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Discrete & Continuous Dynamical Systems - B
July 2007 , Volume 8 , Issue 1
Special Issue on
Differential Equations in Mathematical Biology
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2007, 8(1): i-ii
doi: 10.3934/dcdsb.2007.8.1i
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Abstract:
This special issue is the proceedings of the International Workshop on Differential Equations in Mathematical Biology held in Le Havre, France, July 11-13, 2005. The workshop brought together internationals researchers in Differential Equations and Mathematical Biology to communicate with each other about their current work. The topics of the workshop included various types of differential equations and their applications to biology and other related subjects, such as, ecology, epidemiology, medicine, etc. There were more than 60 participants came from Algeria, Canada, Cameroun, Finland, France, Germany, Hungary, Italy, Japan, Lithuania, Mexico, The Netherlands, Portugal, Romania, Spain, South Africa, UK, and USA. The ple- nary speakers were Pierre Auger (IRD Bondy, France), Josef Hofbauer (University College London, UK), Michel Langlais (Bordeaux 2, France), Hal Smith (Arizona State, USA), Horst Thieme (Arizona State, USA), Glenn Webb (Vanderbilt, USA) and Jianhong Wu (York, Canada). There were also more than 40 presentations by other participants.
The 17 articles which appear in this special issue are from the participants of the Workshop and from other leading researchers in these subjects. Topics include malaria intra-host models, stem cell dynamics, tumor invasion, reaction-diffusion systems for competition and predation, traveling waves, optimal control in age structured models, host-parasitoid models, predator-prey models, HIV infection, immune system memory, bacteria infection, innate immune response, and antibiotic treatment.
For the full preface, please click the Full Text "PDF" button above.
This special issue is the proceedings of the International Workshop on Differential Equations in Mathematical Biology held in Le Havre, France, July 11-13, 2005. The workshop brought together internationals researchers in Differential Equations and Mathematical Biology to communicate with each other about their current work. The topics of the workshop included various types of differential equations and their applications to biology and other related subjects, such as, ecology, epidemiology, medicine, etc. There were more than 60 participants came from Algeria, Canada, Cameroun, Finland, France, Germany, Hungary, Italy, Japan, Lithuania, Mexico, The Netherlands, Portugal, Romania, Spain, South Africa, UK, and USA. The ple- nary speakers were Pierre Auger (IRD Bondy, France), Josef Hofbauer (University College London, UK), Michel Langlais (Bordeaux 2, France), Hal Smith (Arizona State, USA), Horst Thieme (Arizona State, USA), Glenn Webb (Vanderbilt, USA) and Jianhong Wu (York, Canada). There were also more than 40 presentations by other participants.
The 17 articles which appear in this special issue are from the participants of the Workshop and from other leading researchers in these subjects. Topics include malaria intra-host models, stem cell dynamics, tumor invasion, reaction-diffusion systems for competition and predation, traveling waves, optimal control in age structured models, host-parasitoid models, predator-prey models, HIV infection, immune system memory, bacteria infection, innate immune response, and antibiotic treatment.
For the full preface, please click the Full Text "PDF" button above.
2007, 8(1): 1-17
doi: 10.3934/dcdsb.2007.8.1
+[Abstract](2101)
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Abstract:
We obtain global stability results for within-host models with $k$ age-class of parasitized cells and two strains of parasites. The stability is determined by the value of the basic reproduction ratio $\mathcal R_0$. A competitive exclusion principle holds. This means that if $\mathcal R_0 >1$ generically an unique equilibrium, corresponding to the extinction of one strain and the survival of the other strain, is globally asymptotically stable on the positive orthant.
We obtain global stability results for within-host models with $k$ age-class of parasitized cells and two strains of parasites. The stability is determined by the value of the basic reproduction ratio $\mathcal R_0$. A competitive exclusion principle holds. This means that if $\mathcal R_0 >1$ generically an unique equilibrium, corresponding to the extinction of one strain and the survival of the other strain, is globally asymptotically stable on the positive orthant.
