# Comparison of Predator-Prey Model and Hawk-Dove Game for Modelling Leukemia

Authors: Mariam Sultana; Fareeha Sami Khan; M. Khalid; Areej A. Al-moneef; Ali Hasan Ali; Omar Bazighifan
Journal: Computational Intelligence and Neuroscience
 (2022)

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## Abstract

Game theory is an excellent mathematical tool to describe the interaction between the immune system and cancerous leukocytesc.leu. The feature of cancerous leukocytes to differentiate and mutate to give rise to leukemia is in the domain of ecological models as well. In this work, the dynamic of leukemia is described and compared by two models: firstly by a simple probabilistic mathematical model using the zero-sum two player game of Hawk and Dove, and secondly by Leslie Predator Prey model of ecology. The main goal of this study is to compare the results of both models and then discuss the treatment of leukemia i.e., Hematopoietic Stem cell transplant with the best model among them. Hawk and Dove model also describes the cell to cell interaction of cancerous leukocytes and healthy leukocytes leu after diagnoses and the condition of the patient before and after treatments. In this work, Hematopoietic Stem cell transplant is discussed by using concepts of a zero-sum three player game. Also, both models will be characterized by determining the stability properties, identifying basins of attraction, and locating the equilibrium points to see, at what extent the patient’s survival is possible with leukemia in its body. Results for both models will be presented graphically.


