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1551-0018
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Mathematical Biosciences & Engineering
2010 , Volume 7 , Issue 3
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2010, 7(3): 479-504
doi: 10.3934/mbe.2010.7.479
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Abstract:
Stability during the biped locomotion and especially humanoid robots walking is a big challenge in robotics modelling. This paper compares the classical and novel methodologies of modelling and algorithmic implementation of the impact/contact dynamics that occurs during a biped motion. Thus, after establishing the free biped locomotion system model, a formulation using variational inequalities theory via a Linear Complementarity Problem then an impedance model are explicitly developed. Results of the numerical simulations are compared to the experimental measurements then the both approaches are discussed.
Stability during the biped locomotion and especially humanoid robots walking is a big challenge in robotics modelling. This paper compares the classical and novel methodologies of modelling and algorithmic implementation of the impact/contact dynamics that occurs during a biped motion. Thus, after establishing the free biped locomotion system model, a formulation using variational inequalities theory via a Linear Complementarity Problem then an impedance model are explicitly developed. Results of the numerical simulations are compared to the experimental measurements then the both approaches are discussed.
2010, 7(3): 505-526
doi: 10.3934/mbe.2010.7.505
+[Abstract](2732)
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Abstract:
The purpose of this paper is to derive and analyze methods for examining the stability of solutions of partial differential equations modeling collections of excitable cells. In particular, we derive methods for estimating the principal eigenvalue of a linearized version of the Luo-Rudy I model close to an equilibrium solution. It has been suggested that the stability of a collection of unstable cells surrounded by a large collection of stable cells can be studied by considering only a collection of unstable cells equipped with a Dirichlet type boundary condition. This method has earlier been applied to analytically assess the stability of a reduced version the Luo-Rudy I model. In this paper we analyze the accuracy of this technique and apply it to the full Luo-Rudy I model. Furthermore, we extend the method to provide analytical results for the FitzHugh-Nagumo model in the case where a collection of unstable cells is surrounded by a collection of stable cells. All our analytical findings are complemented by numerical computations computing the principal eigenvalue of a discrete version of linearized models.
The purpose of this paper is to derive and analyze methods for examining the stability of solutions of partial differential equations modeling collections of excitable cells. In particular, we derive methods for estimating the principal eigenvalue of a linearized version of the Luo-Rudy I model close to an equilibrium solution. It has been suggested that the stability of a collection of unstable cells surrounded by a large collection of stable cells can be studied by considering only a collection of unstable cells equipped with a Dirichlet type boundary condition. This method has earlier been applied to analytically assess the stability of a reduced version the Luo-Rudy I model. In this paper we analyze the accuracy of this technique and apply it to the full Luo-Rudy I model. Furthermore, we extend the method to provide analytical results for the FitzHugh-Nagumo model in the case where a collection of unstable cells is surrounded by a collection of stable cells. All our analytical findings are complemented by numerical computations computing the principal eigenvalue of a discrete version of linearized models.
2010, 7(3): 527-552
doi: 10.3934/mbe.2010.7.527
+[Abstract](2942)
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Abstract:
The purpose of this paper is to use mathematical models to investigate the claim made in the medical literature over a decade ago that the routine rotation of antibiotics in an intensive care unit (ICU) will select against the evolution and spread of antibiotic-resistant pathogens. In contrast, previous theoretical studies addressing this question have demonstrated that routinely changing the drug of choice for a given pathogenic infection may in fact lead to a greater incidence of drug resistance in comparison to the random deployment of different drugs.
Using mathematical models that do not explicitly incorporate the spatial dynamics of pathogen transmission within the ICU or hospital and assuming the antibiotics are from distinct functional groups, we use a control theoretic-approach to prove that one can relax the medical notion of what constitutes an antibiotic rotation and so obtain protocols that are arbitrarily close to the optimum. Finally, we show that theoretical feedback control measures that rotate between different antibiotics motivated directly by the outcome of clinical studies can be deployed to good effect to reduce the prevalence of antibiotic resistance below what can be achieved with random antibiotic use.
