Contributions to Mathematical Reviews

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John Nagy

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The papers I have reviewed for Mathematical Reviews (MR) are listed below (only those already published). For those unfamiliar with MR, the AMS sends papers that are already in the peer reviewed literature to MR reviewers, who then write a short review of the paper. The review is more than a summary and servers as a resource to help researchers navigate the body of mathematical literature. My fields are mathematical biology and mathematical medicine.

The links on the article titles take you to a pdf copy of the review. The text here is the original LaTeX document. The pdf and LaTeX text may not agree entirely, since the latter has been edited by MR.

Betancourt-Mar, Juvencio Alberto; Nieto-Villar, José Manuel. Theoretical models for chronotherapy: periodic perturbations in funnel chaos type. Math. Biosci. Eng. 4

(2007), no. 2, 177--186. [AMS Classifications 92C50 (34C28 37N25)] MR2293760

The concept of "dynamical disease" lays claim to an enormous variety of pathologies, including diseases that exhibit more-or-less periodic or episodic behavior---like malignant tertian malaria (caused by Plasmodium falciparum), lupus erythematosus or sleep apnea---and situations in which an organ or tissue suddenly changes state---like cardiac arrhythmia or seizure. Despite their diversity in pathogenesis and clinical significance, these diseases behave in ways easily mimicked by nonlinear dynamical systems, which suggests that dynamical systems theory may unify this diverse set of pathologies. In this view, the pathological state is interpreted as arising from a bifurcation that occurs as some physiological parameter shifts from one value to another. If true, this hypothesis implies that successful treatment returns parameters to a region of parameter space characterized by "proper" (nonpathological) behavior.

To achieve this goal, so-called "chronotherapy" seems a natural choice. In this context, chronotherapy refers to essentially any attempt to optimize the timing of treatment episodes. Betancourt-Mar and Nieto-Villar illustrate this idea with the R\"{o}ssler model, which is a simple, three-dimensional system of ordinary differential equations that exhibits very rich dynamics, including both spiral and funnel chaos. They represent chronotherapy as a periodic forcing, in particular a sine function, on one of the equations. The authors illustrate numerically that one can move among chaotic and periodic windows by manipulating the amplitude and frequency of the forcing.

Unfortunately, the paper has limited application because the model has no obvious relationship to anything physiological or medical---it merely serves as an illustration. Nevertheless, the paper's main biological point is that treatment for dynamical diseases need not return the parameters back to their original values to restore health. Rather, the system simply has to enter a region in which it exhibits "healthy" dynamics, and such regions can be quite large in parameter space. However, to take advantage of this insight, one requires an accurate understanding of the bifurcation structure of the system. Otherwise, "healthy" regions of parameter space will be difficult to characterize. This creates a very dangerous situation in real applications since the authors show that the bifurcation structure of even a simple model like the R\"{o}ssler system is quite complex. Therefore, any real application would require not only an appropriate model, but also parameter estimates for individual patients of much more precision and accuracy than has been achieved or even approached in reality.

Michel, Philippe. Existence of a solution to the cell division eigenproblem. Math. Models Methods Appl. Sci. 16 (2006), no. 7, suppl., 1125--1153. [AMS Classifications 92C37 (35C10 35F10 35P05)] MR2250122 (2007f:92016)

Cancer arises, partly, because cells lose control of their own cell cycles. That fact has sparked a number of tumor models with explicit descriptions of cell cycle progression, either as a continuous process, a series of discrete stages \ref[A. Bertuzzi et al., Math. Biosci. 177/178 (2002), 103--125; MR1923805 (2003i:92009)], or a combination of the two \ref[B. Basse et al., J. Math. Biol. 47 (2003), no. 4, 295--312; MR2024498 (2004j:92024)]. In the continuous case, the resulting model is a physiologically structured partial differential equation model, where the structuring variable is typically interpreted as cell "age", DNA content or size. In this paper, the author generalizes a result obtained by B. Perthame and L. Ryzhik \ref[J. Differential Equations 210 (2005), no. 1, 155--177; MR2114128 (2006b:35328)] on such a model with symmetric cell division. In particular, Michel relaxes the assumption of symmetric cell division, and proves existence and uniqueness of the Malthus parameter (first eigenvalue) and stable steady-state distribution (first eigenvector) representing the asymptotic behavior of the solution. (In fact, Michel's results can apply to the general fragmentation problem since he allows "cells" to divide into any number of particles.) The author also extends his own result to include non-constant progression through the cell cycle before returning to an explicit solution of the eigenproblem under more restrictive conditions on the (stage-dependent) reproduction function.

