Mathematical modeling of tumorigenesis

Traditionally, scientists from many different backgrounds have been producing interesting and unexpected ideas. Their methods come from various fields, including applied mathematics, statistics, computer science, material science, fluid mechanics, population dynamics and evolutionary game theory.

Broadly speaking, there are three major areas where theory has contributed the most to cancer research:

(i) Modeling in the context of epidemiology and other statistical data. One of the oldest and most successful methodologies in theoretical cancer research is using the available incident statistics and creating models to explain the observations. This field was originated by Armitage and Doll in 1954 [Armitage and Doll (1954)], and then taken to the next level by Moolgavkar and colleagues [Moolgavkar and Knudson (1981)].

(ii) Mechanistic modeling of avascular and vascular tumor growth. An entirely different approach to cancer modeling is to look at the mechanistic aspects of tumor material and use physical properties of biological tissues to describe tumor growth, see [Preziosi (2003)] for review.

(iii) Modeling of cancer initiation and progression as somatic evolution. In this area of research, methods of population dynamics and evolutionary game theory are applied to study cancer. First developed by ecologists and evolutionary biologists, these methods have been used to understand the collective behavior of a population of cancer cells, see [Gatenby and Gawlinski (2003)], [Gatenby and Vincent (2003b)], [Gatenby and Vincent (2003a)].

In this Chapter we review basic mathematical tools necessary to undertake different types of cancer modeling. These are: ordinary differential equations, partial differential equations, stochastic processes, cellular automata and agent based modeling.

2.1 Ordinary differential equations

Mathematical modeling of growth, differentiation and mutations of cells in tumors is one of the oldest and best developed topics in biomathematics. It involves modeling of growth, differentiation and mutation of cells in tumors. Let us view cancer as a population of cells, which has some potential to grow. In the simplest case, we can model cellular growth followed by saturation with the following logistic ordinary differential equation (ODE):

x = rx(l — x/k), x(0) = 1, where dot is the time derivative, x = x(t) is the number of cancer cells at time t, r is the growth rate and k is the carrying capacity, that is, the maximal size the population of cells can reach, defined by the nutrient supply, spatial constraints etc. The solution of the above ODE is a familiar looking "sigmoidal" curve.

Next, let us suppose that the population of cells is heterogeneous, and all cells compete with each other and with surrounding healthy cells for nutrients, oxygen and space. Then we can imagine the following system, equipped with the appropriate number of initial conditions:

Xi = TiXi - (j>Xi, 0 <i <71, Xi(0) = Xi, where Xi is the number of cells of type i, with the corresponding growth rate, r,L. We have the total of n types, and we can model the competition by the term cf> in a variety of ways, e.g. by setting

where N = the total number of cells in the system, which is assumed to be constant in this model.

As a next step, we can allow for mutations in the system. In other words, each cell division (happening with rate r» for each type) has a chance to result in the production of a different type. Let us assume for simplicity that the type i can mutate into type (¿ + 1) only, according to the following simple diagram:

Then the equations become,

%i = Ui-iri-iXi-i + r»( 1 - Ui)xi - <j>Xi, 1 < i < n - 1,

Xj(0) = 0 < i < n, where <f> is defined as before, and u, is the probability that a cell of type (i + 1) is created as a result of a division of a cell of type i. The above equations are called the quasispecies equations. These were introduced by Manfred Eigen in 1971 as a way to model the evolutionary dynamics of single-stranded RNA molecules in in vitro evolution experiments. Since Eigen's original paper, the quasispecies model has been extended to viruses, bacteria, and even to simple models of the immune system. Quasispecies equations are nonlinear, like most differential equations in cancer modeling. However, there is a simple and elegant way to solve these equations, which we review in Chapter 7. In a more general case, the mutation network can be more complicated, allowing mutations from each type to any other type. This is done by introducing a mutation matrix with entries, Uij, for mutation rates from type i to type j. Examples of using quasispecies equations in recent literature are [Sole and Deisboeck (2004)] and [Gatenby and Vincent (2003b)].

Other ordinary differential equations used to study the dynamics of cancerous cells are similar to predator-prey systems in ecology. For instance, Gatenby and Vincent [Gatenby and Vincent (2003a)] used the following competition model, f, x + axyy\ . ( y + avxx\ * = j— Jx, y = rv[l-—j—jy, where x and y describe the populations of cancerous and healthy cells, respectively. Moore and Li [Moore and Li (2004)] used a model in a similar spirit to describe the interactions between chronic myelogenous leukemia (CML) and T cells. They considered a system of 3 ODEs, for naive T cells, effector T cells and CML cancer cells.