The purpose of this paper is to use mathematical models to investigate the claim made in the medical literature over a decade ago that the routine rotation of antibiotics in an intensive care unit (ICU) will select against the evolution and spread of antibiotic-resistant pathogens. In contrast, previous theoretical studies addressing this question have demonstrated that routinely changing the drug of choice for a given pathogenic infection may in fact lead to a greater incidence of drug resistance in comparison to the random deployment of different drugs.
Using mathematical models that do not explicitly incorporate the spatial dynamics of pathogen transmission within the ICU or hospital and assuming the antibiotics are from distinct functional groups, we use a control theoretic-approach to prove that one can relax the medical notion of what constitutes an antibiotic rotation and so obtain protocols that are arbitrarily close to the optimum. Finally, we show that theoretical feedback control measures that rotate between different antibiotics motivated directly by the outcome of clinical studies can be deployed to good effect to reduce the prevalence of antibiotic resistance below what can be achieved with random antibiotic use.
2010, 7(3): 553-560
doi: 10.3934/mbe.2010.7.553
+[Abstract](2084)
+[PDF](388.9KB)
Abstract:
In this work, a hyperchaotic system was used as a model for chronotherapy. We applied a periodic perturbation to a variable, varying the period and amplitude of forcing. The system, five-dimensional, has until three positive Lyapunov exponents. As a result, we get small periodical windows, but it was possible to get large areas of hyperchaos of two positive Lyapunov exponents from a chaotic behavior. In this chronotherapy model, chaos could be considered as a dynamical disease, and therapy goal must be to restore the hyperchaotic state.
In this work, a hyperchaotic system was used as a model for chronotherapy. We applied a periodic perturbation to a variable, varying the period and amplitude of forcing. The system, five-dimensional, has until three positive Lyapunov exponents. As a result, we get small periodical windows, but it was possible to get large areas of hyperchaos of two positive Lyapunov exponents from a chaotic behavior. In this chronotherapy model, chaos could be considered as a dynamical disease, and therapy goal must be to restore the hyperchaotic state.
2010, 7(3): 561-578
doi: 10.3934/mbe.2010.7.561
+[Abstract](2186)
+[PDF](290.3KB)
Abstract:
'Rational' exemption to vaccination is due to a pseudo-rational comparison between the low risk of infection, and the perceived risk of side effects from the vaccine. Here we consider rational exemption in an SI model with information dependent vaccination where individuals use information on the disease's spread as their information set. Using suitable assumptions, we show the dynamic implications of the interaction between rational exemption, current and delayed information. In particular, if vaccination decisions are based on delayed informations, we illustrate both global attractivity to an endemic state, and the onset, through Hopf bifurcations, of general Yakubovich oscillations. Moreover, in some relevant cases, we plot the Hopf bifurcation curves and we give a behavioural interpretation of their meaning.
'Rational' exemption to vaccination is due to a pseudo-rational comparison between the low risk of infection, and the perceived risk of side effects from the vaccine. Here we consider rational exemption in an SI model with information dependent vaccination where individuals use information on the disease's spread as their information set. Using suitable assumptions, we show the dynamic implications of the interaction between rational exemption, current and delayed information. In particular, if vaccination decisions are based on delayed informations, we illustrate both global attractivity to an endemic state, and the onset, through Hopf bifurcations, of general Yakubovich oscillations. Moreover, in some relevant cases, we plot the Hopf bifurcation curves and we give a behavioural interpretation of their meaning.
2010, 7(3): 579-602
doi: 10.3934/mbe.2010.7.579
+[Abstract](2163)
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Abstract:
In this work, we generalize the Pugliese-Gandolfi Model [A. Pugliese and A. Gandolfi, Math Biosc, 214,73 (2008)] of interaction between an exponentially replicating pathogen and the immune system. After the generalization, we study the properties of boundedness and unboundedness of the solutions, and we also give a condition for the global eradication as well as for the onset of sustained oscillations. Then, we study the condition for the uniqueness of the arising limit cycle, with numerical applications to the Pugliese-Gandolfi model. By means of simulations, we also show some alternative ways to reaching the elimination of the pathogen and interesting effects linked to variations in aspecific immune response. After shortly studying some pathological cases of interest, we include in our model distributed and constant delays and we show that also delays may unstabilize the equilibria.
