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Mathematical Biosciences & Engineering

2014 , Volume 11 , Issue 4

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A structured model for the spread of Mycobacterium marinum: Foundations for a numerical approximation scheme
Azmy S. Ackleh, Mark L. Delcambre, Karyn L. Sutton and Don G. Ennis
2014, 11(4): 679-721 doi: 10.3934/mbe.2014.11.679 +[Abstract](61) +[PDF](693.9KB)
We develop a finite difference scheme to approximate the solution of a novel size-structured mathematical model of the transmission dynamics of Mycobacterium marinum (Mm) in an aquatic environment. The model consists of a system of nonlinear hyperbolic partial differential equations coupled with three nonlinear ordinary differential equations. Existence and uniqueness results are established and convergence of the finite difference approximation to the unique bounded variation weak solution of the model is obtained. Numerical simulations demonstrating the accuracy of the method are presented. We also conducted preliminary studies on the key features of this model, such as various forms of growth rates (indicative of possible theories of development), and conditions for competitive exclusion or coexistence as determined by reproductive fitness and genetic spread in the population.
Critical transitions in a model of a genetic regulatory system
Jesse Berwald and Marian Gidea
2014, 11(4): 723-740 doi: 10.3934/mbe.2014.11.723 +[Abstract](59) +[PDF](1515.8KB)
We consider a model for substrate-depletion oscillations in genetic systems, based on a stochastic differential equation with a slowly evolving external signal. We show the existence of critical transitions in the system. We apply two methods to numerically test the synthetic time series generated by the system for early indicators of critical transitions: a detrended fluctuation analysis method, and a novel method based on topological data analysis (persistence diagrams).
On the estimation of sequestered infected erythrocytes in Plasmodium falciparum malaria patients
Derdei Bichara, Nathalie Cozic and Abderrahman Iggidr
2014, 11(4): 741-759 doi: 10.3934/mbe.2014.11.741 +[Abstract](67) +[PDF](908.8KB)
The aim of this paper is to give a method for the estimation of total parasite burden of the patient and the rate of infection in a malaria's intra-host model by using control theory tools. More precisely, we use an auxiliary system, called observer or estimator, whose solutions tend exponentially to those of the original model. This observer uses only the available measurable data, namely, the values of peripheral infected erythrocytes. It provides estimates of the sequestered infected erythrocytes, that cannot be measured by clinical methods. Therefore this method allows to estimate the total parasite burden within a malaria patient. Moreover, our constructed observer does not use the uncertain infection rate parameter $\beta$. In fact, we derive a simple method to estimate this parameter $\beta$. We apply this estimation method using real data that have been collected when malaria was used as therapy for neurosyphilis by the US Public Health Service.
A SEIR model for control of infectious diseases with constraints
M. H. A. Biswas, L. T. Paiva and MdR de Pinho
2014, 11(4): 761-784 doi: 10.3934/mbe.2014.11.761 +[Abstract](194) +[PDF](680.4KB)
Optimal control can be of help to test and compare different vaccination strategies of a certain disease. In this paper we propose the introduction of constraints involving state variables on an optimal control problem applied to a compartmental SEIR (Susceptible. Exposed, Infectious and Recovered) model. We study the solution of such problems when mixed state control constraints are used to impose upper bounds on the available vaccines at each instant of time. We also explore the possibility of imposing upper bounds on the number of susceptible individuals with and without limitations on the number of vaccines available. In the case of mere mixed constraints a numerical and analytical study is conducted while in the other two situations only numerical results are presented.