The equations shown above are toy models to illustrate general principles, rather than actual tools to study real biological phenomena. However, by modifying these equations and incorporating particular properties of a biological system in question, we can describe certain aspects of cancer dynamics. Like any other method, the method of ODEs has advantages and drawbacks. Among the advantages is its simplicity. The disadvantages include the absence of detail. For instance, no spatial interactions can be described by ODEs, thus imposing the assumption of "mass-action"-type interactions. Stochastic effects are not included, restricting the applicability to large systems with no "extinction" effects.

Finally, because of an empirical nature of this kind of modeling, this method (like most other empirical methods) presents a problem when trying to find ways to measure coefficients in the equations. Several methods of robustness analysis have been developed. The main idea is as follows. If the number of equations is in the tens, and the number of coefficients is in the hundreds, one could argue that almost any kind of behavior can be reproduced if we tune the parameters in the right way. Therefore, it appears desirable to reduce the number of unknown parameters and also to design some sort of reliability measure of the system. In the paper by [Moore and Li (2004)], Latin hypercube sampling on large ranges of the parameters is employed, which is a method for systems with large uncertainties in parameters. This involves choosing parameters randomly from a range and solving the resulting system numerically, trying to identify the parameters to which the behavior is the most sensitive. In the paper by Evans et al [Evans et al. (2004)], "structural identifiability analysis" is discussed, which determines whether model outputs can uniquely determine all of the unknown parameters. This is related to (but is not the same as) the confidence with which we view parameter estimation from experimental data. In general, questions of robustness and reliability are studied in mathematical control theory.

2.2 Partial differential equations

The next method that we will mention here is partial differential equations (PDEs). They can be a very useful tool when studying tumor growth and invasion into surrounding tissue. In many models, tumor tissue is described as a mechanistic system, for instance, as a fluid (with a production term proportional to the concentration of nutrients) [Evans et al. (2004)], or as a mixture of solid (tumor) and liquid (extracellular fluid with nutrients) phases [Byrne and Preziosi (2003)]. As an example, we quote the system used by Pranks et al. [Fauth and Speicher (2001); Pranks et al. (2003)]. These authors view an avascular tumor as a coherent mass whose behavior is similar to that of a viscous fluid. The variables n(x, t), m(x. t) and p(x, i) describe the concentration of tumor cells, dead cells and surrounding material, respectively. The nutrient concentration is c(x, t). and the velocity of cells is denoted by v(x,i). Applying the principle of mass balance to different kinds of material, we arrive at the following system:

Here, we have production terms given by the rate of mitosis, kd(c), and cell death, kd(c), which are both given empirical functions of nutrient concentration. The nutrients are governed by a similar mass transport equation, where D is the diffusion coefficient and 7km(c)n represents the rate of nutrient consumption. In order to fully define the system, we also need to use the mass conservation law for the cells, modeled as incompressible, continuous fluid, n + m + p = 1. Finally, a constitutive law for material deformation must be added to define the relation between concentration (stress) and velocity. Also, the complete set of boundary conditions must be imposed to make the system well defined. We skip the details here, referring the reader to the original papers. Our goal in this chapter is to give the flavor of the method.

As our next example, we will mention the paper by Owen et al. [Owen et al. (2004)] which modeled the use of macrophages as vehicles for drug delivery to hypoxic tumors. This model is based on a growing avascular tumor spheroid, in which volume is filled by tumor cells, macrophages and extracellular material, and tumor cell proliferation and death is regulated by nutrient diffusion. It also includes terms representing the induced death of tumor cells by hypoxic engineered macrophages. Matzavinos et al. [Matzavinos et al. (2004)] used four nonlinear PDEs for tumor-infiltrating cytotoxic lymphocytes, tumor cells, complexes and chemokines to describe the interaction of an early, prevascular tumor with the immune system.