In this work, we generalize the Pugliese-Gandolfi Model [A. Pugliese and A. Gandolfi, Math Biosc, 214,73 (2008)] of interaction between an exponentially replicating pathogen and the immune system. After the generalization, we study the properties of boundedness and unboundedness of the solutions, and we also give a condition for the global eradication as well as for the onset of sustained oscillations. Then, we study the condition for the uniqueness of the arising limit cycle, with numerical applications to the Pugliese-Gandolfi model. By means of simulations, we also show some alternative ways to reaching the elimination of the pathogen and interesting effects linked to variations in aspecific immune response. After shortly studying some pathological cases of interest, we include in our model distributed and constant delays and we show that also delays may unstabilize the equilibria.
2010, 7(3): 603-622
doi: 10.3934/mbe.2010.7.603
+[Abstract](2244)
+[PDF](1379.7KB)
Abstract:
In this paper, we formulate a three-species ecological community model consisting of two aphid species ( Acyrthosiphon pisum and Megoura viciae) and a specialist parasitoid ( Aphidius ervi) that attacks only one of the aphids ( A pisum). The model incorporates both density-mediated and trait-mediated host-parasitoid interactions. Our analysis shows that the model possesses much richer and more realistic dynamics than earlier models. Our theoretical results reveal a new mechanism for stable coexistence in a three-species community in which any two species alone do not co-exist. More specifically, it is known that, when a predator is introduced into a community of two competing species, if the predator only predates on the strong competitor, it can allow the weak competitor to survive, but may drive the strong competitor to extinction through over-exploitation. We show that if the weak competitor interferes the predation on the strong competitor through trait-mediated indirect effects, then all three species can stably co-exist.
In this paper, we formulate a three-species ecological community model consisting of two aphid species ( Acyrthosiphon pisum and Megoura viciae) and a specialist parasitoid ( Aphidius ervi) that attacks only one of the aphids ( A pisum). The model incorporates both density-mediated and trait-mediated host-parasitoid interactions. Our analysis shows that the model possesses much richer and more realistic dynamics than earlier models. Our theoretical results reveal a new mechanism for stable coexistence in a three-species community in which any two species alone do not co-exist. More specifically, it is known that, when a predator is introduced into a community of two competing species, if the predator only predates on the strong competitor, it can allow the weak competitor to survive, but may drive the strong competitor to extinction through over-exploitation. We show that if the weak competitor interferes the predation on the strong competitor through trait-mediated indirect effects, then all three species can stably co-exist.
2010, 7(3): 623-639
doi: 10.3934/mbe.2010.7.623
+[Abstract](2658)
+[PDF](437.4KB)
Abstract:
Relationships between local stability and synchronization in networks of identical dynamical systems are established through the Master Stability Function approach. First, it is shown that stable equilibria of the local dynamics correspond to stable stationary synchronous regimes of the entire network if the coupling among the systems is sufficiently weak or balanced (in other words, stationary synchronous regimes can be unstable only if the coupling is sufficiently large and unbalanced). Then, it is shown that [de]stabilizing factors at local scale are [de]synchronizing at global scale again if the coupling is sufficiently weak or balanced. These results allow one to transfer, with almost no effort, what is known for simple prototypical models in biology and engineering to complex networks composed of these models. This is shown through a series of applications ranging from networks of electrical circuits to various problems in ecology and sociology involving migrations of plants, animal and human populations.
