations with two discrete time delays. With explicit delays, the model is more realistic in physiology, more accurate in mathematics, and more robust in applications. We study this model analytically and perform carefully designed numerical simulations by allowing two parameters to vary. Our analytical and numerical results confirm most current existing physiological observations and reveal more insightful information. The following factors are critical for ensuring the sustained oscillatory regulation and insulin secretion: (1) the time lag for insulin secretion stimulated by glucose and the newly synthesized insulin becoming "remote insulin" (Theorem 4.2 (b) and Theorem 5.6); (2) the delayed effect of hepatic glucose production (Theorem 4.2 (c) and Theorem 5.6); (3) moderate insulin clearance rate (Theorem 5.6 and simulations in section 6.4); and (4) nonoverwhelming glucose infusion (simulations in section 6.2, 6.3, and 6.4).

Dynamics and Growth Rate Implications of Ribosomes and mRNAs Interaction in <i>E. coli</i>

Social Science Research Network, 2021

Dynamics and growth rate implications of ribosomes and mRNAs interaction in E. coli

Heliyon

Biodiversity, Habitat Area, Resource Growth Rate and Interference Competition

Bulletin of Mathematical Biology, May 1, 2003

This highlights the crucial role that resource growth rates may play in promoting coexistence of ... more

Tumour-immune dynamics with an immune checkpoint inhibitor

Letters in Biomathematics, 2018

The use of immune checkpoint inhibitors is becoming more commonplace in clinical trials across th... more The use of immune checkpoint inhibitors is becoming more commonplace in clinical trials across the nation. Two important factors in the tumour-immune response are the checkpoint protein programmed death-1 (PD-1) and its ligand PD-L1. We propose a mathematical tumour-immune model using a system of ordinary differential equations to study dynamics with and without the use of anti-PD-1. A sensitivity analysis is conducted, and series of simulations are performed to investigate the effects of intermittent and continuous treatments on the tumour-immune dynamics. We consider the system without the anti-PD-1 drug to conduct a mathematical analysis to determine the stability of the tumourfree and tumorous equilibria. Through simulations, we found that a normally functioning immune system may control tumour. We observe treatment with anti-PD-1 alone may not be sufficient to eradicate tumour cells. Therefore, it may be beneficial to combine single agent treatments with additional therapies to obtain a better antitumour response.

Oscillatory dynamics of an intravenous glucose tolerance test model with delay interval

Chaos: An Interdisciplinary Journal of Nonlinear Science, 2017

Type 2 diabetes mellitus (T2DM) has become prevalent pandemic disease in view of the modern life ... more Type 2 diabetes mellitus (T2DM) has become prevalent pandemic disease in view of the modern life style. Both diabetic population and health expenses grow rapidly according to American Diabetes Association. Detecting the potential onset of T2DM is an essential focal point in the research of diabetes mellitus. The intravenous glucose tolerance test (IVGTT) is an effective protocol to determine the insulin sensitivity, glucose effectiveness, and pancreatic β-cell functionality, through the analysis and parameter estimation of a proper differential equation model. Delay differential equations have been used to study the complex physiological phenomena including the glucose and insulin regulations. In this paper, we propose a novel approach to model the time delay in IVGTT modeling. This novel approach uses two parameters to simulate not only both discrete time delay and distributed time delay in the past interval, but also the time delay distributed in a past sub-interval. Normally, lar...

Mathematical Models of Androgen Resistance in Prostate Cancer Patients under Intermittent Androgen Suppression Therapy

Applied Sciences, 2016

Predicting the timing of a castrate resistant prostate cancer is critical to lowering medical cos... more Predicting the timing of a castrate resistant prostate cancer is critical to lowering medical costs and improving the quality of life of advanced prostate cancer patients. We formulate, compare and analyze two mathematical models that aim to forecast future levels of prostate-specific antigen (PSA). We accomplish these tasks by employing clinical data of locally advanced prostate cancer patients undergoing androgen deprivation therapy (ADT). While these models are simplifications of a previously published model, they fit data with similar accuracy and improve forecasting results. Both models describe the progression of androgen resistance. Although Model 1 is simpler than the more realistic Model 2, it can fit clinical data to a greater precision. However, we found that Model 2 can forecast future PSA levels more accurately. These findings suggest that including more realistic mechanisms of androgen dynamics in a two population model may help androgen resistance timing prediction.