Avascular growth is relevant only when studying very small lesions, or tumor spheroids grown in vitro. To describe realistically tumorigenesis at later stages, one needs to look at the vascular stage and consider mecha n + V • (nv) = (fcm(c) - kd{c))n, m + V ■ (mv) = kd(c)n, p + V(pv) = 0.

c + V(cv) = _DV2c - 7km(c)n, nisms responsible for angiogenesis. An extension of avascular mechanistic models which includes angiogenesis can be found for example in the paper by Breward et al. [Breward et al. (2003)]. There, cells divide and migrate in response to stress and lack of nutrition. Here we present a different model developed by Anderson and Chaplain [Anderson and Chaplain (1997)]. It describes the dynamics of endothelial cell (EC) density, migrating toward a tumor and forming neovasculature in response to specific chemical signals, tumor angiogenic factors (TAF). If we denote by n(x, t) the EC density, then their migration can be described as n = DW2n - V(x(c)nVc) + g(n, /, c), where D and x(c) are the diffusion and the chemotactic parameter respectively, c(x, t) is a specific chemical (TAF) responsible for chemotaxis, and g(n, c) is the proliferation function. In the simplest case, the chemicals can be assumed to be in a steady-state (that is, i-independent), or they can satisfy a PDE:

c = Z)cV2n + v(c, n), with v(n, /) being a specific production/uptake function.

As with any system of nonlinear PDEs, one should be careful about well-posedness of the problem. The appropriate boundary conditions must be imposed, depending on the dimensionality and geometry of the problem. Then, either numerical solutions can be investigated, or a stability analysis of a simple solution performed. An example of a simple solution could be, for instance, a spherically symmetrical or planar tumor.

These could be interesting exercises in applied mathematics, but in our experience, description of most biologically relevant phenomena does not readily follow from a stability analysis and requires a specific, directed question. Mechanistic models are necessarily an idealization of reality, and the only way to judge how "bad" such a description is comes from formulating a specific biological question and figuring out what is necessary and what is secondary with respect to the phenomenon in question.

As an example of an interesting usage of the mechanistic tumor modeling by PDEs, where experience of physical sciences is used to study specific biological phenomena, we will mention the paper on the phenomenon of vascular collapse [Araujo and McElwain (2004)]. In this study, the distribution of stress is calculated throughout the tumor, as it changes in time as a result of cell division. The vascular collapse is modeled by identifying the region where stresses exceed a critical value. At this point, a collapse occurs and the inner regions of the tumor become cut out of the central blood system. The growth of tumor obtained in the model resembles experimentally observed patterns, with exponential growth before and retardation after the collapse. Another paper that we would like to mention addressed the question of the precise origin of neovascularization [Stoll et al. (2003)]. The traditional view of angiogenesis emphasizes proliferation and migration of local endothelial cells. However, circulating endothelial progenitor cells have recently been shown to contribute to tumor angiogenesis. The paper quantified the relative contributions of endothelial and endothelial progenitor cells to angiogenesis using a mathematical model, with implications for the rational design of antiangiogenic therapy.

At the next level of complexity, we have integro-differential equations. They can be used to describe nonlocal effects or inhomogeneity of the population of cells, such as age structure. For an example of integro-differential equations in tumor modeling, see [Dallon and Sherratt (1998)].

To summarize the method of partial differential equations, applied to mechanistic modeling of tumor growth, we note that it is significantly more powerful than the method of ODEs, as it allows us to make a dynamic description of spatial variations in the system. We have a large, well-established apparatus of mathematical physics, fluid mechanics and material science working for us, as long as we model biological tissue as a "material". We have the comforting convenience of laws as long as we are willing to stay within the realm of physics (complex biological systems cannot boast having any laws whatsoever). The problem is that we do not exactly know to what extent a tumor behaves as an incompressible fluid (or homogeneous porous medium, or any other physical idealization), and to what extent its behavior is governed by those mysterious biological mechanisms that we cannot fit into a neat theory. Any researcher using the framework of mechanistic modeling should be prepared to adapt the model to allow for more biology.

On a more down-to-earth note, there is one obvious limitation of PDEs which comes from the very nature of differential equations: they describe continuous functions. If the cellular structure of an organ is important, then we need to use a different method, and this is what we consider next.

