Showing papers in "Journal of Mathematical Biology in 2012"
TL;DR: Analytical and numerical results indicate that pulsed SIT with small and frequent releases can be an alternative to chemical control tools, but only if it is used or applied early after the beginning of the epidemic or as a preventive tool.
Abstract: Chikungunya is an arthropod-borne disease caused by the Asian tiger mosquito, Aedes albopictus. It can be an important burden to public health and a great cause of morbidity and, sometimes, mortality. Understanding if and when disease control measures should be taken is key to curtail its spread. Dumont and Chiroleu (Math Biosc Eng 7(2):315-348, 2010) showed that the use of chemical control tools such as adulticide and larvicide, and mechanical control, which consists of reducing the breeding sites, would have been useful to control the explosive 2006 epidemic in Reunion Island. Despite this, chemical control tools cannot be of long-time use, because they can induce mosquito resistance, and are detrimental to the biodiversity. It is therefore necessary to develop and test new control tools that are more sustainable, with the same efficacy (if possible). Mathematical models of sterile insect technique (SIT) to prevent, reduce, eliminate or stop an epidemic of Chikungunya are formulated and analysed. In particular, we propose a new model that considers pulsed periodic releases, which leads to a hybrid dynamical system. This pulsed SIT model is coupled with the human population at different epidemiological states in order to assess its efficacy. Numerical simulations for the pulsed SIT, using an appropriate numerical scheme are provided. Analytical and numerical results indicate that pulsed SIT with small and frequent releases can be an alternative to chemical control tools, but only if it is used or applied early after the beginning of the epidemic or as a preventive tool.
120 citations
TL;DR: This structural approach allows us to discuss the complexity of the biological problems investigated, and the necessity for deriving complex mathematical models that would explain the observed spatial and spatiotemporal group patterns.
Abstract: We briefly review hyperbolic and kinetic models for self-organized biological aggregations and traffic-like movement. We begin with the simplest models described by an advection-reaction equation in one spatial dimension. We then increase the complexity of models in steps. To this end, we begin investigating local hyperbolic systems of conservation laws with constant velocity. Next, we proceed to investigate local hyperbolic systems with density-dependent speed, systems that consider population dynamics (i.e., birth and death processes), and nonlocal hyperbolic systems. We conclude by discussing kinetic models in two spatial dimensions and their limiting hyperbolic models. This structural approach allows us to discuss the complexity of the biological problems investigated, and the necessity for deriving complex mathematical models that would explain the observed spatial and spatiotemporal group patterns.
115 citations
TL;DR: A new definition of R0 is introduced based on the generation evolution operator (GEO), which has intuitively clear biological meaning and can be applied to structured populations in any heterogeneous environment.
Abstract: Although its usefulness and possibility of the well-known definition of the basic reproduction number R0 for structured populations by Diekmann, Heesterbeek and Metz (J Math Biol 28:365-382, 1990) (the DHM definition) have been widely recognized mainly in the context of epidemic models, originally it deals with population dynamics in a constant environment, so it cannot be applied to formulate the threshold principle for population growth in time-heterogeneous environments. Since the mid-1990s, several authors proposed some ideas to extend the definition of R0 to the case of a periodic environment. In particular, the definition of R0 in a periodic environment by Bacaer and Guernaoui (J Math Biol 53:421-436, 2006) (the BG definition) is most important, because their definition of periodic R0 can be interpreted as the asymptotic per generation growth rate, which is an essential feature of the DHM definition. In this paper, we introduce a new definition of R0 based on the generation evolution operator (GEO), which has intuitively clear biological meaning and can be applied to structured populations in any heterogeneous environment. Using the generation evolution operator, we show that the DHM definition and the BG definition completely allow the generational interpretation and, in those two cases, the spectral radius of GEO equals the spectral radius of the next generation operator, so it gives the basic reproduction number. Hence the new definition is an extension of the DHM definition and the BG definition. Finally we prove a weak sign relation that if the average Malthusian parameter exists, it is nonnegative when R0>1 and it is nonpositive when R0<1.