Mathematical Biology

Global analysis of a stoichiometric producer–grazer model with Holling type functional responses

Rich Dynamics of a General Producer–Grazer Interaction Model under Shared Multiple Resource Limitations

Applied Sciences

Organism growth is often determined by multiple resources interdependently. However, growth model... more Organism growth is often determined by multiple resources interdependently. However, growth models based on the Droop cell quota framework have historically been built using threshold formulations, which means they intrinsically involve single-resource limitations. In addition, it is a daunting task to study the global dynamics of these models mathematically, since they employ minimum functions that are non-smooth (not differentiable). To provide an approach to encompass interactions of multiple resources, we propose a multiple-resource limitation growth function based on the Droop cell quota concept and incorporate it into an existing producer–grazer model. The formulation of the producer’s growth rate is based on cell growth process time-tracking, while the grazer’s growth rate is constructed based on optimal limiting nutrient allocation in cell transcription and translation phases. We show that the proposed model captures a wide range of experimental observations, such as the par...

Rich dynamics of a simple delay host-pathogen model of cell-to-cell infection for plant virus

Discrete & Continuous Dynamical Systems - B, 2017

Viral dynamics within plant hosts can be important for understanding plant disease prevalence and... more Viral dynamics within plant hosts can be important for understanding plant disease prevalence and impacts. However, few mathematical modeling efforts aim to characterize within-plant viral dynamics. In this paper, we derive a simple system of delay differential equations that describes the spread of infection throughout the plant by barley and cereal yellow dwarf viruses via the cell-to-cell mechanism. By incorporating ratio-dependent incidence function and logistic growth of the healthy cells, the model can capture a wide range of biologically relevant phenomena via the disease-free, endemic, mutual extinction steady states, and a stable periodic orbit. We show that when the basic reproduction number is less than 1 (R 0 < 1), the disease-free steady state is asymptotically stable. When R 0 > 1, the dynamics either converge to the endemic equilibrium or enter a periodic orbit. Using a ratiodependent transformation, we show that if the infection rate is very high relative to the growth rate of healthy cells, then the system collapses to the mutual extinction steady state. Numerical and bifurcation simulations are provided to demonstrate our theoretical results. Finally, we carry out parameter estimation using experimental data to characterize the effects of varying nutrients on the dynamics of the system. Our parameter estimates suggest that varying the nutrient supply of nitrogen and phosphorous can alter the dynamics of the infection in plants, specifically reducing the rate of viral production and the rate of infection in certain cases.

Mathematical modeling of an immune checkpoint inhibitor and its synergy with an immunostimulant

Discrete and Continuous Dynamical Systems - Series B, 2017

Immune checkpoint inhibitors (ICIs) are a novel cancer therapy that may induce tumor regression a... more Immune checkpoint inhibitors (ICIs) are a novel cancer therapy that may induce tumor regression across multiple types of cancer. There has recently been interest in combining the ICIs with other forms of treatments, as not all patients benefit from monotherapy. We propose a mathematical model consisting of ordinary differential equations to investigate the combination treatments of the ICI avelumab and the immunostimulant NHS-muIL12. We validated the model using the average tumor volume curves provided in Xu et al. (2017). We initially analyzed a simple generic model without the use of any drug, which provided us with mathematical conditions for local stability for both the tumorous and tumor-free equilibrium. This enabled us to adapt these conditions for special cases of the model. Additionally, we conducted systematic mathematical analysis for the case that both drugs are applied continuously. Numerical simulations suggest that the two drugs act synergistically, such that, compared to monotherapy, only about one-third the dose of both drugs is required in combination for tumor control.