2.3 Discrete, cellular automaton models

Cellular automaton models are based on a spatial grid, where the dynamics is defined by some local rules of interaction among neighboring nodes. The interaction rules can be deterministic or stochastic (that is, dictated by some random processes, with probabilities imposed). Each grid point may represent an individual cell, or a cluster of cells; for simplicity we will refer to them as "cells". To begin, we present a very simple model of tumor growth which illustrates the method. We start from a rectangular, two-dimensional grid. Let us refer to a grid point as x ij ^ where i and j are the horizontal and vertical coordinates of the point. Each node, iCg j j can be a healthy cell, a cancer cell, or a dead cell. We start from an initial distribution of tumor cells, healthy cells and dead cells. For each time-step, we update the grid values according to some local interaction rules. Let us denote the discrete time variable as n = 1,2, Here is an example of an update rule. At each time step, we update one site, in a fixed order;

+ 1) is given by the following:

• If Xij is surrounded by a layer of thickness 8 of tumor cells, it dies.

£C j j IS du tumor cell not surrounded by a ¿-layer of tumor cells, it reproduces. This means that the site x,l3 remains a tumor cell, and in addition, one of its neighbors (chosen at random) becomes a tumor cell. As a result, all the non-dead cells on that side of Xij between xVJ and the nearest dead cell, are shifted away from x, j by one position.

• If x^ is a healthy cell, it remains a healthy cell.

Of course, this is only a toy model. Cellular automata models used to describe realistic situations are more complicated as they have to grasp many aspects of tumor biology. For instance, Kansal et al. [Kansal et al. (2000)] used a very sophisticated three-dimensional cellular automaton model of brain tumor growth. It included both proliferative and non-proliferative cells, an isotropic lattice, and an adaptive grid lattice.

Cellular automata have been used to study a variety of questions. Alar-con et al. [Alarcon et al. (2003)] studied how inhomogeneous environments can affect tumor growth. They considered a network of normal healthy blood vessels and used an (engineering in spirit) approach to model the dynamics of blood flow through this fixed network. The outcome of this part of the model was the distribution of oxygen (red blood cells) throughout the network. Next, a cellular automaton model was run where, like in our toy model above, each element of the discrete spatial grid could take one of three values: "unoccupied", "has a normal cell", or "has a cancerous cell". The concentration of oxygen was fed into the local interaction rules.

In the paper by Gatenby and Gawlinski [Gatenby and Gawlinski (2003)], the acid-mediated tumor invasion hypothesis was studied. This hypothesis states that tumors are invasive because they perturb the environment in such a way that it is optimal for their proliferation, and toxic to the normal cells with which they compete for space and substrate. The authors considered a spatial tumor invasion model, using PDEs and cellular automata. The model was based on the competition of healthy and tumor cells, with elements of acid production by tumor cells, acid reabsorption, buffering and spatial diffusion of acid and cells. The authors propose that the associated glycolytic phenotype represents a successful adaptation to environmental selection parameters because it confers the ability for the tumor to invade.

A cellular automaton model of tumor angiogenesis was designed by [Anderson and Chaplain (1998)]. In their discrete model, the movement rules between states are based directly on a discretized form of the continuous model, which was considered in the previous section. The discretization is performed by using the Euler finite difference approximation to the PDEs. Then, numerical simulations allow for tracking the dynamics of individual endothelial cells, as they build blood vessels in response to TAFs. A qualitatively novel feature of this model is its ability to describe branching of new vessels by imposing some simple local rules. In particular, it is assumed that if the density of TAFs is above critical, there is enough space for branching, and the current sprout is sufficiently "old", then there is a finite probability for the vessel to branch and form a new sprout. This behavior cannot be grasped by the continuous, PDE-based models.

Finally, we would like to mention that apart from cellular automaton type models, there is an emerging area of agent-based modeling applied to tumor growth. In such models, each cell is modeled as an agent with some "strategy". In a paper by [Mansury and Deisboeck (2003)], an agent-based model was applied to investigate migration of tumor cells. It was assumed that tumor cells can proliferate or migrate, depending on nutrition and toxicity of their environment, using a local search algorithm. It turned out that the search precision did not have to be 100% to ensure the maximum invasion velocity. It had a saturation level, after which it did not pay to overexpress the genes involved in the search.

To conclude the section on discrete modeling, we would like to mention that the cellular automaton approach gives rise to a new class of behaviors which can hardly be seen in continuous, PDE-based models. It allows to track individual cells, and reproduce the dynamics of emerging structures such as tumor vasculature. A drawback of this approach is that it is almost universally numerical. It is difficult to perform any analysis of such models, which leaves the researcher without an ability to generalize the behavioral trends.