Relationships between local stability and synchronization in networks of identical dynamical systems are established through the Master Stability Function approach. First, it is shown that stable equilibria of the local dynamics correspond to stable stationary synchronous regimes of the entire network if the coupling among the systems is sufficiently weak or balanced (in other words, stationary synchronous regimes can be unstable only if the coupling is sufficiently large and unbalanced). Then, it is shown that [de]stabilizing factors at local scale are [de]synchronizing at global scale again if the coupling is sufficiently weak or balanced. These results allow one to transfer, with almost no effort, what is known for simple prototypical models in biology and engineering to complex networks composed of these models. This is shown through a series of applications ranging from networks of electrical circuits to various problems in ecology and sociology involving migrations of plants, animal and human populations.
2010, 7(3): 641-656
doi: 10.3934/mbe.2010.7.641
+[Abstract](3840)
+[PDF](227.1KB)
Abstract:
A mathematical model involving the syntrophic relationship of two major populations of bacteria (acetogens and methanogens), each responsible for a stage of the methane fermentation process is proposed. A detailed qualitative analysis is carried out. The local and global stability analyses of the equilibria are performed. We demonstrate, under general assumptions of monotonicity, relevant from an applied point of view, the global asymptotic stability of a positive equilibrium point which corresponds to the coexistence of acetogenic and methanogenic bacteria.
A mathematical model involving the syntrophic relationship of two major populations of bacteria (acetogens and methanogens), each responsible for a stage of the methane fermentation process is proposed. A detailed qualitative analysis is carried out. The local and global stability analyses of the equilibria are performed. We demonstrate, under general assumptions of monotonicity, relevant from an applied point of view, the global asymptotic stability of a positive equilibrium point which corresponds to the coexistence of acetogenic and methanogenic bacteria.
2010, 7(3): 657-673
doi: 10.3934/mbe.2010.7.657
+[Abstract](2521)
+[PDF](228.4KB)
Abstract:
The parasite Trypanosoma cruzi, which causes Chagas' disease, is typically transmitted through a cycle in which vectors become infected through bloodmeals on infected hosts and then infect other hosts through defecation at the sites of subsequent feedings. The vectors native to the southeastern United States, however, are inefficient at transmitting T. cruzi in this way, which suggests that alternative transmission modes may be responsible for maintaining the established sylvatic infection cycle. Vertical and oral transmission of sylvatic hosts, as well as differential behavior of infected vectors, have been observed anecdotally. This study develops a model which accounts for these alternative modes of transmission, and applies it to transmission between raccoons and the vector Triatoma sanguisuga. Analysis of the system of nonlinear differential equations focuses on endemic prevalence levels and on the infection's basic reproductive number, whose form may account for how a combination of traditionally secondary infection routes can maintain the transmission cycle when the usual primary route becomes ineffective.
The parasite Trypanosoma cruzi, which causes Chagas' disease, is typically transmitted through a cycle in which vectors become infected through bloodmeals on infected hosts and then infect other hosts through defecation at the sites of subsequent feedings. The vectors native to the southeastern United States, however, are inefficient at transmitting T. cruzi in this way, which suggests that alternative transmission modes may be responsible for maintaining the established sylvatic infection cycle. Vertical and oral transmission of sylvatic hosts, as well as differential behavior of infected vectors, have been observed anecdotally. This study develops a model which accounts for these alternative modes of transmission, and applies it to transmission between raccoons and the vector Triatoma sanguisuga. Analysis of the system of nonlinear differential equations focuses on endemic prevalence levels and on the infection's basic reproductive number, whose form may account for how a combination of traditionally secondary infection routes can maintain the transmission cycle when the usual primary route becomes ineffective.
2010, 7(3): 675-685
doi: 10.3934/mbe.2010.7.675
+[Abstract](2878)
+[PDF](167.9KB)
Abstract:
Global stability is analyzed for a general mathematical model of HIV-1 pathogenesis proposed by Nelson and Perelson [11]. The general model include two distributed intracellular delays and a combination therapy with a reverse transcriptase inhibitor and a protease inhibitor. It is shown that the model exhibits a threshold dynamics: if the basic reproduction number is less than or equal to one, then the HIV-1 infection is cleared from the T-cell population; whereas if the basic reproduction number is larger than one, then the HIV-1 infection persists and the viral concentration maintains at a constant level.