Tumor growth dynamics with nutrient limitation and cell proliferation time delay

Discrete and Continuous Dynamical Systems - Series B, 2017

It is known that avascular spherical solid tumors grow monotonically, often tends to a limiting f... more It is known that avascular spherical solid tumors grow monotonically, often tends to a limiting final size. This is repeatedly confirmed by various mathematical models consisting of mostly ordinary differential equations. However, cell growth is limited by nutrient and its proliferation incurs a time delay. In this paper, we formulate a nutrient limited compartmental model of avascular spherical solid tumor growth with cell proliferation time delay and study its limiting dynamics. The nutrient is assumed to enter the tumor proportional to its surface area. This model is a modification of a recent model which is built on a two-compartment model of cancer cell growth with transitions between proliferating and quiescent cells. Due to the limitation of resources, it is imperative that the population values or densities of a population model be nonnegative and bounded without any technical conditions. We confirm that our model meets this basic requirement. From an explicit expression of the tumor final size we show that the ratio of proliferating cells to the total tumor cells tends to zero as the death rate of quiescent cells tends to zero. We also study the stability of the tumor at steady states even though there is no Jacobian at the trivial steady state. The characteristic equation at the positive steady state is complicated so we made an initial effort to study some special cases in details. We find that delay may not destabilize the positive steady state in a very extreme situation. However, in a more general case, we show that sufficiently long cell proliferation delay can produce oscillatory solutions.

Modeling B cell Dysfunction in HIV infection

Progression from infection with the Human Immunodeficiency Virus (HIV) to AIDS is a complex proce... more Progression from infection with the Human Immunodeficiency Virus (HIV) to AIDS is a complex process that remains poorly understood. While mathematical models representing the ongoing battle between HIV and the immune system have been successful, they remain focused on cellular, as opposed to humoral, immunity. This situation remains in spite of the fact that recent evidence has shown the mediators of humoral immunity, the B cells, to represent a significant factor in the progression to AIDS. We propose here a ...

A Simple Si Model with Two Age Groups and Its Application to Us Hiv Epidemics: To Treat or Not to Treat?

Journal of Biological Systems, 2007

The objective of this paper is to study the global dynamics of a simple SI model with two explici... more The objective of this paper is to study the global dynamics of a simple SI model with two explicit age groups and apply the findings to the HIV dynamics in the United States. Specifically, we would like to explore the long term HIV dynamics to answer questions such as what will happen to human population level if all the demographical and epidemiological parameters stay constant. We also wonder if treatment alone will actually slow the spread of HIV or not. We divide the population into juvenile and adult groups. Only adults are assumed to be sexually active and we implicitly assume that the sex ratio is constant. We also assume that infected adults may produce both susceptible newborns and infected newborns. The model is fit with parameters from the HIV epidemic in the US. It produces an optimistic outcome: if nothing changes, the USA infected population may be halved in about 20 years. However, if treatment is found to extend the life expectance of infected individuals to 30 years...

A Simple Si Model with Two Age Groups and Its Application to Us Hiv Epidemics: To Treat or Not to Treat?

Journal of Biological Systems, 2007

The objective of this paper is to study the global dynamics of a simple SI model with two explici... more The objective of this paper is to study the global dynamics of a simple SI model with two explicit age groups and apply the findings to the HIV dynamics in the United States. Specifically, we would like to explore the long term HIV dynamics to answer questions such as what will happen to human population level if all the demographical and epidemiological parameters stay constant. We also wonder if treatment alone will actually slow the spread of HIV or not. We divide the population into juvenile and adult groups. Only adults are assumed to be sexually active and we implicitly assume that the sex ratio is constant. We also assume that infected adults may produce both susceptible newborns and infected newborns. The model is fit with parameters from the HIV epidemic in the US. It produces an optimistic outcome: if nothing changes, the USA infected population may be halved in about 20 years. However, if treatment is found to extend the life expectance of infected individuals to 30 years...