2.4 Stochastic modeling

Next we review one of the most important tools in biological modeling, which is stochastic processes. The need for stochastic modeling arises because many of the phenomena in biology have characteristics of random variables. That is, as a process develops, we cannot predict exactly the state of the system at any given moment of time, but there are certain trends which can be deduced, and, if repeated, an experiment will lead to a similar in some sense (but not identical) outcome. In this chapter we are not aiming at a comprehensive introduction to stochastic processes. Rather, we will give several examples where various stochastic methods are used to describe tumorigenesis.

The process where the stochastic nature of events can be seen very clearly is the accumulation of mutations. This process is central to cancer progression, and therefore developing tools to describe it is of vital importance for modeling. In the simplest case, we can envisage cell division as a binary (or branching) process, where at regular instances of time, each cell divides into two identical cells with probability 1 — u, and it results in creating one mutant and one wild type cell with probability u. To complete the description of this simplified model, we assume that a mutant cell can only give rise to two mutant daughter cells. Let us start from one wild type cell and denote the number of mutants at time n as zn. The random variable zn can take nonnegative integer values; another way to say this is that the state space is {0} U I. This is a simple branching process, which is a discrete state space, discrete time process. We could ask the question: what is the probability distribution of the variable zn? Possible modifications of this process can come from the existence of several consecutive mutations, a possibility of having one or both daughter cells mutate as a result of cell division, allowing for cell death, or from distinguishing among different kinds of mutations. As an example of a branching process type model, we will mention the recent paper by Frank [Frank (2003)] which addressed the accumulation of somatic mutation during the embryonic (developmental) stage, where cells divide in a binary fashion, similar to the branching process. Two recessive mutations to the retinoblastoma locus are required to initiate tumors. In this paper, a mathematical framework is developed for somatic mosaicism in which two recessive mutations cause cancer. The following question is asked: given the observed frequency of cells with two mutations, what is the conditional frequency distribution of cells carrying one mutation (thus rendering them susceptible to transformation by a second mutation)? Luria-Delbruck-type analysis is used to calculate a conditional distribution of single somatic mutations.

Next, we consider another important process, the birth and death process. Suppose that we have a population of cells, whose number changes from time t to time t + At, where At, is a short time interval, according to the following rules:

• With probability LAt a cell reproduces, creating an identical copy of itself,

• With probability DAt a cell dies.

All other events have a vanishingly small probability. The number of cells, X(t), can take positive integer values, and it depends on the continuous time variable. That is, it can change at any time, and not just at prescribed intervals. Therefore, this is a continuous time, discrete state space process. One obvious modification to the above rules is to include mutations. Say, instead of LAt, we could have the probability L( 1 — u)At to reproduce faithfully, and probability LuAt to create a mutant. Further, we could consider a chain of mutations, and describe the evolution of the number of cells of each type. This resembles Moolgavkar's description of multistage carcinogenesis [Moolgavkar and Knudson (1981)] which is reviewed in Chapter 3.

In the birth-death type processes, the population of cells may become extinct, or it could grow indefinitely. Another type of process that is very common in tumor modeling corresponds to constant population size. An example is the Moran process. Whenever a cell reproduces (with the probability weighted with the cell's fitness), another cell is chosen to die to keep a constant population size. If we include a possibility of mutations (or sequences of mutations), which lead to a change of fitness in cells, we can model an emergence and invasion of malignant cells. Models of this kind are relevant for the description of cellular compartments [Komarova et al. (2003)] or organs of adult organisms. In a series of stochastic models, Frank and Nowak [Frank and Nowak (2003)] discussed how the architecture of renewing epithelial tissues could affect the accumulation of mutations. They showed that a hierarchy of stem cells could reduce the accumulation of mutations by the mechanism that they term stochastic flushing. They assume that each compartment retains a pool of nearly quiescent proto-stem cells. The renewal of tissue happens in the usual way by stem cell divisions. If a stem cell dies, it is replaced from the pool of proto-stem cells. This process is characterized by the absence of long stem cell lineages, which protects tissue from accumulating mutations [Michor et al. (2003b)]. The interesting question of the role of compartment size on the accumulation of somatic mutations in cancer was addressed by [Michor et al. (2003a)]. They assumed that the total number of cells in the organ is fixed, and divided the population into compartments of variable size, N. Then they used a Moran process to calculate the optimum value, N, which minimizes the rate of accumulation of mutant cells.