113 citations
TL;DR: The present paper shows that in demography and epidemiology, the basic reproduction number R0 is the asymptotic ratio of total births in two successive generations of the family tree.
Abstract: An adaptation of the definition of the basic reproduction number R0 to time-periodic seasonal models was suggested a few years ago However, its biological interpretation remained unclear The present paper shows that in demography, this R0 is the asymptotic ratio of total births in two successive generations of the family tree In epidemiology, it is the asymptotic ratio of total infections in two successive generations of the infection tree This result is compared with other recent work
102 citations
TL;DR: How small the mutation rate should be to make the embedded Markov chain a good approximation by calculating the difference between the real stationary distribution and the approximated one is analyzed.
Abstract: We consider evolutionary game dynamics in a finite population of size N. When mutations are rare, the population is monomorphic most of the time. Occasionally a mutation arises. It can either reach fixation or go extinct. The evolutionary dynamics of the process under small mutation rates can be approximated by an embedded Markov chain on the pure states. Here we analyze how small the mutation rate should be to make the embedded Markov chain a good approximation by calculating the difference between the real stationary distribution and the approximated one. While for a coexistence game, where the best reply to any strategy is the opposite strategy, it is necessary that the mutation rate μ is less than N
−1/2exp[−N] to ensure that the approximation is good, for all other games, it is sufficient if the mutation rate is smaller than (N ln N)−1. Our results also hold for a wide class of imitation processes under arbitrary selection intensity.
100 citations
TL;DR: An efficient algorithm for computing transition probabilities in a general birth–death process with arbitrary birth and death rates is developed and this error-controlled method agrees with known solutions and outperforms previous approaches to computing these probabilities.
Abstract: A birth-death process is a continuous-time Markov chain that counts the number of particles in a system over time. In the general process with n current particles, a new particle is born with instantaneous rate λ(n) and a particle dies with instantaneous rate μ(n). Currently no robust and efficient method exists to evaluate the finite-time transition probabilities in a general birth-death process with arbitrary birth and death rates. In this paper, we first revisit the theory of continued fractions to obtain expressions for the Laplace transforms of these transition probabilities and make explicit an important derivation connecting transition probabilities and continued fractions. We then develop an efficient algorithm for computing these probabilities that analyzes the error associated with approximations in the method. We demonstrate that this error-controlled method agrees with known solutions and outperforms previous approaches to computing these probabilities. Finally, we apply our novel method to several important problems in ecology, evolution, and genetics.
90 citations
TL;DR: A generic gene-expression model of this kind is studied which explicitly includes the representations of the processes of transcription and translation and determines the generating function of the steady-state distribution of mRNA and protein counts.
Abstract: Gene expression at the single-cell level incorporates reaction mechanisms which are intrinsically stochastic as they involve molecular species present at low copy numbers. The dynamics of these mechanisms can be described quantitatively using stochastic master-equation modelling; in this paper we study a generic gene-expression model of this kind which explicitly includes the representations of the processes of transcription and translation. For this model we determine the generating function of the steady-state distribution of mRNA and protein counts and characterise the underlying probability law using a combination of analytic, asymptotic and numerical approaches, finding that the distribution may assume a number of qualitatively distinct forms. The results of the analysis are suitable for comparison with single-molecule resolution gene-expression data emerging from recent experimental studies.
88 citations
TL;DR: Continuous-time and discrete-space models are used to determine when the dispersal strategy with no movement is evolutionarily stable and when an ideal free dispersal Strategy is evolutionally stable, both in a spatially heterogeneous but temporally constant environment.