2007, 8(1): 19-38
doi: 10.3934/dcdsb.2007.8.19
+[Abstract](1776)
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Abstract:
Under the action of growth factors, proliferating and nonproliferating hematopoietic stem cells differentiate and divide, so as to produce blood cells. Growth factors act at different levels in the differentiation process, and we consider their action on the mortality rate (apoptosis) of the proliferating cell population. We propose a mathematical model describing the evolution of a hematopoietic stem cell population under the action of growth factors. It consists of a system of two age-structured evolution equations modeling the dynamics of the stem cell population coupled with a delay differential equation describing the evolution of the growth factor concentration. We first reduce our system of three differential equations to a system of two nonlinear differential equations with two delays and a distributed delay. We investigate some positivity and boundedness properties of the solutions, as well as the existence of steady states. We then analyze the asymptotic stability of the two steady states by studying the characteristic equation with delay-dependent coefficients obtained while linearizing our system. We obtain necessary and sufficient conditions for the global stability of the steady state describing the cell population's dying out, using a Lyapunov function, and we prove the existence of periodic solutions about the other steady state through the existence of a Hopf bifurcation.
Under the action of growth factors, proliferating and nonproliferating hematopoietic stem cells differentiate and divide, so as to produce blood cells. Growth factors act at different levels in the differentiation process, and we consider their action on the mortality rate (apoptosis) of the proliferating cell population. We propose a mathematical model describing the evolution of a hematopoietic stem cell population under the action of growth factors. It consists of a system of two age-structured evolution equations modeling the dynamics of the stem cell population coupled with a delay differential equation describing the evolution of the growth factor concentration. We first reduce our system of three differential equations to a system of two nonlinear differential equations with two delays and a distributed delay. We investigate some positivity and boundedness properties of the solutions, as well as the existence of steady states. We then analyze the asymptotic stability of the two steady states by studying the characteristic equation with delay-dependent coefficients obtained while linearizing our system. We obtain necessary and sufficient conditions for the global stability of the steady state describing the cell population's dying out, using a Lyapunov function, and we prove the existence of periodic solutions about the other steady state through the existence of a Hopf bifurcation.
2007, 8(1): 39-44
doi: 10.3934/dcdsb.2007.8.39
+[Abstract](1490)
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Abstract:
We consider a two-component competition-diffusion system with equal diffusion coefficients and inhomogeneous Dirichlet boundary conditions. Provided certain conditions on a limit problem hold and provided that the competition rate is sufficiently large, all non-negative solutions of the system converge to a stationary solution of the system as $ t \rightarrow \infty$.
We consider a two-component competition-diffusion system with equal diffusion coefficients and inhomogeneous Dirichlet boundary conditions. Provided certain conditions on a limit problem hold and provided that the competition rate is sufficiently large, all non-negative solutions of the system converge to a stationary solution of the system as $ t \rightarrow \infty$.
2007, 8(1): 45-60
doi: 10.3934/dcdsb.2007.8.45
+[Abstract](1554)
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Abstract:
A model of tumor growth into surrounding tissue is analyzed. The model consists of a system of nonlinear partial differential equations for the populations of tumor cells, extracellular matrix macromolecules, oxygen concentration, and extracellular matrix degradative enzyme concentration. The spatial growth of the tumor involves the directed movement of tumor cells toward the extracellular matrix through haptotaxis. Cell age is used to track progression of cells through the cell cycle. The existence of unique global solutions is proved using the theory of fractional powers of analytic semigroup generators.
A model of tumor growth into surrounding tissue is analyzed. The model consists of a system of nonlinear partial differential equations for the populations of tumor cells, extracellular matrix macromolecules, oxygen concentration, and extracellular matrix degradative enzyme concentration. The spatial growth of the tumor involves the directed movement of tumor cells toward the extracellular matrix through haptotaxis. Cell age is used to track progression of cells through the cell cycle. The existence of unique global solutions is proved using the theory of fractional powers of analytic semigroup generators.