Stability and bifurcation analysis of epidemic models with saturated incidence rates: An application to a nonmonotone incidence rate
Yoichi Enatsu and Yukihiko Nakata
2014, 11(4): 785-805 doi: 10.3934/mbe.2014.11.785 +[Abstract](136) +[PDF](1585.6KB)
We analyze local asymptotic stability of an SIRS epidemic model with a distributed delay. The incidence rate is given by a general saturated function of the number of infective individuals. Our first aim is to find a class of nonmonotone incidence rates such that a unique endemic equilibrium is always asymptotically stable. We establish a characterization for the incidence rate, which shows that nonmonotonicity with delay in the incidence rate is necessary for destabilization of the endemic equilibrium. We further elaborate the stability analysis for a specific incidence rate. Here we improve a stability condition obtained in [Y. Yang and D. Xiao, Influence of latent period and nonlinear incidence rate on the dynamics of SIRS epidemiological models, Disc. Cont. Dynam. Sys. B 13 (2010) 195-211], which is illustrated in a suitable parameter plane. Two-parameter plane analysis together with an application of the implicit function theorem facilitates us to obtain an exact stability condition. It is proven that as increasing a parameter, measuring saturation effect, the number of infective individuals at the endemic steady state decreases, while the equilibrium can be unstable via Hopf bifurcation. This can be interpreted as that reducing a contact rate may cause periodic oscillation of the number of infective individuals, thus disease can not be eradicated completely from the host population, though the level of the endemic equilibrium for the infective population decreases. Numerical simulations are performed to illustrate our theoretical results.
On a diffusive predator-prey model with nonlinear harvesting
Peng Feng
2014, 11(4): 807-821 doi: 10.3934/mbe.2014.11.807 +[Abstract](129) +[PDF](953.2KB)
In this paper, we study the dynamics of a diffusive Leslie-Gower model with a nonlinear harvesting term on the prey. We analyze the existence of positive equilibria and their dynamical behaviors. In particular, we consider the model with a weak harvesting term and find the conditions for the local and global asymptotic stability of the interior equilibrium. The global stability is established by considering a proper Lyapunov function. In contrast, the model with strong harvesting term has two interior equilibria and bi-stability may occur for this system. We also give the conditions of Turing instability and perform a series of numerical simulations and find that the model exhibits complex patterns.
Coexistence and asymptotic stability in stage-structured predator-prey models
Wei Feng, Michael T. Cowen and Xin Lu
2014, 11(4): 823-839 doi: 10.3934/mbe.2014.11.823 +[Abstract](67) +[PDF](302.6KB)
In this paper we analyze the effects of a stage-structured predator-prey system where the prey has two stages, juvenile and adult. Three different models (where the juvenile or adult prey populations are vulnerable) are studied to evaluate the impacts of this structure to the stability of the system and coexistence of the species. We assess how various ecological parameters, including predator mortality rate and handling times on prey, prey growth rate and death rate, prey capture rate and nutritional values in two stages, affect the existence and stability of all possible equilibria in each of the models, as well as the ultimate bounds and the dynamics of the populations. The main focus of this paper is to find general conditions to ensure the presence and stability of the coexistence equilibrium where both the predator and prey can co-exist Through specific examples, we demonstrate the stability of the trivial and co-existence equilibrium as well as the dynamics in each system.
Effects of nutrient enrichment on coevolution of a stoichiometric producer-grazer system
Lina Hao, Meng Fan and Xin Wang
2014, 11(4): 841-875 doi: 10.3934/mbe.2014.11.841 +[Abstract](166) +[PDF](1785.1KB)
A simple producer-grazer model based on adaptive evolution and ecological stoichiometry is proposed and well explored to examine the patterns and consequences of adaptive changes for the evolutionary trait (i.e., body size), and also to investigate the effect of nutrient enrichment on the coevolutin of the producer and the grazer. The analytical and numerical results indicate that this simple model predicts a wide range of evolutionary dynamics and that the total nutrient concentration in the ecosystem plays a pivotal role in determining the outcome of producer-grazer coevolution. Nutrient enrichment may yield evolutionary branching, trait cycles or sensitive dependence on the initial values, depending on how much nutrient is present in the ecosystem. In the absence of grazing, the lower nutrient density facilitates the continuously stable strategy while the higher nutrient density induces evolutionary branching. When the grazer is present, with the increasing of nutrient level, the evolutionary dynamics is very complicated. The evolutionary dynamics sequentially undergo continuously stable strategy, evolutionary branching, evolutionary cycle, and sensitive dependence on the initial values. Nutrient enrichment asserts not only stabilizing but also destabilizing impact on the evolutionary dynamics. The evolutionary dynamics potentially show the paradox of nutrient enrichment. This study well documents the interplay and co-effect of the ecological and evolutionary processes.