Abstract: A central question in the study of the evolution of dispersal is what kind of dispersal strategies are evolutionarily stable. Hastings (Theor Pop Biol 24:244–251, 1983) showed that among unconditional dispersal strategies in a spatially heterogeneous but temporally constant environment, the dispersal strategy with no movement is convergent stable. McPeek and Holt’s (Am Nat 140:1010–1027, 1992) work suggested that among conditional dispersal strategies in a spatially heterogeneous but temporally constant environment, an ideal free dispersal strategy, which results in the ideal free distribution for a single species at equilibrium, is evolutionarily stable. We use continuous-time and discrete-space models to determine when the dispersal strategy with no movement is evolutionarily stable and when an ideal free dispersal strategy is evolutionarily stable, both in a spatially heterogeneous but temporally constant environment.
79 citations
TL;DR: This paper formalises the link between a well known pairwise model and the exact Markovian formulation of the exact ODE model at the level of pairs in terms of the expected number of pairs and triples and shows that the existing pairwise models are a good fit for certain types of network but have to be used with caution as higher-order network structures may compromise their effectiveness.
Abstract: Many if not all models of disease transmission on networks can be linked to the exact state-based Markovian formulation. However the large number of equations for any system of realistic size limits their applicability to small populations. As a result, most modelling work relies on simulation and pairwise models. In this paper, for a simple SIS dynamics on an arbitrary network, we formalise the link between a well known pairwise model and the exact Markovian formulation. This involves the rigorous derivation of the exact ODE model at the level of pairs in terms of the expected number of pairs and triples. The exact system is then closed using two different closures, one well established and one that has been recently proposed. A new interpretation of both closures is presented, which explains several of their previously observed properties. The closed dynamical systems are solved numerically and the results are compared to output from individual-based stochastic simulations. This is done for a range of networks with the same average degree and clustering coefficient but generated using different algorithms. It is shown that the ability of the pairwise system to accurately model an epidemic is fundamentally dependent on the underlying large-scale network structure. We show that the existing pairwise models are a good fit for certain types of network but have to be used with caution as higher-order network structures may compromise their effectiveness.
78 citations
TL;DR: An infectious disease model containing a fixed latency and non-local infection caused by the mobility of the latent individuals in a continuous bounded domain is derived and it is shown that the model has the global threshold dynamics which predicts whether the disease will die out or persist.
Abstract: In this paper, we derive and analyze an infectious disease model containing a fixed latency and non-local infection caused by the mobility of the latent individuals in a continuous bounded domain. The model is given by a spatially non-local reaction–diffusion system carrying a discrete delay associated with the zero-flux condition on the boundary. By applying some existing abstract results in dynamical systems theory, we prove the existence of a global attractor for the model system. By appealing to the theory of monotone dynamical systems and uniform persistence, we show that the model has the global threshold dynamics which can be described either by the principal eigenvalue of a linear non-local scalar reaction diffusion equation or equivalently by the basic reproduction number \({\mathcal{R}_0}\) for the model. Such threshold dynamics predicts whether the disease will die out or persist. We identify the next generation operator, the spectral radius of which defines basic reproduction number. When all model parameters are constants, we are able to find explicitly the principal eigenvalue and \({\mathcal{R}_0}\). In addition to computing the spectral radius of the next generation operator, we also discuss an alternative way to compute \({\mathcal{R}_0}\).
75 citations
TL;DR: In this paper, a mathematical model that couples a reaction-diffusion system in the inner volume to a reaction diffusion system on the membrane via a flux condition and an attachment/detachment law at the membrane is presented.
Abstract: GTPase molecules are important regulators in cells that continuously run through an activation/deactivation and membrane-attachment/membrane-detachment cycle. Activated GTPase is able to localize in parts of the membranes and to induce cell polarity. As feedback loops contribute to the GTPase cycle and as the coupling between membrane-bound and cytoplasmic processes introduces different diffusion coefficients a Turing mechanism is a natural candidate for this symmetry breaking. We formulate a mathematical model that couples a reaction–diffusion system in the inner volume to a reaction–diffusion system on the membrane via a flux condition and an attachment/detachment law at the membrane. We present a reduction to a simpler non-local reaction–diffusion model and perform a stability analysis and numerical simulations for this reduction. Our model in principle does support Turing instabilities but only if the lateral diffusion of inactivated GTPase is much faster than the diffusion of activated GTPase.