2007, 8(1): 61-72
doi: 10.3934/dcdsb.2007.8.61
+[Abstract](1446)
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Abstract:
This note is dedicated to the question of global existence for solutions to a two component singular system of reaction-diffusion equations modeling predator-prey interactions in insular environments. Depending on a 2D parameter space, positive orbits of the underlying ODE system undergo interesting dynamics, e.g., finite time existence and global existence may coexist. These results are partially extended to the reaction-diffusion system in the case of identical diffusivities. Our analysis relies on an auxiliary non singular reaction-diffusion system whose solutions may or may not blow up in finite time. Numerical simulations illustrate our analysis, including a numerical evidence of spatio-temporal oscillations.
This note is dedicated to the question of global existence for solutions to a two component singular system of reaction-diffusion equations modeling predator-prey interactions in insular environments. Depending on a 2D parameter space, positive orbits of the underlying ODE system undergo interesting dynamics, e.g., finite time existence and global existence may coexist. These results are partially extended to the reaction-diffusion system in the case of identical diffusivities. Our analysis relies on an auxiliary non singular reaction-diffusion system whose solutions may or may not blow up in finite time. Numerical simulations illustrate our analysis, including a numerical evidence of spatio-temporal oscillations.
2007, 8(1): 73-93
doi: 10.3934/dcdsb.2007.8.73
+[Abstract](1801)
+[PDF](316.9KB)
Abstract:
We show that stability of the equilibrium of a family of interconnected scalar systems can be proved by using a sum of monotonic $C^0$ functions as a Lyapunov function. We prove this result in the general framework of nonlinear systems and then in the special case of Kolmogorov systems. As an application, it is then used to show that intra-specific competition can explain coexistence of several species in a chemostat where they compete for a single substrate. This invalidates the Competitive Exclusion Principle, that states that in the classical case (without this intra-specific competition), it is indeed known that only one of the species will survive.
We show that stability of the equilibrium of a family of interconnected scalar systems can be proved by using a sum of monotonic $C^0$ functions as a Lyapunov function. We prove this result in the general framework of nonlinear systems and then in the special case of Kolmogorov systems. As an application, it is then used to show that intra-specific competition can explain coexistence of several species in a chemostat where they compete for a single substrate. This invalidates the Competitive Exclusion Principle, that states that in the classical case (without this intra-specific competition), it is indeed known that only one of the species will survive.
2007, 8(1): 95-105
doi: 10.3934/dcdsb.2007.8.95
+[Abstract](1631)
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Abstract:
For reaction-diffusion equations with delay, the joint effects of diffusion and delay are studied. In particular, for two-dimensional systems where only the interaction between species is delayed, the interdependence of stability against delay and against diffusion (Turing instability) can be clearly exhibited. Turing instabilities occur largely independent of delay. But periodic oscillations, constant in space or with low spatial frequency, can be achieved via increasing the delay or changing the diffusion rates.
For reaction-diffusion equations with delay, the joint effects of diffusion and delay are studied. In particular, for two-dimensional systems where only the interaction between species is delayed, the interdependence of stability against delay and against diffusion (Turing instability) can be clearly exhibited. Turing instabilities occur largely independent of delay. But periodic oscillations, constant in space or with low spatial frequency, can be achieved via increasing the delay or changing the diffusion rates.
2007, 8(1): 107-114
doi: 10.3934/dcdsb.2007.8.107
+[Abstract](1522)
+[PDF](128.5KB)
Abstract:
We consider the McKendrick linear model for the evolution of an age structured population. Usually the birth rate is given through a linear functional of the present population using the fertility rate. We are investigating the question of the existence of an objective function, depending on the control and some observation of the state, for which the associated optimal control problem using the birth rate as a control would yield the previous relation using the fertility rate as the optimal closed loop form. Then we consider adaption mechanisms that we model by including a desired value of the observation in the objective function. A modified fertility rate is derived.
We consider the McKendrick linear model for the evolution of an age structured population. Usually the birth rate is given through a linear functional of the present population using the fertility rate. We are investigating the question of the existence of an objective function, depending on the control and some observation of the state, for which the associated optimal control problem using the birth rate as a control would yield the previous relation using the fertility rate as the optimal closed loop form. Then we consider adaption mechanisms that we model by including a desired value of the observation in the objective function. A modified fertility rate is derived.
