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1551-0018
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Mathematical Biosciences & Engineering
2004 , Volume 1 , Issue 2
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2004, 1(2): 215-222
doi: 10.3934/mbe.2004.1.215
+[Abstract](3214)
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Abstract:
The purpose of this note is to mechanistically formulate a mathematically tractable model that specifically deals with the dynamics of plant-herbivore interaction in a closed phosphorous (P) limiting environment. The key to our approach is the employment of the plant cell P quota and the Droop equation for its growth. Our model takes the simple form of a system of two autonomous ordinary differential equations. It can be shown that our model includes the LKE model (Loladze, Kuang and Elser (2000)) as a special case. Our study reveals that the details of ecological stoichiometry models really matter for quantitative predictions of plant-herbivore dynamics, especially at intermediate ranges of the carrying capacity.
The purpose of this note is to mechanistically formulate a mathematically tractable model that specifically deals with the dynamics of plant-herbivore interaction in a closed phosphorous (P) limiting environment. The key to our approach is the employment of the plant cell P quota and the Droop equation for its growth. Our model takes the simple form of a system of two autonomous ordinary differential equations. It can be shown that our model includes the LKE model (Loladze, Kuang and Elser (2000)) as a special case. Our study reveals that the details of ecological stoichiometry models really matter for quantitative predictions of plant-herbivore dynamics, especially at intermediate ranges of the carrying capacity.
2004, 1(2): 223-241
doi: 10.3934/mbe.2004.1.223
+[Abstract](3921)
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Abstract:
We formulate a dynamic mathematical model that describes the interaction of the immune system with the human immunodeficiency virus (HIV) and that permits drug ''cocktail'' therapies. We derive HIV therapeutic strategies by formulating and analyzing an optimal control problem using two types of dynamic treatments representing reverse transcriptase (RT) inhibitors and protease inhibitors (PIs). Continuous optimal therapies are found by solving the corresponding optimality systems. In addition, using ideas from dynamic programming, we formulate and derive suboptimal structured treatment interruptions (STI) in antiviral therapy that include drug-free periods of immune-mediated control of HIV. Our numerical results support a scenario in which STI therapies can lead to long-term control of HIV by the immune response system after discontinuation of therapy.
We formulate a dynamic mathematical model that describes the interaction of the immune system with the human immunodeficiency virus (HIV) and that permits drug ''cocktail'' therapies. We derive HIV therapeutic strategies by formulating and analyzing an optimal control problem using two types of dynamic treatments representing reverse transcriptase (RT) inhibitors and protease inhibitors (PIs). Continuous optimal therapies are found by solving the corresponding optimality systems. In addition, using ideas from dynamic programming, we formulate and derive suboptimal structured treatment interruptions (STI) in antiviral therapy that include drug-free periods of immune-mediated control of HIV. Our numerical results support a scenario in which STI therapies can lead to long-term control of HIV by the immune response system after discontinuation of therapy.
2004, 1(2): 243-266
doi: 10.3934/mbe.2004.1.243
+[Abstract](2260)
+[PDF](236.0KB)
Abstract:
Although in the broadly defined genetic algebra, multiplication suggests a forward direction of from parents to progeny, when looking from the reverse direction, it also suggests to us a new algebraic structure --- coalgebraic structure, which we call genetic coalgebras. It is not the dual coalgebraic structure and can be used in the construction of phylogenetic trees. Mathematically, to construct phylogenetic trees means we need to solve equations x[n] =a , or x(n)=b. It is generally impossible to solve these equations in algebras. However, we can solve them in coalgebras in the sense of tracing back for their ancestors. A thorough exploration of coalgebraic structure in genetics is apparently necessary. Here, we develop a theoretical framework of the coalgebraic structure of genetics. From biological viewpoint, we defined various fundamental concepts and examined their elementary properties that contain genetic significance. Mathematically, by genetic coalgebra, we mean any coalgebra that occurs in genetics. They are generally noncoassociative and without counit; and in the case of non-sex-linked inheritance, they are cocommutative. Each coalgebra with genetic realization has a baric property. We have also discussed the methods to construct new genetic coalgebras, including cocommutative duplication, the tensor product, linear combinations and the skew linear map, which allow us to describe complex genetic traits. We also put forward certain theorems that state the relationship between gametic coalgebra and gametic algebra. By Brower's theorem in topology, we prove the existence of equilibrium state for the in-evolution operator.