Analysis of tumor-immune dynamics in a delayed dendritic cell therapy model

Chaos: An Interdisciplinary Journal of Nonlinear Science, 2020

We formulate a tumor-immune interaction model with constant delay to capture the behavior followi... more We formulate a tumor-immune interaction model with constant delay to capture the behavior following application of a dendritic cell therapy. The model is validated using experimental data from melanoma-induced mice. Through theoretical and numerical analyses, the model is shown to produce rich dynamics, such as a Hopf bifurcation and bistability. We provide thresholds for tumor existence and, in a special case, tumor elimination. Our work indicates a sensitivity in model outcomes to the immune response time. We provide a stability analysis for the high tumor equilibrium. For small delays in response, the tumor and immune system coexist at a low level. Large delays give rise to fatally high tumor levels. Our computational and theoretical work reveals there exists an intermediate region of delay that generates complex oscillatory, even chaotic, behavior. The model then reflects uncertainty in treatment outcomes for varying initial tumor burdens, as well as tumor dormancy followed by uncontrolled growth to a lethal size, a phenomenon seen in vivo. Theoretical and computational analyses suggest efficacious treatments to use in conjunction with the dendritic cell vaccine. Additional analysis of a highly aggressive tumor additionally confirms the importance of representation with a time delay, as periodic solutions are strictly able to be generated when a delay is present. Delay differential equations (DDEs) are routinely used to reflect intricate behaviors caused by naturally occurring delays. We propose a set of differential equations to examine the tumor-immune dynamics following application of a dendritic cell therapy. Delay is used to reflect the immune system response time. This model provides a good fit to clinical data and exhibits naturally observed complexities, such as oscillatory behavior and bistability. Sensitivity in model outcomes to initial conditions is reflected and dependent on the immune system response time. The model additionally reveals regions of complex, chaotic behavior and establishes thresholds for tumor elimination and existence.

A review on delay differential equation models in diabetes

This is a sequel to the paper by Makroglou et al (2010) which gave a review of diabetes modeling ... more This is a sequel to the paper by Makroglou et al (2010) which gave a review of diabetes modeling in the form of delay differential equations associated with ultradian oscillations and the intravenous glucose tolerance test (IVGTT). This sequel includes a review of diabetes modeling in the form of delay differential equations associated with insulin therapies and models involving intracellular activities of β-cells. The statement of the models is accompanied with some computational results and brief summaries of theoretical results.

Solutions and properties of some degenerate systems of difference equations

Journal of Computational Analysis and Applications, 2015

This paper is devoted to obtain the form of the solution and the qualitative properties of the fo... more This paper is devoted to obtain the form of the solution and the qualitative properties of the following systems of a rational di erence equations of order two +1 = 1 (±1± 1) +1 = 1 (±1± 1) with positive initial conditions 1 0 1 and 0 are nonzero real numbers. If we let = 1 and = 1, then these systems can be viewed as special cases of the system of the form +1 = ( ) +1 = ( ) This system has applications in modeling population growth with age structure or the dynamics of plant-herbivore interaction. Let = 2 , we have +1 = ( ( )) ( ). At a nonzero steady state of the last di erence equation, we have | 0 )| = | ( ( )) ( )| = 1 indicating that the system is degenerate at this steady state.