2007, 8(1): 115-125
doi: 10.3934/dcdsb.2007.8.115
+[Abstract](1853)
+[PDF](442.2KB)
Abstract:
This paper investigates the existence of travelling waves for the two component higher order autocatalytic reaction-diffusion systems by the phase plane analysis. We prove the existence of travelling waves for the system without decay for two extreme cases: the non-diffusive reactant case and the equal diffusive case. We further discuss the existence problem for the system with higher order decay when the reactant does not diffuse. Our analysis also gives the estimate of the minimal propagation speeds in terms of the order of autocatalysis.
This paper investigates the existence of travelling waves for the two component higher order autocatalytic reaction-diffusion systems by the phase plane analysis. We prove the existence of travelling waves for the system without decay for two extreme cases: the non-diffusive reactant case and the equal diffusive case. We further discuss the existence problem for the system with higher order decay when the reactant does not diffuse. Our analysis also gives the estimate of the minimal propagation speeds in terms of the order of autocatalysis.
2007, 8(1): 127-143
doi: 10.3934/dcdsb.2007.8.127
+[Abstract](1533)
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Abstract:
We develop a simple mathematical model of a bacterial colonization of host tissue which takes account of nutrient availability and innate immune response. The model features an infection-free state which is locally but not globally attracting implying that a super-threshold bacterial inoculum is required for successful colonization and tissue infection. A subset $B$ of the domain of attraction of the disease-free state is explicitly identified. The dynamics of antibiotic treatment of the infection is also considered. Successful treatment results if the antibiotic dosing regime drives the state of the system into $B$.
We develop a simple mathematical model of a bacterial colonization of host tissue which takes account of nutrient availability and innate immune response. The model features an infection-free state which is locally but not globally attracting implying that a super-threshold bacterial inoculum is required for successful colonization and tissue infection. A subset $B$ of the domain of attraction of the disease-free state is explicitly identified. The dynamics of antibiotic treatment of the infection is also considered. Successful treatment results if the antibiotic dosing regime drives the state of the system into $B$.
2007, 8(1): 145-159
doi: 10.3934/dcdsb.2007.8.145
+[Abstract](1847)
+[PDF](192.5KB)
Abstract:
We study a single-species population model with two stages, adults and juveniles, and the model with Allee effects. In these models, the fertility rate of an adult individual is assumed to be density dependent on the total adult population size and the transition probability from juvenile to adult over each time unit is assumed to be a constant. Both models exhibit a boundary $2$-cycle. Population persistence can occur for the model without the Allee effects. However, there exists a population threshold below which the population will go to extinction if the Allee effects are considered. We also propose a host-parasitoid model with stage structure in the host. Both populations can coexist with each other under some conditions if Allee effects are ignored. On the other hand, there exists a host population threshold below which both populations become extinct if Allee effects are incorporated into the interaction.
We study a single-species population model with two stages, adults and juveniles, and the model with Allee effects. In these models, the fertility rate of an adult individual is assumed to be density dependent on the total adult population size and the transition probability from juvenile to adult over each time unit is assumed to be a constant. Both models exhibit a boundary $2$-cycle. Population persistence can occur for the model without the Allee effects. However, there exists a population threshold below which the population will go to extinction if the Allee effects are considered. We also propose a host-parasitoid model with stage structure in the host. Both populations can coexist with each other under some conditions if Allee effects are ignored. On the other hand, there exists a host population threshold below which both populations become extinct if Allee effects are incorporated into the interaction.
2007, 8(1): 161-173
doi: 10.3934/dcdsb.2007.8.161
+[Abstract](1589)
+[PDF](242.6KB)
Abstract:
A simple epidemic model with a nonlinear incidence rate and two compartments is studied. The backward bifurcation is described and the corresponding threshold is calculated. The Hopf bifurcation and Bogdanov-Takens bifurcation are analyzed and numerical evidences for the stable or unstable limit cycle are provided.
A simple epidemic model with a nonlinear incidence rate and two compartments is studied. The backward bifurcation is described and the corresponding threshold is calculated. The Hopf bifurcation and Bogdanov-Takens bifurcation are analyzed and numerical evidences for the stable or unstable limit cycle are provided.