Although in the broadly defined genetic algebra, multiplication suggests a forward direction of from parents to progeny, when looking from the reverse direction, it also suggests to us a new algebraic structure --- coalgebraic structure, which we call genetic coalgebras. It is not the dual coalgebraic structure and can be used in the construction of phylogenetic trees. Mathematically, to construct phylogenetic trees means we need to solve equations x[n] =a , or x(n)=b. It is generally impossible to solve these equations in algebras. However, we can solve them in coalgebras in the sense of tracing back for their ancestors. A thorough exploration of coalgebraic structure in genetics is apparently necessary. Here, we develop a theoretical framework of the coalgebraic structure of genetics. From biological viewpoint, we defined various fundamental concepts and examined their elementary properties that contain genetic significance. Mathematically, by genetic coalgebra, we mean any coalgebra that occurs in genetics. They are generally noncoassociative and without counit; and in the case of non-sex-linked inheritance, they are cocommutative. Each coalgebra with genetic realization has a baric property. We have also discussed the methods to construct new genetic coalgebras, including cocommutative duplication, the tensor product, linear combinations and the skew linear map, which allow us to describe complex genetic traits. We also put forward certain theorems that state the relationship between gametic coalgebra and gametic algebra. By Brower's theorem in topology, we prove the existence of equilibrium state for the in-evolution operator.
2004, 1(2): 267-288
doi: 10.3934/mbe.2004.1.267
+[Abstract](3422)
+[PDF](285.6KB)
Abstract:
Mathematical models of HIV-1 infection can help interpret drug treatment experiments and improve our understanding of the interplay between HIV-1 and the immune system. We develop and analyze an age-structured model of HIV-1 infection that allows for variations in the death rate of productively infected T cells and the production rate of viral particles as a function of the length of time a T cell has been infected. We show that this model is a generalization of the standard differential equation and of delay models previously used to describe HIV-1 infection, and provides a means for exploring fundamental issues of viral production and death. We show that the model has uninfected and infected steady states, linked by a transcritical bifurcation. We perform a local stability analysis of the nontrivial equilibrium solution and provide a general stability condition for models with age structure. We then use numerical methods to study solutions of our model focusing on the analysis of primary HIV infection. We show that the time to reach peak viral levels in the blood depends not only on initial conditions but also on the way in which viral production ramps up. If viral production ramps up slowly, we find that the time to peak viral load is delayed compared to results obtained using the standard (constant viral production) model of HIV infection. We find that data on viral load changing over time is insufficient to identify the functions specifying the dependence of the viral production rate or infected cell death rate on infected cell age. These functions must be determined through new quantitative experiments.
Mathematical models of HIV-1 infection can help interpret drug treatment experiments and improve our understanding of the interplay between HIV-1 and the immune system. We develop and analyze an age-structured model of HIV-1 infection that allows for variations in the death rate of productively infected T cells and the production rate of viral particles as a function of the length of time a T cell has been infected. We show that this model is a generalization of the standard differential equation and of delay models previously used to describe HIV-1 infection, and provides a means for exploring fundamental issues of viral production and death. We show that the model has uninfected and infected steady states, linked by a transcritical bifurcation. We perform a local stability analysis of the nontrivial equilibrium solution and provide a general stability condition for models with age structure. We then use numerical methods to study solutions of our model focusing on the analysis of primary HIV infection. We show that the time to reach peak viral levels in the blood depends not only on initial conditions but also on the way in which viral production ramps up. If viral production ramps up slowly, we find that the time to peak viral load is delayed compared to results obtained using the standard (constant viral production) model of HIV infection. We find that data on viral load changing over time is insufficient to identify the functions specifying the dependence of the viral production rate or infected cell death rate on infected cell age. These functions must be determined through new quantitative experiments.
2004, 1(2): 289-305
doi: 10.3934/mbe.2004.1.289
+[Abstract](2194)
+[PDF](1339.3KB)
Abstract:
T-lymphocyte (T-cell) development constitutes one of the basic and most vital processes in immunology. The process is profoundly affected by the thymic microenvironment, the dysregulation of which may be the pathogenesis or the etiology of some diseases. On the basis of a general conceptual framework, we have designed the first biophysical model to describe thymocyte development. The microclimate within the thymus, which is shaped by various cytokines, is first conceptualized into a growth field $\lambda$ and a differentiation field $\mu$, under the influence of which the thymocytes mature. A partial differential equation is then derived through the analysis of an infinitesimal element of the flow of thymocytes. A general method is presented to estimate the two fields based on experimental data obtained by flow cytometric analysis of the thymus. Numerical examples are given for both normal and pathologic conditions. Our results are quite good, and even the time varying fields can be accurately estimated. Our method has demonstrated its great potential for the study of immunopathogenesis. The plan for implementation of the method is addressed.