TL;DR: An optimal control problem for cancer chemotherapy is considered that includes immunological activity and, employing a Gompertzian growth model for the cancer cells, for various scenarios optimal controls and corresponding responses of the system are calculated.
Abstract: An optimal control problem for cancer chemotherapy is considered that includes immunological activity. In the objective a weighted average of several quantities that describe the effectiveness of treatment is minimized. These terms include (i) the number of cancer cells at the terminal time, (ii) a measure for the immunocompetent cell densities at the terminal point (included as a negative term), (iii) the overall amount of cytotoxic agents given as a measure for the side effects of treatment and (iv) a small penalty on the terminal time that limits the overall therapy horizon which is assumed to be free. This last term is essential in obtaining a well-posed problem formulation. Employing a Gompertzian growth model for the cancer cells, for various scenarios optimal controls and corresponding responses of the system are calculated. Solutions initially follow a full dose treatment, but then at one point switch to a singular regimen that only applies partial dosages. This structure is consistent with protocols that apply an initial burst to reduce the tumor volume and then maintain a small volume through lower dosages. Optimal controls end with either a prolonged period of no dose treatment or, in a small number of scenarios, this no dose interval is still followed by one more short burst of full dose treatment.
TL;DR: This paper uses an abstract theorem about persistence by Fonda to address the persistence of a class of seasonally forced epidemiological models.
Abstract: In this paper we address the persistence of a class of seasonally forced epidemiological models. We use an abstract theorem about persistence by Fonda. Five different examples of application are given.
TL;DR: This work presents examples of how the master equation for gene products such as mRNA and proteins can be reduced to a simpler problem using asymptotic methods, and analyses the relationship between the reduced models and the original.
Abstract: Stochastic phenomena in gene regulatory networks can be modelled by the chemical master equation for gene products such as mRNA and proteins. If some of these elements are present in significantly higher amounts than the rest, or if some of the reactions between these elements are substantially faster than others, it is often possible to reduce the master equation to a simpler problem using asymptotic methods. We present examples of such a procedure and analyse the relationship between the reduced models and the original.
TL;DR: A metapopulation malaria model is proposed using SI and SIRS models for the vectors and hosts, respectively, and type reproduction numbers are used to identify the reservoirs of infection and evaluate the effect of control measures.
Abstract: A metapopulation malaria model is proposed using SI and SIRS models for the vectors and hosts, respectively. Recovered hosts are partially immune to the disease and while they cannot directly become infectious again, they can still transmit the parasite to vectors. The basic reproduction number \({\mathcal{R}_0}\) is shown to govern the local stability of the disease free equilibrium but not the global behavior of the system because of the potential occurrence of a backward bifurcation. Using type reproduction numbers, we identify the reservoirs of infection and evaluate the effect of control measures. Applications to the spread to non-endemic areas and the interaction between rural and urban areas are given.
TL;DR: A 2D mathematical model of the initiation and development of atherosclerosis is suggested which takes into account the concentration of blood cells inside the intima and of pro- and anti-inflammatory cytokines, and proves the existence of travelling waves described by this system and confirms the previous results which suggest that Atherosclerosis develops as a reaction–diffusion wave.