Stochastic models of stem cell dynamics have been proposed by many authors. [Nowak et al. (2003)] employ a linear process of somatic evolution to mimic the dynamics of tissue renewal. There, cells in a constant population are thought to be put in a straight line. The first one is the symmetrically dividing stem cell, which places its offspring next to itself and moves the other cells by one position. The last cell is taken out of the system. This process has the property of canceling out selective differences among cells yet retaining the protective function of apoptosis. It is shown that this design can slow down the rate of somatic evolution and therefore delay the onset of cancer. A different constant population model is employed by [Calabrese et al. (2004); Kim et al. (2004)], where precancerous mutations in colon stem cell compartments (niches) are studied. Each niche contains multiple stem cells, and niche stem cells are lost at random with replacement. It is assumed that each stem cell can either divide asymmetrically, or give rise to two stem cells, or to two differentiated cells. This loss and replacement dynamics eventually leads to the loss of all stem cell lineages except one. The average time to cancer is calculated with this model, using five successive mutational steps. The results are compared with the existing age-incidence statistics. We will briefly mention statistical methods in the next subsection.

2.5 Statistics and parameter fitting

The idea is the following. A multi-stage model of carcinogenesis is formulated as a stochastic process, which includes a series of mutational events and clonal expansions. The mutation rates, the average rates of clonal expansions for each stage, and even the number of stages are variables of the model. Then, the probability of developing cancer by a certain age is calculated (usually, by means of numerical simulations), as a function of all the unknown parameters. The outcome of such calculations, for each set of parameters, is then compared with the existing data on cancer incidence, and the set of parameters which gives the best fit is identified. In their excellent paper, Luebeck and Moolgavkar [Luebeck and Moolgavkar (2002)] use the data on the incidence of colorectal cancers in the Surveillance, Epidemiology, and End Results (SEER) registry. They conclude that the statistics are most consistent with a model with two rare events followed by a high-frequency event in the conversion of a normal stem cell into an initiated cell that expands clonally, which is followed by one more rare event. The two rare events involved in the initiation are interpreted to represent the homozygous loss of the APC gene.

Many authors have analyzed age-incidence curves [Prank (2004); Hay-lock and Muirhead (2004); Krewski et al. (2003)] and death statistics [Filoche and Schwartz (2004)]. In the latter paper, the statistics of fluctuations in cancer deaths per year lead to an intriguing discovery: there is a big difference between cancers of young ages and cancers after 40. The authors suggest that cancers attacking older people behave like "critical systems" in physics and can be considered as an avalanche of "malfunctions" in the entire organism.

Another interesting way of looking at cancer statistics (of a different kind) is due to Mitelman and his colleagues. They study distributions of chromosome aberrations in various cancers and use statistical tools to analyze the emerging patterns [Hoglund et al. (2002a); Hoglund et al. (2002b); Hoglund et al. (2001); Mitelman (2000)]. In one paper [Frigyesi et al. (2003)], they found that the number of chromosomal imbalances per tumor follows a power law (with the exponent one). The main idea of a model explaining this behavior is as follows. The karyotype in unstable cancers evolves gradually, in such a way that the variability is proportional to the number of changes that already exist. The authors propose two possible interpretations of the model. One is that the rate at which changes accumulate, increases as cancer progresses (this is consistent with the notion of a "mutator phenotype" due to Loeb [Loeb (2001)]). The other is the evolving and increasingly permissive tumor environment. Similar theoretical and computational tools were applied to testicular germ cell tumor karyotypes [Frigyesi et al. (2004)]. It was shown that two distinct processes are operating in the karyotypic evolution of these tumors; whole-chromosome changes originate from a multipolar cell division of a tetraploid cell, whereas imbalances accumulate in a stepwise manner.

2.6 Concluding remarks

In the rest of the book, we will employ many of the methods mentioned here, and explain models and their solutions in detail. In particular, Chapter 3 talks about birth-death processes; Chapter 5 uses a branching process; the Moran process is employed in Chapters 3 and 4; Chapter 6 combines a stochastic Moran process with a deterministic ODE; Chapter 7 studies quasispecies ODEs; Chapter 8 uses PDEs and pattern formation-type analysis; and Chapters 9-12 talk about analytical and numerical solutions of ODEs.

Chapter 3

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