Dynamics of a predator-prey system with prey subject to Allee effects and disease
Yun Kang, Sourav Kumar Sasmal, Amiya Ranjan Bhowmick and Joydev Chattopadhyay
2014, 11(4): 877-918 doi: 10.3934/mbe.2014.11.877 +[Abstract](144) +[PDF](2746.9KB)
In this article, we propose a general predator-prey system where prey is subject to Allee effects and disease with the following unique features: (i) Allee effects built in the reproduction process of prey where infected prey (I-class) has no contribution; (ii) Consuming infected prey would contribute less or negatively to the growth rate of predator (P-class) in comparison to the consumption of susceptible prey (S-class). We provide basic dynamical properties for this general model and perform the detailed analysis on a concrete model (SIP-Allee Model) as well as its corresponding model in the absence of Allee effects (SIP-no-Allee Model); we obtain the complete dynamics of both models: (a) SIP-Allee Model may have only one attractor (extinction of all species), two attractors (bi-stability either induced by small values of reproduction number of both disease and predator or induced by competition exclusion), or three attractors (tri-stability); (b) SIP-no-Allee Model may have either one attractor (only S-class survives or the persistence of S and I-class or the persistence of S and P-class) or two attractors (bi-stability with the persistence of S and I-class or the persistence of S and P-class). One of the most interesting findings is that neither models can support the coexistence of all three S, I, P-class. This is caused by the assumption (ii), whose biological implications are that I and P-class are at exploitative competition for S-class whereas I-class cannot be superior and P-class cannot gain significantly from its consumption of I-class. In addition, the comparison study between the dynamics of SIP-Allee Model and SIP-no-Allee Model lead to the following conclusions: 1) In the presence of Allee effects, species are prone to extinction and initial condition plays an important role on the surviving of prey as well as its corresponding predator; 2) In the presence of Allee effects, disease may be able to save prey from the predation-driven extinction and leads to the coexistence of S and I-class while predator can not save the disease-driven extinction. All these findings may have potential applications in conservation biology.
A continuous phenotype space model of RNA virus evolution within a host
Andrei Korobeinikov and Conor Dempsey
2014, 11(4): 919-927 doi: 10.3934/mbe.2014.11.919 +[Abstract](86) +[PDF](223.5KB)
Due to their very high replication and mutation rates, RNA viruses can serve as an excellent testing model for verifying hypothesis and addressing questions in evolutionary biology. In this paper, we suggest a simple deterministic mathematical model of the within-host viral dynamics, where a possibility for random mutations incorporates. This model assumes a continuous distribution of viral strains in a one-dimensional phenotype space where random mutations are modelled by Brownian motion (that is, by diffusion). Numerical simulations show that random mutations combined with competition for a resource result in evolution towards higher Darwinian fitness: a stable pulse traveling wave of evolution, moving towards higher levels of fitness, is formed in the phenotype space. The advantage of this model, compared with the previously constructed models, is that this model is mechanistic and is based on commonly accepted model of virus dynamics within a host, and thus it allows an incorporation of features of the real-life host-virus system such as immune response, antiviral therapy, etc.
$R_0$ and the global behavior of an age-structured SIS epidemic model with periodicity and vertical transmission
Toshikazu Kuniya and Mimmo Iannelli
2014, 11(4): 929-945 doi: 10.3934/mbe.2014.11.929 +[Abstract](99) +[PDF](414.9KB)
In this paper, we study an age-structured SIS epidemic model with periodicity and vertical transmission. We show that the spectral radius of the Fréchet derivative of a nonlinear integral operator plays the role of a threshold value for the global behavior of the model, that is, if the value is less than unity, then the disease-free steady state of the model is globally asymptotically stable, while if the value is greater than unity, then the model has a unique globally asymptotically stable endemic (nontrivial) periodic solution. We also show that the value can coincide with the well-know epidemiological threshold value, the basic reproduction number $\mathcal{R}_0$.