Abstract: Atherosclerosis begins as an inflammation in blood vessel walls (intima). The inflammatory response of the organism leads to the recruitment of monocytes. Trapped in the intima, they differentiate into macrophages and foam cells leading to the production of inflammatory cytokines and further recruitment of white blood cells. This self-accelerating process, strongly influenced by low-density lipoproteins (cholesterol), results in a dramatic increase of the width of blood vessel walls, formation of an atherosclerotic plaque and, possibly, of its rupture. We suggest a 2D mathematical model of the initiation and development of atherosclerosis which takes into account the concentration of blood cells inside the intima and of pro- and anti-inflammatory cytokines. The model represents a reaction-diffusion system in a strip with nonlinear boundary conditions which describe the recruitment of monocytes as a function of the concentration of inflammatory cytokines. We prove the existence of travelling waves described by this system and confirm our previous results which suggest that atherosclerosis develops as a reaction-diffusion wave. The theoretical results are confirmed by the results of numerical simulations.
TL;DR: This work presents necessary and sufficient conditions for multistationarity that take the form of linear inequality systems for mass action networks with certain structural properties, expressed in terms of the stoichiometric matrix and the reaction rate-exponent matrix.
Abstract: Ordinary Differential Equations (ODEs) are an important tool in many areas of Quantitative Biology. For many ODE systems multistationarity (i.e. the existence of at least two positive steady states) is a desired feature. In general establishing multistationarity is a difficult task as realistic biological models are large in terms of states and (unknown) parameters and in most cases poorly parameterized (because of noisy measurement data of few components, a very small number of data points and only a limited number of repetitions). For mass action networks establishing multistationarity hence is equivalent to establishing the existence of at least two positive solutions of a large polynomial system with unknown coefficients. For mass action networks with certain structural properties, expressed in terms of the stoichiometric matrix and the reaction rate-exponent matrix, we present necessary and sufficient conditions for multistationarity that take the form of linear inequality systems. Solutions of these inequality systems define pairs of steady states and parameter values. We also present a sufficient condition to identify networks where the aforementioned conditions hold. To show the applicability of our results we analyse an ODE system that is defined by the mass action network describing the extracellular signal-regulated kinase (ERK) cascade (i.e. ERK-activation).
TL;DR: Two-parameter bifurcation analysis of an in-host model that describes infections of many viruses including HIV-I, HBV and HTLV-I shows that, while both τ and r can destabilize E* and cause Hopf bifircations, they do behave differently.
Abstract: To understand joint effects of logistic growth in target cells and intracellular delay on viral dynamics in vivo, we carry out two-parameter bifurcation analysis of an in-host model that describes infections of many viruses including HIV-I, HBV and HTLV-I. The bifurcation parameters are the mitosis rate r of the target cells and an intracellular delay τ in the incidence of viral infection. We describe the stability region of the chronic-infection equilibrium E* in the two-dimensional (r, τ) parameter space, as well as the global Hopf bifurcation curves as each of τ and r varies. Our analysis shows that, while both τ and r can destabilize E* and cause Hopf bifurcations, they do behave differently. The intracellular delay τ can cause Hopf bifurcations only when r is positive and sufficiently large, while r can cause Hopf bifurcations even when τ = 0. Intracellular delay τ can cause stability switches in E* while r does not.
TL;DR: It is shown that the system displays chaotic behaviour under the combination of high per capita recruitment and small survivorship rates and the phenomenon of bubbling and the hydra effect, which means that the stock size gets larger as harvesting rate increases.
Abstract: We analyze the effects of a strategy of constant effort harvesting in the global dynamics of a one-dimensional discrete population model that includes density-independent survivorship of adults and overcompensating density dependence. We discuss the phenomenon of bubbling (which indicates that harvesting can magnify fluctuations in population abundance) and the hydra effect, which means that the stock size gets larger as harvesting rate increases. Moreover, we show that the system displays chaotic behaviour under the combination of high per capita recruitment and small survivorship rates.
TL;DR: A simplified finite-element model for wound healing takes into account the sequential steps of dermal regeneration, wound contraction, angiogenesis and wound closure, and confirms the clinical observation that epidermal closure proceeds by a crawling and climbing mechanism at the early stages, and by a stratification process in layers parallel to the skin surface at the later stages.