Global stability is analyzed for a general mathematical model of HIV-1 pathogenesis proposed by Nelson and Perelson [11]. The general model include two distributed intracellular delays and a combination therapy with a reverse transcriptase inhibitor and a protease inhibitor. It is shown that the model exhibits a threshold dynamics: if the basic reproduction number is less than or equal to one, then the HIV-1 infection is cleared from the T-cell population; whereas if the basic reproduction number is larger than one, then the HIV-1 infection persists and the viral concentration maintains at a constant level.
2010, 7(3): 687-717
doi: 10.3934/mbe.2010.7.687
+[Abstract](3916)
+[PDF](951.5KB)
Abstract:
A classical epidemiological framework is used to provide a preliminary cost analysis of the effects of quarantine and isolation on the dynamics of infectious diseases for which no treatment or immediate diagnosis tools are available. Within this framework we consider the cost incurred from the implementation of three types of dynamic control strategies. Taking the context of the 2003 SARS outbreak in Hong Kong as an example, we use a simple cost function to compare the total cost of each mixed (quarantine and isolation) control strategy from a public health resource allocation perspective. The goal is to extend existing epi-economics methodology by developing a theoretical framework of dynamic quarantine strategies aimed at emerging diseases, by drawing upon the large body of literature on the dynamics of infectious diseases. We find that the total cost decreases with increases in the quarantine rates past a critical value, regardless of the resource allocation strategy. In the case of a manageable outbreak resources must be used early to achieve the best results whereas in case of an unmanageable outbreak, a constant-effort strategy seems the best among our limited plausible sets.
A classical epidemiological framework is used to provide a preliminary cost analysis of the effects of quarantine and isolation on the dynamics of infectious diseases for which no treatment or immediate diagnosis tools are available. Within this framework we consider the cost incurred from the implementation of three types of dynamic control strategies. Taking the context of the 2003 SARS outbreak in Hong Kong as an example, we use a simple cost function to compare the total cost of each mixed (quarantine and isolation) control strategy from a public health resource allocation perspective. The goal is to extend existing epi-economics methodology by developing a theoretical framework of dynamic quarantine strategies aimed at emerging diseases, by drawing upon the large body of literature on the dynamics of infectious diseases. We find that the total cost decreases with increases in the quarantine rates past a critical value, regardless of the resource allocation strategy. In the case of a manageable outbreak resources must be used early to achieve the best results whereas in case of an unmanageable outbreak, a constant-effort strategy seems the best among our limited plausible sets.
2010, 7(3): 719-728
doi: 10.3934/mbe.2010.7.719
+[Abstract](2238)
+[PDF](207.8KB)
Abstract:
In some species, an inducible secondary phenotype will develop some time after the environmental change that evokes it. Nishimura (2006) [4] showed how an individual organism should optimize the time it takes to respond to an environmental change ("waiting time''). If the optimal waiting time is considered to act over the population, there are implications for the expected value of the mean fitness in that population. A stochastic predator-prey model is proposed in which the prey have a fixed initial energy budget. Fitness is the product of survival probability and the energy remaining for non-defensive purposes. The model is placed in the stochastic domain by assuming that the waiting time in the population is a normally distributed random variable because of biological variance inherent in mounting the response. It is found that the value of the mean waiting time that maximises fitness depends linearly on the variance of the waiting time.
In some species, an inducible secondary phenotype will develop some time after the environmental change that evokes it. Nishimura (2006) [4] showed how an individual organism should optimize the time it takes to respond to an environmental change ("waiting time''). If the optimal waiting time is considered to act over the population, there are implications for the expected value of the mean fitness in that population. A stochastic predator-prey model is proposed in which the prey have a fixed initial energy budget. Fitness is the product of survival probability and the energy remaining for non-defensive purposes. The model is placed in the stochastic domain by assuming that the waiting time in the population is a normally distributed random variable because of biological variance inherent in mounting the response. It is found that the value of the mean waiting time that maximises fitness depends linearly on the variance of the waiting time.
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