T-lymphocyte (T-cell) development constitutes one of the basic and most vital processes in immunology. The process is profoundly affected by the thymic microenvironment, the dysregulation of which may be the pathogenesis or the etiology of some diseases. On the basis of a general conceptual framework, we have designed the first biophysical model to describe thymocyte development. The microclimate within the thymus, which is shaped by various cytokines, is first conceptualized into a growth field $\lambda$ and a differentiation field $\mu$, under the influence of which the thymocytes mature. A partial differential equation is then derived through the analysis of an infinitesimal element of the flow of thymocytes. A general method is presented to estimate the two fields based on experimental data obtained by flow cytometric analysis of the thymus. Numerical examples are given for both normal and pathologic conditions. Our results are quite good, and even the time varying fields can be accurately estimated. Our method has demonstrated its great potential for the study of immunopathogenesis. The plan for implementation of the method is addressed.
2004, 1(2): 307-324
doi: 10.3934/mbe.2004.1.307
+[Abstract](2528)
+[PDF](569.6KB)
Abstract:
A symmetrical cubic discrete coupled logistic equation is proposed to model the symbiotic interaction of two isolated species. The coupling depends on the population size of both species and on a positive constant $\lambda$, called the mutual benefit. Different dynamical regimes are obtained when the mutual benefit is modified. For small $\lambda$, the species become extinct. For increasing $\lambda$, the system stabilizes in a synchronized state or oscillates in a two-periodic orbit. For the greatest permitted values of $\lambda$, the dynamics evolves into a quasiperiodic, into a chaotic scenario, or into extinction. The basins for these regimes are visualized as colored figures on the plane. These patterns suffer different changes as consequence of basins' bifurcations. The use of the critical curves allows us to determine the influence of the zones with different numbers of first-rank preimages in those bifurcation mechanisms.
A symmetrical cubic discrete coupled logistic equation is proposed to model the symbiotic interaction of two isolated species. The coupling depends on the population size of both species and on a positive constant $\lambda$, called the mutual benefit. Different dynamical regimes are obtained when the mutual benefit is modified. For small $\lambda$, the species become extinct. For increasing $\lambda$, the system stabilizes in a synchronized state or oscillates in a two-periodic orbit. For the greatest permitted values of $\lambda$, the dynamics evolves into a quasiperiodic, into a chaotic scenario, or into extinction. The basins for these regimes are visualized as colored figures on the plane. These patterns suffer different changes as consequence of basins' bifurcations. The use of the critical curves allows us to determine the influence of the zones with different numbers of first-rank preimages in those bifurcation mechanisms.
2004, 1(2): 325-338
doi: 10.3934/mbe.2004.1.325
+[Abstract](2108)
+[PDF](500.8KB)
Abstract:
The scientific importance of understanding programmed cell death is undeniable; however, the complexity of death signal propagation and the formerly incomplete knowledge of apoptotic pathways has left this topic virtually untouched by mathematical modeling. In this paper, we use a mechanistic approach to frame the current understanding of receptor-mediated apoptosis with an immediate goal of isolating the role receptor trimerization plays in this process. Analysis and simulation suggest that if the death signal is to be successful at low-receptor, high-ligand concentration, Fas trimerization is unlikely to be the driving force in the signal propagation. However at high-receptor and low-ligand concentrations, the mathematical model illustrates how the ability of FasL to cluster three Fas receptors can be crucially important for downstream events that propagate the apoptotic signal.
The scientific importance of understanding programmed cell death is undeniable; however, the complexity of death signal propagation and the formerly incomplete knowledge of apoptotic pathways has left this topic virtually untouched by mathematical modeling. In this paper, we use a mechanistic approach to frame the current understanding of receptor-mediated apoptosis with an immediate goal of isolating the role receptor trimerization plays in this process. Analysis and simulation suggest that if the death signal is to be successful at low-receptor, high-ligand concentration, Fas trimerization is unlikely to be the driving force in the signal propagation. However at high-receptor and low-ligand concentrations, the mathematical model illustrates how the ability of FasL to cluster three Fas receptors can be crucially important for downstream events that propagate the apoptotic signal.