Periodic Solutions of Systems of Delay Differential Equations

We establish several criteria for the existence, multiplicity and nonexistence of positive period... more

Dynamics of a mechanistically derived stoichiometric producer-grazer model

Journal of Biological Dynamics, Jul 1, 2008

One of the simplest predator-prey models that tracks the quantity and the quality of prey is the ... more

Analysis of a Model of the Glucose‐Insulin Regulatory System with Two Delays

Siam Journal on Applied Mathematics, 2007

We continue a recent attempt to better understand the glucose-insulin regulatory system via a mat... more We continue a recent attempt to better understand the glucose-insulin regulatory system via a mathematical model of delay differential equations with two discrete time delays. With explicit delays, the model is more realistic in physiology, more accurate in mathematics, and more robust in applications. We study this model analytically and perform carefully designed numerical simulations by allowing two parameters to vary. Our analytical and numerical results confirm most current existing physiological observations and reveal more insightful information. The following factors are critical for ensuring the sustained oscillatory regulation and insulin secretion: (1) the time lag for insulin secretion stimulated by glucose and the newly synthesized insulin becoming "remote insulin" (Theorem 4.2 (b) and Theorem 5.6); (2) the delayed effect of hepatic glucose production (Theorem 4.2 (c) and Theorem 5.6); (3) moderate insulin clearance rate (Theorem 5.6 and simulations in section 6.4); and (4) nonoverwhelming glucose infusion (simulations in section 6.2, 6.3, and 6.4).

Dynamics and Growth Rate Implications of Ribosomes and mRNAs Interaction in <i>E. coli</i>

Social Science Research Network, 2021

Dynamics and growth rate implications of ribosomes and mRNAs interaction in E. coli

Heliyon

Biodiversity, Habitat Area, Resource Growth Rate and Interference Competition

Bulletin of Mathematical Biology, May 1, 2003

This highlights the crucial role that resource growth rates may play in promoting coexistence of ... more

Tumour-immune dynamics with an immune checkpoint inhibitor

Letters in Biomathematics, 2018

The use of immune checkpoint inhibitors is becoming more commonplace in clinical trials across th... more The use of immune checkpoint inhibitors is becoming more commonplace in clinical trials across the nation. Two important factors in the tumour-immune response are the checkpoint protein programmed death-1 (PD-1) and its ligand PD-L1. We propose a mathematical tumour-immune model using a system of ordinary differential equations to study dynamics with and without the use of anti-PD-1. A sensitivity analysis is conducted, and series of simulations are performed to investigate the effects of intermittent and continuous treatments on the tumour-immune dynamics. We consider the system without the anti-PD-1 drug to conduct a mathematical analysis to determine the stability of the tumourfree and tumorous equilibria. Through simulations, we found that a normally functioning immune system may control tumour. We observe treatment with anti-PD-1 alone may not be sufficient to eradicate tumour cells. Therefore, it may be beneficial to combine single agent treatments with additional therapies to obtain a better antitumour response.

Oscillatory dynamics of an intravenous glucose tolerance test model with delay interval

Chaos: An Interdisciplinary Journal of Nonlinear Science, 2017

Type 2 diabetes mellitus (T2DM) has become prevalent pandemic disease in view of the modern life ... more Type 2 diabetes mellitus (T2DM) has become prevalent pandemic disease in view of the modern life style. Both diabetic population and health expenses grow rapidly according to American Diabetes Association. Detecting the potential onset of T2DM is an essential focal point in the research of diabetes mellitus. The intravenous glucose tolerance test (IVGTT) is an effective protocol to determine the insulin sensitivity, glucose effectiveness, and pancreatic β-cell functionality, through the analysis and parameter estimation of a proper differential equation model. Delay differential equations have been used to study the complex physiological phenomena including the glucose and insulin regulations. In this paper, we propose a novel approach to model the time delay in IVGTT modeling. This novel approach uses two parameters to simulate not only both discrete time delay and distributed time delay in the past interval, but also the time delay distributed in a past sub-interval. Normally, lar...

