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Mathematical Biosciences & Engineering

2012 , Volume 9 , Issue 3

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Stochastic models for competing species with a shared pathogen
Linda J. S. Allen and Vrushali A. Bokil
2012, 9(3): 461-485 doi: 10.3934/mbe.2012.9.461 +[Abstract](12861) +[PDF](1380.4KB)
The presence of a pathogen among multiple competing species has important ecological implications. For example, a pathogen may change the competitive outcome, resulting in replacement of a native species by a non-native species. Alternately, if a pathogen becomes established, there may be a drastic reduction in species numbers. Stochastic variability in the birth, death and pathogen transmission processes plays an important role in determining the success of species or pathogen invasion. We investigate these phenomena while studying the dynamics of deterministic and stochastic models for $n$ competing species with a shared pathogen. The deterministic model is a system of ordinary differential equations for $n$ competing species in which a single shared pathogen is transmitted among the $n$ species. There is no immunity from infection, individuals either die or recover and become immediately susceptible, an SIS disease model. Analytical results about pathogen persistence or extinction are summarized for the deterministic model for two and three species and new results about stability of the infection-free state and invasion by one species of a system of $n-1$ species are obtained. New stochastic models are derived in the form of continuous-time Markov chains and stochastic differential equations. Branching process theory is applied to the continuous-time Markov chain model to estimate probabilities for pathogen extinction or species invasion. Finally, numerical simulations are conducted to explore the effect of disease on two-species competition, to illustrate some of the analytical results and to highlight some of the differences in the stochastic and deterministic models.
A comparison of computational efficiencies of stochastic algorithms in terms of two infection models
H. Thomas Banks, Shuhua Hu, Michele Joyner, Anna Broido, Brandi Canter, Kaitlyn Gayvert and Kathryn Link
2012, 9(3): 487-526 doi: 10.3934/mbe.2012.9.487 +[Abstract](3155) +[PDF](5962.6KB)
In this paper, we investigate three particular algorithms: a stochastic simulation algorithm (SSA), and explicit and implicit tau-leaping algorithms. To compare these methods, we used them to analyze two infection models: a Vancomycin-resistant enterococcus (VRE) infection model at the population level, and a Human Immunodeficiency Virus (HIV) within host infection model. While the first has a low species count and few transitions, the second is more complex with a comparable number of species involved. The relative efficiency of each algorithm is determined based on computational time and degree of precision required. The numerical results suggest that all three algorithms have the similar computational efficiency for the simpler VRE model, and the SSA is the best choice due to its simplicity and accuracy. In addition, we have found that with the larger and more complex HIV model, implementation and modification of tau-Leaping methods are preferred.
Fast two dimensional to three dimensional registration of fluoroscopy and CT-scans using Octrees on segmentation maps
Luca Bertelli and Frédéric Gibou
2012, 9(3): 527-537 doi: 10.3934/mbe.2012.9.527 +[Abstract](2786) +[PDF](1925.8KB)
We introduce a computationally efficient approach to the generation of Digital Reconstructed Radiographs (DRRs) needed to perform three dimensional to two dimensional medical image registration and apply this algorithm to virtual surgery. The DRR generation process is the bottleneck of any three dimensional to two dimensional registration system, since its computational complexity scales with the number of voxels in the Computed Tomography Data, which can be of the order of tens to hundreds of millions. Our approach originates from the segmentation of the volumetric data into multiple regions, which allows a compact representation via Octree Data Structures. This, in turn, yields efficient storage and access of the attenuation indexes of the volumetric cells, required in the projection procedure that generates the DRR. A functional based on Mutual Information is then maximized to obtain the alignment of the DRR with the two dimensional X-ray fluoroscopy scans acquired during the operation. Promising experimental results on real data are presented.
Impact of vaccine arrival on the optimal control of a newly emerging infectious disease: A theoretical study
Bruno Buonomo and Eleonora Messina
2012, 9(3): 539-552 doi: 10.3934/mbe.2012.9.539 +[Abstract](3114) +[PDF](466.2KB)
When a newly emerging human infectious disease spreads through a host population, it may be that public health authorities must begin facing the outbreaks and planning an intervention campaign when not all intervention tools are readily available. In such cases, the problem of finding optimal intervention strategies to minimize both the disease burden and the intervention costs may be addressed by considering multiple intervention regimes. In this paper, we consider the scenario in which authorities may rely initially only on non-pharmaceutical interventions at the beginning of the campaign, knowing that a vaccine will later be available, at an exogenous and known switching time. We use a two-stage optimal control problem over a finite time horizon to analyze the optimal intervention strategies during the whole campaign, and to assess the effects of the new intervention tool on the preceding stage of the campaign. We obtain the optimality systems of two connected optimal control problems, and show the solution profiles through numerical simulations.