Abstract: A simplified finite-element model for wound healing is proposed. The model takes into account the sequential steps of dermal regeneration, wound contraction, angiogenesis and wound closure. An innovation in the present study is the combination of the aforementioned partially overlapping processes, which can be used to deliver novel insights into the process of wound healing, such as geometry related influences, as well as the influence of coupling between the various existing subprocesses on the actual healing behavior. The model confirms the clinical observation that epidermal closure proceeds by a crawling and climbing mechanism at the early stages, and by a stratification process in layers parallel to the skin surface at the later stages. The local epidermal oxygen content may play an important role here. The model can also be used to investigate the influence of local injection of hormones that stimulate partial processes occurring during wound healing. These insights can be used to improve wound healing treatments.
TL;DR: This paper investigates the time optimal control problem for periodically firing neurons, represented by different one-dimensional phase models, and finds analytical expressions for the minimum and maximum values of inter-spike intervals achievable with small bounded control stimuli.
Abstract: By injecting an electrical current control stimulus into a neuron, one can change its inter-spike intervals. In this paper, we investigate the time optimal control problem for periodically firing neurons, represented by different one-dimensional phase models, and find analytical expressions for the minimum and maximum values of inter-spike intervals achievable with small bounded control stimuli. We consider two cases: with a charge-balance constraint on the input, and without it. The analytical calculations are supported with numerical results for examples of qualitatively different neuron models.
TL;DR: It is shown that depending on the allometry coefficient the limit behavior of the population process can be approximated by nonlinear Lévy flights of different nature: either deterministic, in the form of non-local fractional reaction–diffusion equations, or stochastic, as nonlinear super-processes with the underlying reaction and a fractional diffusion operator.
Abstract: We are interested in modeling Darwinian evolution resulting from the interplay of phenotypic variation and natural selection through ecological interactions. The population is modeled as a stochastic point process whose generator captures the probabilistic dynamics over continuous time of birth, mutation, and death, as influenced by each individual’s trait values, and interactions between individuals. An offspring usually inherits the trait values of her progenitor, except when a random mutation causes the offspring to take an instantaneous mutation step at birth to new trait values. In the case we are interested in, the probability distribution of mutations has a heavy tail and belongs to the domain of attraction of a stable law and the corresponding diffusion admits jumps. This could be seen as an alternative to Gould and Eldredge’s model of evolutionary punctuated equilibria. We investigate the large-population limit with allometric demographies: larger populations made up of smaller individuals which reproduce and die faster, as is typical for micro-organisms. We show that depending on the allometry coefficient the limit behavior of the population process can be approximated by nonlinear Levy flights of different nature: either deterministic, in the form of non-local fractional reaction–diffusion equations, or stochastic, as nonlinear super-processes with the underlying reaction and a fractional diffusion operator. These approximation results demonstrate the existence of such non-trivial fractional objects; their uniqueness is also proved.
TL;DR: A comprehensive computational framework within which the effects of chemical signalling factors on growing epithelial tissues can be studied, and may be adapted to a range of potential application areas, and to other cell-based models with designated node movements, to accurately probe the role of morphogens.
Abstract: In this paper we present a comprehensive computational framework within which the effects of chemical signalling factors on growing epithelial tissues can be studied. The method incorporates a vertex-based cell model, in conjunction with a solver for the governing chemical equations. The vertex model provides a natural mesh for the finite element method (FEM), with node movements determined by force laws. The arbitrary Lagrangian–Eulerian formulation is adopted to account for domain movement between iterations. The effects of cell proliferation and junctional rearrangements on the mesh are also examined. By implementing refinements of the mesh we show that the finite element (FE) approximation converges towards an accurate numerical solution. The potential utility of the system is demonstrated in the context of Decapentaplegic (Dpp), a morphogen which plays a crucial role in development of the Drosophila imaginal wing disc. Despite the presence of a Dpp gradient, growth is uniform across the wing disc. We make the growth rate of cells dependent on Dpp concentration and show that the number of proliferation events increases in regions of high concentration. This allows hypotheses regarding mechanisms of growth control to be rigorously tested. The method we describe may be adapted to a range of potential application areas, and to other cell-based models with designated node movements, to accurately probe the role of morphogens in epithelial tissues.