Mathematical Models of Androgen Resistance in Prostate Cancer Patients under Intermittent Androgen Suppression Therapy

Applied Sciences, 2016

Predicting the timing of a castrate resistant prostate cancer is critical to lowering medical cos... more Predicting the timing of a castrate resistant prostate cancer is critical to lowering medical costs and improving the quality of life of advanced prostate cancer patients. We formulate, compare and analyze two mathematical models that aim to forecast future levels of prostate-specific antigen (PSA). We accomplish these tasks by employing clinical data of locally advanced prostate cancer patients undergoing androgen deprivation therapy (ADT). While these models are simplifications of a previously published model, they fit data with similar accuracy and improve forecasting results. Both models describe the progression of androgen resistance. Although Model 1 is simpler than the more realistic Model 2, it can fit clinical data to a greater precision. However, we found that Model 2 can forecast future PSA levels more accurately. These findings suggest that including more realistic mechanisms of androgen dynamics in a two population model may help androgen resistance timing prediction.

Mathematical Biology

Global analysis of a stoichiometric producer–grazer model with Holling type functional responses

Rich Dynamics of a General Producer–Grazer Interaction Model under Shared Multiple Resource Limitations

Applied Sciences

Organism growth is often determined by multiple resources interdependently. However, growth model... more Organism growth is often determined by multiple resources interdependently. However, growth models based on the Droop cell quota framework have historically been built using threshold formulations, which means they intrinsically involve single-resource limitations. In addition, it is a daunting task to study the global dynamics of these models mathematically, since they employ minimum functions that are non-smooth (not differentiable). To provide an approach to encompass interactions of multiple resources, we propose a multiple-resource limitation growth function based on the Droop cell quota concept and incorporate it into an existing producer–grazer model. The formulation of the producer’s growth rate is based on cell growth process time-tracking, while the grazer’s growth rate is constructed based on optimal limiting nutrient allocation in cell transcription and translation phases. We show that the proposed model captures a wide range of experimental observations, such as the par...

Rich dynamics of a simple delay host-pathogen model of cell-to-cell infection for plant virus

Discrete & Continuous Dynamical Systems - B, 2017

Viral dynamics within plant hosts can be important for understanding plant disease prevalence and... more Viral dynamics within plant hosts can be important for understanding plant disease prevalence and impacts. However, few mathematical modeling efforts aim to characterize within-plant viral dynamics. In this paper, we derive a simple system of delay differential equations that describes the spread of infection throughout the plant by barley and cereal yellow dwarf viruses via the cell-to-cell mechanism. By incorporating ratio-dependent incidence function and logistic growth of the healthy cells, the model can capture a wide range of biologically relevant phenomena via the disease-free, endemic, mutual extinction steady states, and a stable periodic orbit. We show that when the basic reproduction number is less than 1 (R 0 < 1), the disease-free steady state is asymptotically stable. When R 0 > 1, the dynamics either converge to the endemic equilibrium or enter a periodic orbit. Using a ratiodependent transformation, we show that if the infection rate is very high relative to the growth rate of healthy cells, then the system collapses to the mutual extinction steady state. Numerical and bifurcation simulations are provided to demonstrate our theoretical results. Finally, we carry out parameter estimation using experimental data to characterize the effects of varying nutrients on the dynamics of the system. Our parameter estimates suggest that varying the nutrient supply of nitrogen and phosphorous can alter the dynamics of the infection in plants, specifically reducing the rate of viral production and the rate of infection in certain cases.

Mathematical modeling of an immune checkpoint inhibitor and its synergy with an immunostimulant

Discrete and Continuous Dynamical Systems - Series B, 2017

Immune checkpoint inhibitors (ICIs) are a novel cancer therapy that may induce tumor regression a... more Immune checkpoint inhibitors (ICIs) are a novel cancer therapy that may induce tumor regression across multiple types of cancer. There has recently been interest in combining the ICIs with other forms of treatments, as not all patients benefit from monotherapy. We propose a mathematical model consisting of ordinary differential equations to investigate the combination treatments of the ICI avelumab and the immunostimulant NHS-muIL12. We validated the model using the average tumor volume curves provided in Xu et al. (2017). We initially analyzed a simple generic model without the use of any drug, which provided us with mathematical conditions for local stability for both the tumorous and tumor-free equilibrium. This enabled us to adapt these conditions for special cases of the model. Additionally, we conducted systematic mathematical analysis for the case that both drugs are applied continuously. Numerical simulations suggest that the two drugs act synergistically, such that, compared to monotherapy, only about one-third the dose of both drugs is required in combination for tumor control.