Parameter estimation and uncertainty quantification for an epidemic model
Alex Capaldi, Samuel Behrend, Benjamin Berman, Jason Smith, Justin Wright and Alun L. Lloyd
2012, 9(3): 553-576 doi: 10.3934/mbe.2012.9.553 +[Abstract](3991) +[PDF](503.5KB)
We examine estimation of the parameters of Susceptible-Infective-Recovered (SIR) models in the context of least squares. We review the use of asymptotic statistical theory and sensitivity analysis to obtain measures of uncertainty for estimates of the model parameters and the basic reproductive number ($R_0$)---an epidemiologically significant parameter grouping. We find that estimates of different parameters, such as the transmission parameter and recovery rate, are correlated, with the magnitude and sign of this correlation depending on the value of $R_0$. Situations are highlighted in which this correlation allows $R_0$ to be estimated with greater ease than its constituent parameters. Implications of correlation for parameter identifiability are discussed. Uncertainty estimates and sensitivity analysis are used to investigate how the frequency at which data is sampled affects the estimation process and how the accuracy and uncertainty of estimates improves as data is collected over the course of an outbreak. We assess the informativeness of individual data points in a given time series to determine when more frequent sampling (if possible) would prove to be most beneficial to the estimation process. This technique can be used to design data sampling schemes in more general contexts.
Multiple endemic states in age-structured $SIR$ epidemic models
Andrea Franceschetti, Andrea Pugliese and Dimitri Breda
2012, 9(3): 577-599 doi: 10.3934/mbe.2012.9.577 +[Abstract](3145) +[PDF](1719.6KB)
$SIR$ age-structured models are very often used as a basic model of epidemic spread. Yet, their behaviour, under generic assumptions on contact rates between different age classes, is not completely known, and, in the most detailed analysis so far, Inaba (1990) was able to prove uniqueness of the endemic equilibrium only under a rather restrictive condition.
    Here, we show an example in the form of a $3 \times 3$ contact matrix in which multiple non-trivial steady states exist. This instance of non-uniqueness of positive equilibria differs from most existing ones for epidemic models, since it arises not from a backward transcritical bifurcation at the disease free equilibrium, but through two saddle-node bifurcations of the positive equilibrium. The dynamical behaviour of the model is analysed numerically around the range where multiple endemic equilibria exist; many other features are shown to occur, from coexistence of multiple attractive periodic solutions, some with extremely long period, to quasi-periodic and chaotic attractors.
    It is also shown that, if the contact rates are in the form of a $2 \times 2$ WAIFW matrix, uniqueness of non-trivial steady states always holds, so that 3 is the minimum dimension of the contact matrix to allow for multiple endemic equilibria.
Modeling the effects of introducing a new antibiotic in a hospital setting: A case study
Michele L. Joyner, Cammey C. Manning and Brandi N. Canter
2012, 9(3): 601-625 doi: 10.3934/mbe.2012.9.601 +[Abstract](3259) +[PDF](2094.6KB)
The increase in antibiotic resistance continues to pose a public health risk as very few new antibiotics are being produced, and bacteria resistant to currently prescribed antibiotics is growing. Within a typical hospital setting, one may find patients colonized with bacteria resistant to a single antibiotic, or, of a more emergent threat, patients may be colonized with bacteria resistant to multiple antibiotics. Precautions have been implemented to try to prevent the growth and spread of antimicrobial resistance such as a reduction in the distribution of antibiotics and increased hand washing and barrier preventions; however, the rise of this resistance is still evident. As a result, there is a new movement to try to re-examine the need for the development of new antibiotics. In this paper, we use mathematical models to study the possible benefits of implementing a new antibiotic in this setting; through these models, we examine the use of a new antibiotic that is distributed in various ways and how this could reduce total resistance in the hospital. We compare several different models in which patients colonized with both single and dual-resistant bacteria are present, including a model with no additional treatment protocols for the population colonized with dual-resistant bacteria as well as models including isolation and/or treatment with a new antibiotic. We examine the benefits and limitations of each scenario in the simulations presented.
The mathematical analysis of a syntrophic relationship between two microbial species in a chemostat
Tewfik Sari, Miled El Hajji and Jérôme Harmand
2012, 9(3): 627-645 doi: 10.3934/mbe.2012.9.627 +[Abstract](4395) +[PDF](280.3KB)
A mathematical model involving a syntrophic relationship between two populations of bacteria in a continuous culture is proposed. A detailed qualitative analysis is carried out as well as the analysis of the local and global stability of the equilibria. We demonstrate, under general assumptions of monotonicity which are relevant from an applied point of view, the asymptotic stability of the positive equilibrium point which corresponds to the coexistence of the two bacteria. A syntrophic relationship in the anaerobic digestion process is proposed as a real candidate for this model.