TL;DR: A rigorous definition of the hydra effect in population models is proposed and results show that hydra effects typically occur in the well-known Gause-type models whenever the system dynamics are cyclic.
Abstract: The seemingly paradoxical increase of a species population size in response to an increase in its mortality rate has been observed in several continuous-time and discrete-time models. This phenomenon has been termed the “hydra effect”. In light of the fact that there is almost no empirical evidence yet for hydra effects in natural and laboratory populations, we address the question whether the examples that have been put forward are exceptions, or whether hydra effects are in fact a common feature of a wide range of models. We first propose a rigorous definition of the hydra effect in population models. Our results show that hydra effects typically occur in the well-known Gause-type models whenever the system dynamics are cyclic. We discuss the apparent discrepancy between the lack of hydra effects in natural populations and their occurrence in this standard class of predator–prey models.
TL;DR: In this chapter, the logistic model of population growth is derived by first describing the behaviour of the individuals and then using it to derive a model of the population, which shows that the authors cannot meaningfully vary r and K independently.
Abstract: Everyone knows the logistic model of population growth. Populations cannot grow indefinitely; hence the per capita growth rate must decrease as the population density N increases; the simplest decreasing function is linear; and so we have d N dt = [r(1 − N/K )]N , where r is the intrinsic growth rate and K is called the carrying capacity of the environment. Conscientious textbooks will add in parentheses, “(assuming r > 0)”. Indeed, with r < 0, the equation predicts unbounded growth, a biological nonsense, for populations whose initial size exceeds K . But an experimentalist is free to choose any positive initial density, and r < 0 is certainly not nonsense: r < 0 holds when the death rate exceeds the birth rate even at low population densities, as for an unfit mutant or a species outside its region of viability. This shows that we cannot meaningfully vary r and K independently. But how they should be connected is unclear: The information is simply not in the equation. A population is an ensemble of individuals, and its behaviour is ultimately a consequence of the behaviour of the individuals. In order to predict population behaviour, we should first describe the behaviour of the individuals and then use it to derive a model of the population. When the logistic model is derived in this way, it is clear that K is not an externally fixed parameter (the “carrying capacity of the environment”).
TL;DR: The results show that only the drug-resistant strain can dominate (the first-line treatment program guided by the Manuals) or both strains may be rapidly eliminated (the second-line Treatment program), thus the work indicates the importance of implementing the second- line treatment program as soon as possible.
Abstract: The purposes of this paper are twofold: to develop a rigorous approach to analyze the threshold behaviors of nonlinear virus dynamics models with impulsive drug effects and to examine the feasibility of virus clearance following the Manuals of National AIDS Free Antiviral Treatment in China. An impulsive system of differential equations is developed to describe the within-host virus dynamics of both wild-type and drug-resistant strains when a combination of antiretroviral drugs is used to induce instantaneous drug effects at a sequence of dosing times equally spaced while drug concentrations decay exponentially after the dosing time. Threshold parameters are derived using the basic reproduction number of periodic epidemic models, and are used to depict virus clearance/persistence scenarios using the theory of asymptotic periodic systems and the persistence theory of discrete dynamical systems. Numerical simulations using model systems parametrized in terms of the antiretroviral therapy recommended in the aforementioned Manuals illustrate the theoretical threshold virus dynamics, and examine conditions under which the impulsive antiretroviral therapy leads to treatment success. In particular, our results show that only the drug-resistant strain can dominate (the first-line treatment program guided by the Manuals) or both strains may be rapidly eliminated (the second-line treatment program), thus the work indicates the importance of implementing the second-line treatment program as soon as possible.
TL;DR: It is shown that, even when resource-supply has a continuous distribution in trait space, a positive continuous distribution of consumer trait is impossible and global convergence to the evolutionarily stable distribution is proved.