Global dynamics for two-species competition in patchy environment
Kuang-Hui Lin, Yuan Lou, Chih-Wen Shih and Tze-Hung Tsai
2014, 11(4): 947-970 doi: 10.3934/mbe.2014.11.947 +[Abstract](78) +[PDF](534.6KB)
An ODE system modeling the competition between two species in a two-patch environment is studied. Both species move between the patches with the same dispersal rate. It is shown that the species with larger birth rates in both patches drives the other species to extinction, regardless of the dispersal rate. The more interesting case is when both species have the same average birth rate but each species has larger birth rate in one patch. It has previously been conjectured by Gourley and Kuang that the species that can concentrate its birth in a single patch wins if the diffusion rate is large enough, and two species will coexist if the diffusion rate is small. We solve these two conjectures by applying the monotone dynamics theory, incorporated with a complete characterization of the positive equilibrium and a thorough analysis on the stability of the semi-trivial equilibria with respect to the dispersal rate. Our result on the winning strategy for sufficiently large dispersal rate might explain the group breeding behavior that is observed in some animals under certain ecological conditions.
Theoretical assessment of the relative incidences of sensitive and resistant tuberculosis epidemic in presence of drug treatment
Silvia Martorano Raimundo, Hyun Mo Yang and Ezio Venturino
2014, 11(4): 971-993 doi: 10.3934/mbe.2014.11.971 +[Abstract](69) +[PDF](596.6KB)
Despite the availability of effective treatment, tuberculosis (TB) remains a major global cause of mortality. Multidrug-resistant tuberculosis (MDR-TB) is a form of TB that is resistant to at least two drugs used for the treatment of TB, and originally is developed when a case of drug-susceptible TB is improperly or incompletely treated. This work is concerned with a mathematical model to evaluate the effect of MDR-TB on TB epidemic and its control. The model assessing the transmission dynamics of both drug-sensitive and drug-resistant TB includes slow TB (cases that result from endogenous reactivation of susceptible and resistant latent infections). We identify the steady states of the model to analyse their stability. We establish threshold conditions for possible scenarios: elimination of sensitive and resistant strains and coexistence of both. We find that the effective reproductive number is composed of two critical values, relative reproductive number for drug-sensitive and drug-resistant strains. Our results imply that the potential for the spreading of the drug-resistant strain should be evaluated within the context of several others factors. We have also found that even the considerably less fit drug-resistant strains can lead to a high MDR-TB incidence, because the treatment is less effective against them.
A note on global stability for malaria infections model with latencies
Jinliang Wang, Jingmei Pang and Toshikazu Kuniya
2014, 11(4): 995-1001 doi: 10.3934/mbe.2014.11.995 +[Abstract](100) +[PDF](298.2KB)
A recent paper [Y. Xiao and X. Zou, On latencies in malaria infections and their impact on the disease dynamics, Math. Biosci. Eng., 10(2) 2013, 463-481.] presented a mathematical model to investigate the spread of malaria. The model is obtained by modifying the classic Ross-Macdonald model by incorporating latencies both for human beings and female mosquitoes. It is realistic to consider the new model with latencies differing from individuals to individuals. However, the analysis in that paper did not resolve the global malaria disease dynamics when $\Re_0>1$. The authors just showed global stability of endemic equilibrium for two specific probability functions: exponential functions and step functions. Here, we show that if there is no recovery, the endemic equilibrium is globally stable for $\Re_0>1$ without other additional conditions. The approach used here, is to use a direct Lyapunov functional and Lyapunov- LaSalle invariance principle.
Stochastic dynamics of SIRS epidemic models with random perturbation
Qingshan Yang and Xuerong Mao
2014, 11(4): 1003-1025 doi: 10.3934/mbe.2014.11.1003 +[Abstract](86) +[PDF](430.4KB)
In this paper, we consider a stochastic SIRS model with parameter perturbation, which is a standard technique in modeling population dynamics. In our model, the disease transmission coefficient and the removal rates are all affected by noise. We show that the stochastic model has a unique positive solution as is essential in any population model. Then we establish conditions for extinction or persistence of the infectious disease. When the infective part is forced to expire, the susceptible part converges weakly to an inverse-gamma distribution with explicit shape and scale parameters. In case of persistence, by new stochastic Lyapunov functions, we show the ergodic property and positive recurrence of the stochastic model. We also derive the an estimate for the mean of the stationary distribution. The analytical results are all verified by computer simulations, including examples based on experiments in laboratory populations of mice.

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