Tumor growth dynamics with nutrient limitation and cell proliferation time delay

Discrete and Continuous Dynamical Systems - Series B, 2017

It is known that avascular spherical solid tumors grow monotonically, often tends to a limiting f... more It is known that avascular spherical solid tumors grow monotonically, often tends to a limiting final size. This is repeatedly confirmed by various mathematical models consisting of mostly ordinary differential equations. However, cell growth is limited by nutrient and its proliferation incurs a time delay. In this paper, we formulate a nutrient limited compartmental model of avascular spherical solid tumor growth with cell proliferation time delay and study its limiting dynamics. The nutrient is assumed to enter the tumor proportional to its surface area. This model is a modification of a recent model which is built on a two-compartment model of cancer cell growth with transitions between proliferating and quiescent cells. Due to the limitation of resources, it is imperative that the population values or densities of a population model be nonnegative and bounded without any technical conditions. We confirm that our model meets this basic requirement. From an explicit expression of the tumor final size we show that the ratio of proliferating cells to the total tumor cells tends to zero as the death rate of quiescent cells tends to zero. We also study the stability of the tumor at steady states even though there is no Jacobian at the trivial steady state. The characteristic equation at the positive steady state is complicated so we made an initial effort to study some special cases in details. We find that delay may not destabilize the positive steady state in a very extreme situation. However, in a more general case, we show that sufficiently long cell proliferation delay can produce oscillatory solutions.

Modeling B cell Dysfunction in HIV infection

Progression from infection with the Human Immunodeficiency Virus (HIV) to AIDS is a complex proce... more Progression from infection with the Human Immunodeficiency Virus (HIV) to AIDS is a complex process that remains poorly understood. While mathematical models representing the ongoing battle between HIV and the immune system have been successful, they remain focused on cellular, as opposed to humoral, immunity. This situation remains in spite of the fact that recent evidence has shown the mediators of humoral immunity, the B cells, to represent a significant factor in the progression to AIDS. We propose here a ...

A Simple Si Model with Two Age Groups and Its Application to Us Hiv Epidemics: To Treat or Not to Treat?

Journal of Biological Systems, 2007

The objective of this paper is to study the global dynamics of a simple SI model with two explici... more The objective of this paper is to study the global dynamics of a simple SI model with two explicit age groups and apply the findings to the HIV dynamics in the United States. Specifically, we would like to explore the long term HIV dynamics to answer questions such as what will happen to human population level if all the demographical and epidemiological parameters stay constant. We also wonder if treatment alone will actually slow the spread of HIV or not. We divide the population into juvenile and adult groups. Only adults are assumed to be sexually active and we implicitly assume that the sex ratio is constant. We also assume that infected adults may produce both susceptible newborns and infected newborns. The model is fit with parameters from the HIV epidemic in the US. It produces an optimistic outcome: if nothing changes, the USA infected population may be halved in about 20 years. However, if treatment is found to extend the life expectance of infected individuals to 30 years...

A Simple Si Model with Two Age Groups and Its Application to Us Hiv Epidemics: To Treat or Not to Treat?