Abstract: To understand the evolution of diverse species, theoretical studies using a Lotka–Volterra type direct competition model had shown that concentrated distributions of species in continuous trait space often occurs. However, a more mechanistic approach is preferred because the competitive interaction of species usually occurs not directly but through competition for resource. We consider a chemostat-type model where species consume resource that are constantly supplied. Continuous traits in both consumer species and resource are incorporated. Consumers utilize resource whose trait values are similar with their own. We show that, even when resource-supply has a continuous distribution in trait space, a positive continuous distribution of consumer trait is impossible. Self-organized generation of distinct species occurs. We also prove global convergence to the evolutionarily stable distribution.
TL;DR: It is shown that transient periodic oscillations occur when a saddle-type periodic solution exists and can be robust and observable and implications to the dynamics of CTL response to HTLV-I infections in vivo and pathogenesis of HAM/TSP are discussed.
Abstract: The cytotoxic T lymphocyte (CTL) response to the infection of CD4+ T cells by human T cell leukemia virus type I (HTLV-I) has previously been modelled using standard response functions, with relatively simple dynamical outcomes. In this paper, we investigate the consequences of a more general CTL response and show that a sigmoidal response function gives rise to complex behaviours previously unobserved. Multiple equilibria are shown to exist and none of the equilibria is a global attractor during the chronic infection phase. Coexistence of local attractors with their own basin of attractions is the norm. In addition, both stable and unstable periodic oscillations can be created through Hopf bifurcations. We show that transient periodic oscillations occur when a saddle-type periodic solution exists. As a consequence, transient periodic oscillations can be robust and observable. Implications of our findings to the dynamics of CTL response to HTLV-I infections in vivo and pathogenesis of HAM/TSP are discussed.
TL;DR: A surprising ‘50% law’ is proved: if infection by a pathogen induces a partial immunity reducing susceptibility by less than 50%, then, whatever the value of R0 > 1 before the first epidemic, a second epidemic will occur, while if susceptibility is reduced by more than50%, then aSecond epidemic will only occur if R0 is larger than a certain critical value greater than 1.
Abstract: We formulate and study a general epidemic model allowing for an arbitrary distribution of susceptibility in the population. We derive the final-size equation which determines the attack rate of the epidemic, somewhat generalizing previous work. Our main aim is to use this equation to investigate how properties of the susceptibility distribution affect the attack rate. Defining an ordering among susceptibility distributions in terms of their Laplace transforms, we show that a susceptibility distribution dominates another in this ordering if and only if the corresponding attack rates are ordered for every value of the reproductive number R0. This result is used to prove a sharp universal upper bound for the attack rate valid for any susceptibility distribution, in terms of R0 alone, and a sharp lower bound in terms of R0 and the coefficient of variation of the susceptibility distribution. We apply some of these results to study two issues of epidemiological interest in a population with heterogeneous susceptibility: (1) the effect of vaccination of a fraction of the population with a partially effective vaccine, (2) the effect of an epidemic of a pathogen inducing partial immunity on the possibility and size of a future epidemic. In the latter case, we prove a surprising '50% law': if infection by a pathogen induces a partial immunity reducing susceptibility by less than 50%, then, whatever the value of R0>1 before the first epidemic, a second epidemic will occur, while if susceptibility is reduced by more than 50%, then a second epidemic will only occur if R0 is larger than a certain critical value greater than 1.
TL;DR: A complete classification for the global dynamics of a Lotka–Volterra two species competition model with seasonal succession is obtained via the stability analysis of equilibria and the theory of monotone dynamical systems.
Abstract: A complete classification for the global dynamics of a Lotka–Volterra two species competition model with seasonal succession is obtained via the stability analysis of equilibria and the theory of monotone dynamical systems. The effects of two death rates in the bad season and the proportion of the good season on the competition outcomes are also discussed.