Hernández-García, Emilio; López, Cristóbal. Logistic population growth and beyond: the influence of advection and nonlocal effects. The logistic map and the route to chaos, 117--129, Underst. Complex Syst., Springer, Berlin, 2006. [AMS Classifications 92D25 (37N25)] MR2202739

The role that space---geography in its broadest sense---plays in biological population dynamics remains largely an open question and an active area of research in mathematical biology. Typical modeling approaches include discrete patch metapopulation models \ref[M. Martcheva and H. R. Thieme, Natur. Resource Modeling 18 (2005), no. 4, 379--413; MR2199045 (2006i:92039)], in which individuals disperse among habitat patches, and spatially continuous models, in which dispersers move by a diffusion-like process \ref[Y. H. Fan and W. T. Li, J. Comput. Appl. Math. 188 (2006), no. 2, 205--227; MR2201577 (2006h:35137)], advection, or both through a continuous habitat. In this paper, Emilio Hernández-García and Cristobál López focus on the latter approach, including both diffusion and advection. They review two families of spatially continuous population growth models. The first describes a marine predator-prey system, in which the predator and prey are zoo- and phytoplankton, respectively. Basic population dynamics are governed by a standard predator-prey model with a Holling type III predator response function. This basic model is known to exhibit "blooms," in which, following a small perturbation from equilibrium, the populations temporarily flair before returning to equilibrium \ref[J. E. Truscott and J. Brindley. Bull. Math. Biol. 56 (1994), no. 5, 981--998; Zbl 0803.92026]. To this basic population model Hernández-García and López add diffusion and advection via an oscillating oceanic jet. They find boundary conditions and parameters that allow an initial, local bloom to propagate into a permanent, although spatially and temporally dynamic, feature of the system. The second class of models the authors review are "Brownian Bug" models, in which a population of "bugs" move about in space by both diffusion and advection. The bugs reproduce with a probability dependent on the density of bugs in their local area, called the interaction range. The authors survey parameter space numerically and catalogue regions in which the bugs spread out apparently randomly (large diffusivity), or form spatially periodic clumps (low diffusivity) whose centers are separated by a distance on the order of the interaction range. They show that a mean-field analogue of the stochastic model exhibits a similar transition to periodic clumps in similar regions of parameter space. Finally, by adding advection, they characterize parameter regions in which clumps are pulled into tendril-like patterns.

Parthasarathy, P. R.; Dietz, Klaus. Exact representations for tumour incidence for some density-dependent models. Int. J. Math. Math. Sci. 2005, no. 16, 2655--2667. [AMS classifications 92C50 (60J80 60J85 92C60 92D30)]  MR2184839 (2006h:92019)

In this paper the authors introduce a novel technique to the study of stochastic models of tumorigenesis. They use their method to obtain characterizations of the epidemiological incidence of cancer in situations that some commonly used approaches cannot handle. The modeling framework they select, originally developed by Moolgavkar, Venson and Knudson, and hence called the "MVK" model, has become a standard in stochastic approaches to malignant tumorigenesis. Parthasarathy and Dietz extend MKV by introducing density dependent, per capita birth, death and mutation rates to reflect physiological and demographic changes in aging tumors. The authors first address the problem with generating functions, showing that for a specific form of density dependence, one can obtain an expression for the cumulative distribution function of the time before the first cancer cell develops---essentially the cumulative incidence as a function of time for a given cohort---along with expectations, variances and the covariance for the number of cells in each stage of tumorigenesis. However, since generating functions may be unable to handle more general forms of density dependence they introduce another method that can. In particular, they show that one can express as a continued fraction the Laplace transform of $P_{1,0}(t)$, where $P_{mn}(t)$ is the probability of having $m$ precancerous and $n$ cancer cells at time $t$. (In this formulation, healthy cells are ignored.) If one assumes that the tumor-wide mutation rate (not per capita) remains constant, then the Laplace transform takes on a relatively simple explicit expression. One can then invert the Laplace transform, at least numerically, to obtain $P_{1,0}(t)$. With this value one can then calculate the epidemiological incidence. They end the paper with a series of worked examples.