2007, 8(1): 175-185
doi: 10.3934/dcdsb.2007.8.175
+[Abstract](1514)
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Abstract:
By using the continuation theorem of coincidence degree theory, the existence of a positive periodic solution for a delayed predator-prey model with nonmonotonic functional response ${ x^{'}(t)=x(t)[ a(t)-b(t)x(t)] -\frac{x(t)y(t)}{m^2+x^2(t)},$
$y^{'}(t)=y(t)[ \frac{\mu (t)x(t-\tau )}{m^2+x^2(t-\tau )} -d(t)]. \]$ is established, where $a(t), b(t), \mu (t)$ and $d(t)$ are all positive periodic continuous functions with period $\omega >0$, $m>0$ and $\tau \geq 0 $ are constants.
By using the continuation theorem of coincidence degree theory, the existence of a positive periodic solution for a delayed predator-prey model with nonmonotonic functional response ${ x^{'}(t)=x(t)[ a(t)-b(t)x(t)] -\frac{x(t)y(t)}{m^2+x^2(t)},$
$y^{'}(t)=y(t)[ \frac{\mu (t)x(t-\tau )}{m^2+x^2(t-\tau )} -d(t)]. \]$ is established, where $a(t), b(t), \mu (t)$ and $d(t)$ are all positive periodic continuous functions with period $\omega >0$, $m>0$ and $\tau \geq 0 $ are constants.
2007, 8(1): 187-205
doi: 10.3934/dcdsb.2007.8.187
+[Abstract](1235)
+[PDF](583.4KB)
Abstract:
Ecology, Economy and Management Science require a huge interdisciplinary effort to ascertain the hidden mechanisms driving the evolution of communities and firm networks. This article shows that strategic alliances in competitive environments might provoke an explosive increment of productivity and stability through a feedback mechanism promoted by cooperation, while competition causes segregation within cooperative profiles. Some further speciation and radiation mechanisms enhancing innovation, facilitated by environmental heterogeneities, or specific market regulations, might explain the biodiversity of life and the high complexity of industrial and financial markets. Extinctions are likely to occur by the lack of adaptation of the strongest competitors to sudden environmental stress.
Ecology, Economy and Management Science require a huge interdisciplinary effort to ascertain the hidden mechanisms driving the evolution of communities and firm networks. This article shows that strategic alliances in competitive environments might provoke an explosive increment of productivity and stability through a feedback mechanism promoted by cooperation, while competition causes segregation within cooperative profiles. Some further speciation and radiation mechanisms enhancing innovation, facilitated by environmental heterogeneities, or specific market regulations, might explain the biodiversity of life and the high complexity of industrial and financial markets. Extinctions are likely to occur by the lack of adaptation of the strongest competitors to sudden environmental stress.
2007, 8(1): 207-228
doi: 10.3934/dcdsb.2007.8.207
+[Abstract](1785)
+[PDF](598.4KB)
Abstract:
In this paper we analyze the population dynamics of bacteria competing by anti-bacterial toxins (bacteriocins). Three types of bacteria involved in these dynamics can be distinguished: toxin producers, resistant bacteria and sensitive bacteria. Their interplay can be regarded as a Rock-Scissors-Paper - game (RSP). Here, this is modeled by a reasonable three-dimensional Lotka- Volterra ($L$V) type differential equation system. In contrast to earlier approaches to modeling the RSP game such as replicator equations, all interaction terms have negative signs because the interaction between the three different types of bacteria is purely competitive, either by toxification or by competition for nutrients. The model allows one to choose asymmetric parameter values. Depending on parameter values, our model gives rise to a stable steady state, a stable limit cycle or a heteroclinic orbit with three fixed points, each fixed point corresponding to the existence of only one bacteria type. An alternative model, the May - Leonard model, leads to coexistence only under very restricted conditions. We carry out a comprehensive analysis of the generic stability conditions of our model, using, among other tools, the Volterra-Lyapunov method. We also give biological interpretations of our theoretical results, in particular, of the predicted dynamics and of the ranges for parameter values where different dynamic behavior occurs. For example, one result is that the intrinsic growth rate of the producer is lower than that of the resistant while its growth yield is higher. This is in agreement with experimental results for the bacterium Listeria monocytogenes.