2004, 1(2): 339-346
doi: 10.3934/mbe.2004.1.339
+[Abstract](2738)
+[PDF](1961.5KB)
Abstract:
Complete synchronization between two Hele-Shaw cells is examined. The two dynamical systems are chaotic in time and spatially extended in two dimensions. It is shown that a large number of connectors are needed to achieve synchronization. In particular, we have studied how the number of connectors influences the dynamical regime that is set inside the Hele-Shaw cells.
Complete synchronization between two Hele-Shaw cells is examined. The two dynamical systems are chaotic in time and spatially extended in two dimensions. It is shown that a large number of connectors are needed to achieve synchronization. In particular, we have studied how the number of connectors influences the dynamical regime that is set inside the Hele-Shaw cells.
2004, 1(2): 347-359
doi: 10.3934/mbe.2004.1.347
+[Abstract](3282)
+[PDF](1236.4KB)
Abstract:
We consider systems that are well modelled as networks that evolve in time, which we call Moving Neighborhood Networks. These models are relevant in studying cooperative behavior of swarms and other phenomena where emergent interactions arise from ad hoc networks. In a natural way, the time-averaged degree distribution gives rise to a scale-free network. Simulations show that although the network may have many noncommunicating components, the recent weighted time-averaged communication is sufficient to yield robust synchronization of chaotic oscillators. In particular, we contend that such time-varying networks are important to model in the situation where each agent carries a pathogen (such as a disease) in which the pathogen's life-cycle has a natural time-scale which competes with the time-scale of movement of the agents, and thus with the networks communication channels.
We consider systems that are well modelled as networks that evolve in time, which we call Moving Neighborhood Networks. These models are relevant in studying cooperative behavior of swarms and other phenomena where emergent interactions arise from ad hoc networks. In a natural way, the time-averaged degree distribution gives rise to a scale-free network. Simulations show that although the network may have many noncommunicating components, the recent weighted time-averaged communication is sufficient to yield robust synchronization of chaotic oscillators. In particular, we contend that such time-varying networks are important to model in the situation where each agent carries a pathogen (such as a disease) in which the pathogen's life-cycle has a natural time-scale which competes with the time-scale of movement of the agents, and thus with the networks communication channels.
2004, 1(2): 361-404
doi: 10.3934/mbe.2004.1.361
+[Abstract](11679)
+[PDF](2524.1KB)
Abstract:
The reemergence of tuberculosis (TB) from the 1980s to the early 1990s instigated extensive researches on the mechanisms behind the transmission dynamics of TB epidemics. This article provides a detailed review of the work on the dynamics and control of TB. The earliest mathematical models describing the TB dynamics appeared in the 1960s and focused on the prediction and control strategies using simulation approaches. Most recently developed models not only pay attention to simulations but also take care of dynamical analysis using modern knowledge of dynamical systems. Questions addressed by these models mainly concentrate on TB control strategies, optimal vaccination policies, approaches toward the elimination of TB in the U.S.A., TB co-infection with HIV/AIDS, drug-resistant TB, responses of the immune system, impacts of demography, the role of public transportation systems, and the impact of contact patterns. Model formulations involve a variety of mathematical areas, such as ODEs (Ordinary Differential Equations) (both autonomous and non-autonomous systems), PDEs (Partial Differential Equations), system of difference equations, system of integro-differential equations, Markov chain model, and simulation models.
The reemergence of tuberculosis (TB) from the 1980s to the early 1990s instigated extensive researches on the mechanisms behind the transmission dynamics of TB epidemics. This article provides a detailed review of the work on the dynamics and control of TB. The earliest mathematical models describing the TB dynamics appeared in the 1960s and focused on the prediction and control strategies using simulation approaches. Most recently developed models not only pay attention to simulations but also take care of dynamical analysis using modern knowledge of dynamical systems. Questions addressed by these models mainly concentrate on TB control strategies, optimal vaccination policies, approaches toward the elimination of TB in the U.S.A., TB co-infection with HIV/AIDS, drug-resistant TB, responses of the immune system, impacts of demography, the role of public transportation systems, and the impact of contact patterns. Model formulations involve a variety of mathematical areas, such as ODEs (Ordinary Differential Equations) (both autonomous and non-autonomous systems), PDEs (Partial Differential Equations), system of difference equations, system of integro-differential equations, Markov chain model, and simulation models.
2018
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