Song, Xinyu; Cheng, Shuhan. A delay-differential equation model of HIV infection of ${\rm CD4}\sp +$ T-cells. J. Korean Math. Soc. 42 (2005), no. 5, 1071--1086. [AMS classifications 92C30 (34K13 34K20 92D25)] MR2157361 (2006e:92022)

Mathematical modelers studying HIV infection frequently simplify their models by assuming that the moment a cell becomes infected, it begins releasing newly-made viruses. In reality, however, some time elapses between initial infection and productive infection---when infected cells shed virions (virus particles)---because once inside the host cell, the virus has to complete the infection cycle and ramp up the cell's metabolism. Song and Cheng study the effect of this time delay in a simple model of HIV dynamics that is very similar to the model by R. V. Culshaw and S. Ruan \ref[Math. Biosci. 165 (2000), no. 1, 27--39]. Like Culshaw and Ruan's model, Song and Cheng's model comprises a system of delay differential equations expressing dynamics of uninfected CD4$^+$ T lymphocytes, productively infected CD4$^+$ T lymphocytes and free virions. The delay arises when newly infected T cells enter a state in which they cannot reproduce themselves or the virus, although they suffer a constant per capita mortality rate. Infected cells that survive in this state for $\tau$ time units reenter the model as productively infected cells.

Song and Cheng's analysis provides a fairly thorough picture of the model's dynamics. In particular, solutions are always bounded, and the system has two equilibria---a disease-free state, $E_1$, and one representing chronic infection, $E_2$. After defining the infection's basic reproductive ratio, $R_0$, the authors show that $R_0<0$ implies that $E_1$ is globally asymptotically stable, while it is unstable if $R_0>0$. They also show that if $E_1$ is unstable, then the system is permanent, and $E_2$ can be either unstable or locally asymptotically stable (LAS) when $\tau=0$. The stability of $E_2$ can be destroyed by a Hopf bifurcation that occurs as $\tau$ increases past a critical point, $\tau_0$---specifically, $E_2$ is LAS for $0\leq\tau<\tau_0$ and unstable for $\tau>\tau_0$, as is commonly observed in other time-delay systems. Interestingly, they also prove the existence of asymptotically stable periodic orbits when both $E_1$ and $E_2$ are unstable and $\tau=0$. Specific parameter values are provided in each dynamic regime for those wishing to explore the model numerically, although the values are not necessarily biologically relevant.

De Boer, Rob J.; Perelson, Alan S. Estimating division and death rates from CFSE data. J. Comput. Appl. Math. 184 (2005), no. 1, 140--164. [AMS classifications 92C37 (34B45 62F10 62P10)] MR2160061 (2006g:92019)

De Boer and Perelson review the consequences of applying various growth models and calculation procedures to CFSE data. In particular, they compare three methods for calculating the mean number of divisions as a function of time---a raw calculation denoted $µ(t)$; a renormalization due to A. V. Gett and P. D. Hodgkin \ref[Nat. Immunology 1 (2000), no. 3, 239--244] that estimates the proportion of cells in the original population that have divided a certain number of times, including zero ($\mu_2(t)$); and a similar renormalization that includes only cells that have divided at least once ($\widehat{\mu_2}(t)$). They consider a variety of ODE models that differ in their assumed time delay before the first cell reproduces, proportion of cells in the population that will eventually divide ($0 \leq \phi\leq 1$), and a difference in proliferation rates between cells that have never reproduced and cells that have. They compare these variations to the standard cell growth model due to J. A. Smith and L. Martin \ref[Proc. Nat. Acad. Sci. U.S.A. 70 (1973), no. 4, 1263--1267] in which the length of the G$_1$ phase is random with exponential distribution, and the lengths of all other phases (S, G$_2$ and M) are fixed. Fitting all these variations to real cell growth data from Gett and Hodkin and simulations, De Boer and Perelson distill a useful "recipe for interpreting CFSE data" (p. 161) that begins by graphing $µ(t)$, $\mu_2(t)$ and $\widehat{\mu_2}(t)$ and certain population measures, obtained from the data, as points estimating functions of time. All measurements, after an initial transient, should approach a line asymptotically, which can then be estimated with linear regression. The slopes of these regression lines can be used to estimate $\phi$ and parameters for the Smith-Martin model, which in general performs best among all the variants explored, with the caveat that death rates in each phase will still be difficult to estimate but can have a profound impact on estimates of proliferation rates. If confidence in the Smith-Martin model is low, De Boer and Perelson recommend using the rescaling method of S. S. Pilyugin et al. \ref[J. Theoret. Biol. 225 (2003), no. 2, 275--283; MR2077393] "to estimate the invariant `mean generation time of surviving cells' and the `fraction of cells that die in one generation'\," since "[t]hese invariant parameters remain valid even when the cells are not obeying the Smith-Martin model".