In this paper we analyze the population dynamics of bacteria competing by anti-bacterial toxins (bacteriocins). Three types of bacteria involved in these dynamics can be distinguished: toxin producers, resistant bacteria and sensitive bacteria. Their interplay can be regarded as a Rock-Scissors-Paper - game (RSP). Here, this is modeled by a reasonable three-dimensional Lotka- Volterra ($L$V) type differential equation system. In contrast to earlier approaches to modeling the RSP game such as replicator equations, all interaction terms have negative signs because the interaction between the three different types of bacteria is purely competitive, either by toxification or by competition for nutrients. The model allows one to choose asymmetric parameter values. Depending on parameter values, our model gives rise to a stable steady state, a stable limit cycle or a heteroclinic orbit with three fixed points, each fixed point corresponding to the existence of only one bacteria type. An alternative model, the May - Leonard model, leads to coexistence only under very restricted conditions. We carry out a comprehensive analysis of the generic stability conditions of our model, using, among other tools, the Volterra-Lyapunov method. We also give biological interpretations of our theoretical results, in particular, of the predicted dynamics and of the ranges for parameter values where different dynamic behavior occurs. For example, one result is that the intrinsic growth rate of the producer is lower than that of the resistant while its growth yield is higher. This is in agreement with experimental results for the bacterium Listeria monocytogenes.
2007, 8(1): 229-240
doi: 10.3934/dcdsb.2007.8.229
+[Abstract](1442)
+[PDF](202.8KB)
Abstract:
We consider a model of HIV-1 infection with triple drug therapy (HAART) and three delays: the first delay represents the time between the infection and the viral production, the second is associated with the loss of target cells by infection, and the third represents the time for the newly produced virions to become infectious. We show that the incorporation of these delays improves the critical efficacy of the treatment, and destabilizes the infected steady state or leads to an infected steady state with more healthy cells and less infected cells and viruses. Also, we considered the periodic treatment case. We analyze the stability of the viral free steady state and derive an effective strategy for reducing the viral load.
We consider a model of HIV-1 infection with triple drug therapy (HAART) and three delays: the first delay represents the time between the infection and the viral production, the second is associated with the loss of target cells by infection, and the third represents the time for the newly produced virions to become infectious. We show that the incorporation of these delays improves the critical efficacy of the treatment, and destabilizes the infected steady state or leads to an infected steady state with more healthy cells and less infected cells and viruses. Also, we considered the periodic treatment case. We analyze the stability of the viral free steady state and derive an effective strategy for reducing the viral load.
2007, 8(1): 241-259
doi: 10.3934/dcdsb.2007.8.241
+[Abstract](1254)
+[PDF](292.7KB)
Abstract:
A general process of the immune system consists of effector stage and memory stage. Current theoretical studies of the immune system often focus on the memory stage and pay less attention on the function of non-immune system substances such as tissue cells in adjusting the dynamical behavior of the immune system. We propose a mathematical population model to investigate the interaction between influenza A virus(IAV) susceptible tissue cells and generic immune cells when the tissue is invaded by IAV. We carry out a linear stability analysis and numerically study the Neimark-Sacker bifurcation of the models. The behavior of the model system agrees with some important experimental or clinical observations for IAV. However, we show that without considering the space between tissue cells, the expected memory stage does not form. By considering the space which allows antibodies to bind antigens, the memory stage then forms without missing the property of the system in the effector stage.
A general process of the immune system consists of effector stage and memory stage. Current theoretical studies of the immune system often focus on the memory stage and pay less attention on the function of non-immune system substances such as tissue cells in adjusting the dynamical behavior of the immune system. We propose a mathematical population model to investigate the interaction between influenza A virus(IAV) susceptible tissue cells and generic immune cells when the tissue is invaded by IAV. We carry out a linear stability analysis and numerically study the Neimark-Sacker bifurcation of the models. The behavior of the model system agrees with some important experimental or clinical observations for IAV. However, we show that without considering the space between tissue cells, the expected memory stage does not form. By considering the space which allows antibodies to bind antigens, the memory stage then forms without missing the property of the system in the effector stage.
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