A mathematical model for within-host Toxoplasma gondii invasion dynamics
Adam Sullivan, Folashade Agusto, Sharon Bewick, Chunlei Su, Suzanne Lenhart and Xiaopeng Zhao
2012, 9(3): 647-662 doi: 10.3934/mbe.2012.9.647 +[Abstract](3208) +[PDF](465.4KB)
Toxoplasma gondii (T. gondii) is a protozoan parasite that infects a wide range of intermediate hosts, including all mammals and birds. Up to 20% of the human population in the US and 30% in the world are chronically infected. This paper presents a mathematical model to describe intra-host dynamics of T. gondii infection. The model considers the invasion process, egress kinetics, interconversion between fast-replicating tachyzoite stage and slowly replicating bradyzoite stage, as well as the host's immune response. Analytical and numerical studies of the model can help to understand the influences of various parameters to the transient and steady-state dynamics of the disease infection.
A minimal mathematical model for the initial molecular interactions of death receptor signalling
Christian Winkel, Simon Neumann, Christina Surulescu and Peter Scheurich
2012, 9(3): 663-683 doi: 10.3934/mbe.2012.9.663 +[Abstract](4250) +[PDF](585.6KB)
Tumor necrosis factor (TNF) is the name giving member of a large cytokine family mirrored by a respective cell membrane receptor super family. TNF itself is a strong proinflammatory regulator of the innate immune system, but has been also recognized as a major factor in progression of autoimmune diseases. A subgroup of the TNF ligand family, including TNF, signals via so-called death receptors, capable to induce a major form of programmed cell death, called apoptosis. Typical for most members of the whole family, death ligands form homotrimeric proteins, capable to bind up to three of their respective receptor molecules. But also unligated receptors occur on the cell surface as homomultimers due to a homophilic interaction domain. Based on these two interaction motivs (ligand/receptor and receptor/receptor) formation of large ligand/receptor clusters can be postulated which have been also observed experimentally. We use here a mass action kinetics approach to establish an ordinary differential equations model describing the dynamics of primary ligand/receptor complex formation as a basis for further clustering on the cell membrane. Based on available experimental data we develop our model in a way that not only ligand/receptor, but also homophilic receptor interaction is encompassed. The model allows formation of two distict primary ligand/receptor complexes in a ligand concentration dependent manner. At extremely high ligand concentrations the system is dominated by ligated receptor homodimers.
Global properties of a delayed SIR epidemic model with multiple parallel infectious stages
Xia Wang and Shengqiang Liu
2012, 9(3): 685-695 doi: 10.3934/mbe.2012.9.685 +[Abstract](3107) +[PDF](348.7KB)
In this paper, we study the global properties of an SIR epidemic model with distributed delays, where there are several parallel infective stages, and some of the infected cells are detected and treated, which others remain undetected and untreated. The model is analyzed by determining a basic reproduction number $R_0$, and by using Lyapunov functionals, we prove that the infection-free equilibrium $E^0$ of system (3) is globally asymptotically attractive when $R_0\leq 1$, and that the unique infected equilibrium $E^*$ of system (3) exists and it is globally asymptotically attractive when $R_0>1$.
Erratum to: Investigating the steady state of multicellular sheroids by revisiting the two-fluid model
Antonio Fasano, Marco Gabrielli and Alberto Gandolfi
2012, 9(3): 697-697 doi: 10.3934/mbe.2012.9.697 +[Abstract](2703) +[PDF](176.5KB)
In our paper [1], equation (45) has been reported incorrectly. Its actual form is as follows: \begin{align}\label{stresscont3} \frac{2\gamma}{R}=&\frac{1}{\nu(1-\nu)}\frac{\chi R^2}{3K}\left\{\frac{R} {\rho_D^{}} \left[1-\left(\frac{\rho_P^{}}{R}\right)^3\right]-\frac{3}{2}\left[1-\left( \frac{\rho_P^{}}{R}\right)^2\right]\right\} \nonumber\\ +&\frac{4}{3}\eta_C\chi\left(\frac{R}{\rho_D^{}}\right)^3\left[1-\left( \frac{\rho_P^{}}{R}\right)^3\right] \nonumber\\ +&2\sqrt{3}\tau_0\left[\ln\frac{\rho_P^{}}{\rho_D^{}}+\frac{1}{3}\ln \frac{\sqrt{2}(\frac{R}{\rho_P^{}})^3 + \sqrt{1+2(\frac{R}{\rho_P^{}})^6}} {\sqrt{2}+\sqrt{3}}\right]. \end{align} The comments that followed Eq. (45) then change accordingly. It is immediate to realize that the presence of $\tau_0$ has two effects: it increases the minimal value of the surface tension needed for the existence of the steady state, and, given $\gamma$, if (45) has a solution this solution is greater than the solution of (35). However, since it is possible that the surface tension $\gamma$ has a monotone dependence on the yield stress $\tau_0$ (both these quantity have their physical origin in the intercellular adhesion bonds), a partial compensation of the effect of the yield stress on the determination of $R$ can be expected.

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