Journal of Biological Systems, 2007

The objective of this paper is to study the global dynamics of a simple SI model with two explici... more The objective of this paper is to study the global dynamics of a simple SI model with two explicit age groups and apply the findings to the HIV dynamics in the United States. Specifically, we would like to explore the long term HIV dynamics to answer questions such as what will happen to human population level if all the demographical and epidemiological parameters stay constant. We also wonder if treatment alone will actually slow the spread of HIV or not. We divide the population into juvenile and adult groups. Only adults are assumed to be sexually active and we implicitly assume that the sex ratio is constant. We also assume that infected adults may produce both susceptible newborns and infected newborns. The model is fit with parameters from the HIV epidemic in the US. It produces an optimistic outcome: if nothing changes, the USA infected population may be halved in about 20 years. However, if treatment is found to extend the life expectance of infected individuals to 30 years...

Analysis of tumor-immune dynamics in a delayed dendritic cell therapy model

Chaos: An Interdisciplinary Journal of Nonlinear Science, 2020

We formulate a tumor-immune interaction model with constant delay to capture the behavior followi... more We formulate a tumor-immune interaction model with constant delay to capture the behavior following application of a dendritic cell therapy. The model is validated using experimental data from melanoma-induced mice. Through theoretical and numerical analyses, the model is shown to produce rich dynamics, such as a Hopf bifurcation and bistability. We provide thresholds for tumor existence and, in a special case, tumor elimination. Our work indicates a sensitivity in model outcomes to the immune response time. We provide a stability analysis for the high tumor equilibrium. For small delays in response, the tumor and immune system coexist at a low level. Large delays give rise to fatally high tumor levels. Our computational and theoretical work reveals there exists an intermediate region of delay that generates complex oscillatory, even chaotic, behavior. The model then reflects uncertainty in treatment outcomes for varying initial tumor burdens, as well as tumor dormancy followed by uncontrolled growth to a lethal size, a phenomenon seen in vivo. Theoretical and computational analyses suggest efficacious treatments to use in conjunction with the dendritic cell vaccine. Additional analysis of a highly aggressive tumor additionally confirms the importance of representation with a time delay, as periodic solutions are strictly able to be generated when a delay is present. Delay differential equations (DDEs) are routinely used to reflect intricate behaviors caused by naturally occurring delays. We propose a set of differential equations to examine the tumor-immune dynamics following application of a dendritic cell therapy. Delay is used to reflect the immune system response time. This model provides a good fit to clinical data and exhibits naturally observed complexities, such as oscillatory behavior and bistability. Sensitivity in model outcomes to initial conditions is reflected and dependent on the immune system response time. The model additionally reveals regions of complex, chaotic behavior and establishes thresholds for tumor elimination and existence.

A review on delay differential equation models in diabetes

This is a sequel to the paper by Makroglou et al (2010) which gave a review of diabetes modeling ... more This is a sequel to the paper by Makroglou et al (2010) which gave a review of diabetes modeling in the form of delay differential equations associated with ultradian oscillations and the intravenous glucose tolerance test (IVGTT). This sequel includes a review of diabetes modeling in the form of delay differential equations associated with insulin therapies and models involving intracellular activities of β-cells. The statement of the models is accompanied with some computational results and brief summaries of theoretical results.

Solutions and properties of some degenerate systems of difference equations

Journal of Computational Analysis and Applications, 2015

This paper is devoted to obtain the form of the solution and the qualitative properties of the fo... more This paper is devoted to obtain the form of the solution and the qualitative properties of the following systems of a rational di erence equations of order two +1 = 1 (±1± 1) +1 = 1 (±1± 1) with positive initial conditions 1 0 1 and 0 are nonzero real numbers. If we let = 1 and = 1, then these systems can be viewed as special cases of the system of the form +1 = ( ) +1 = ( ) This system has applications in modeling population growth with age structure or the dynamics of plant-herbivore interaction. Let = 2 , we have +1 = ( ( )) ( ). At a nonzero steady state of the last di erence equation, we have | 0 )| = | ( ( )) ( )| = 1 indicating that the system is degenerate at this steady state.

Periodic Solutions of Systems of Delay Differential Equations

We establish several criteria for the existence, multiplicity and nonexistence of positive period... more