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## Body

## 1. Introduction

In blood there are several types of diseases that can occur, among them is “hematological malignancy.” This term refers to the type of malignancy that can affect lymph nodes, bone marrow, and blood as well. Among hematological malignancy, leukemia was discovered in 1847 by Rudolf Virchow. In Leukemia, the production of a certain kind of white blood cells known as “the blasts” increases, and the normal function of blood and bone marrow gets disturbed. Leukemia is mainly categorized into four types: Acute lymphocytic leukemia (ALL), Chronic lymphocytic leukemia (CLL), Acute myelogenous leukemia (AML), and Chronic myelogenous leukemia (CML). Leukemia, up to some extent, is regarded as the treatable disease in the medical world. For this cause, medical and all other branches of science and technology have been continuously trying to find the cure and efficient drugs to tackle this disease in order to relieve the curb and misery of mankind. To examine the root causes and their therapies, there are plenty of approaches. With the available scientifically obtained data from computer aided models, the mathematical and statistical approach has paved the way for new useful medical investigations. Research on Leukemia is underway. Furth & Makhn [1] showed in history, the transmission of leukemia in mice, for the very first time. Later on, many mathematical models and treatments were suggested based on intercellular interactions by [2–4]. A further detailed model demonstrated the interactions of myeloblasts with neutrophils, T-cells and leukemia and also provided the treatment model for acute myeloblastic leukemia chemotherapy, see [5]. The model was proposed by Helen & Natasha [6] based on the usual differential equations explaining the interaction of effector T-cell, a naive T-cell in chronic myelogenous leukemia. Huge changes in the growth and death rates of chronic myelogenous leukemia cells can be observed in the results.Recently, some authors investigated that the patients prognosis of acute myeloid leukemia is due to cytokine-independent leukemic cell proliferation see [7]. Miguel [8] & Rodriguesa et.al [9], also contributed their results for cancer metabolism and presented a detailed mathematical model for chemoimmunotherapy for lymphocytic leukemia describing its normal conditions and stability analysis of immunotherapy. Leukemia and tumor based on several other mathematical models were studied by a few more researchers such as [10–13]. On the other hand, many other models related to differential equations were studied recently by several authors, see [14–26].It is important to mention here, that all discussed models were furnished either with differential equations or numerical methods but the use of Game theory for a such complex disease is a recent advancement. Game theory is very effective in describing the interaction between the immune system and cancer leukocytes. Whereas the interrelation of stem cell after the transplant with leukocytes against cancerous leukocytes or with cancerous leukocytes against leukocytes is in the domain of evolutionary game theory.In this century, a new arising treatment of leukemia i.e., stem-cell transplantation has gained the interest of researchers of different fields. By the hematopoietic stem-cell idea, Pappenheim [27] initiated stem-cell biology. Since they are capable of differentiating into the main three major life cells, but also replicating into another cell of their own kind, until the end of life, the process of differentiation of the same DNA continues. This treatment was first performed by Main & Prehn [28] in the mice. Stem-cells have a history that is very fascinating. The researchers observed after the discovery of stem-cells that leukemia tumours have the ability to develop in vitro and vivo colonies, see [29]. This finding and other postulates led to the result that, just as other normal cells differentiate, stem-cells have the capacity to indefinitely differentiate themselves. Cancer stem-cells were first investigated by Lapidot et.al [30] in acute myeloid leukemia and Cobaleda et.al [31] for acute lymphocytic leukemia. These studies captured the interest of mathematicians so many mathematical models have been developed for the treatment of different diseases. Boni et.al [32] suggested a model using evolutionary game theory for the interaction of stem-cells in bioactive scaffolds (i.e., cardiac muscles that lose elasticity). Mneimneh et. al [33] described a game-theoretic and stochastic survivability mechanism against induced attacks in cognitive radio networks. Torkaman et. al [34, 35] developed a system for classifying leukemia based on a cooperative game based on the population of Iran.With all the available treatments yet, a leukemia patient can not be treated completely but his survival is possible with long term treatments such as chemotherapy, immunotherapy, radiation, and stem cell transplantation. But in all these treatments the major drawback is that these medicines while targeting thec.leudestroy the immune system of the patient too. It is yet a challenge to find a such cure that can only target the cancerous cells and improve the life quality of such patients by lessening their pain and suffering. Therefore, this paper is very important in understanding the cell-cell interaction for medical science to gain capabilities of improving the diagnostic techniques and improve survival rate and quality of medicines.This paper combines the evolutionary game theory and ecology, predator-prey model with life threatening disease i.e., leukemia. The main goal of this study is to apply replicator dynamics and the Leslie predator-prey model to the dynamics of leukemia see Figure1. Then, treatment of leukemia i.e., stem cell transplant is discussed with the concept of evolutionary game theory because it is easier to formulate the cell to cell interaction by this methodology instead of the Leslie Predator Prey model see Figure 2.Figure 1
Flow chart of the methodology for mathematical modeling of leukemia.Figure 2
Flow chart of methodology for mathematical modeling of treatment of leukemia.In this paper, significant contributions have been made in both directions: (1) in terms of the formal analysis of this disease, this work has focused on the proposed extension model of an existing formalism. The viability and effectiveness of this proposal are corroborated by the application of both formalism to the analysis. (2) With regard to the design of leukemia dynamics, this approach is based on applying population dynamics, and to generate a mathematical model which can be proven rationally within an underlying game theoretical framework.In this paper, Section2 is about showing leukemia in a fast time scale using the replicator dynamics model. Section 3 describes the leukemia Dynamics by the Predator Prey model. Section 4 is about a patient’s condition after stem-cell transplant therapy in view of replicator dynamics. Section 5 discusses the results of replicator dynamics for Hematopoietic Stem-Cell Transplantation. Also, discusses the numerical values of parameters (i.e cell count of each cell type) among calculations to show the graphical illustrations. Section 6 is the discussion of the results. In Section 7, Conclusion and Future Work is given.


## 2. Leukemia Dynamics by Hawk and Dove Game

The principles of game theory provide a common framework for creating structures and, eventually, comprehending several significant cancer problems. A game can be defined as a set of comparisons and the opportunity to play the game in line with the game’s rules. The classical game theory is founded on a set of strictly defined axioms about game structures. It is assumed in game theory that each player or agent has predefined goals, choices, and interests, which are described by a function called the “utility function.” Again, utility refers to the most gains that players may receive as an outcome of the game, and each player’s goal, theoretically, is to yield maximum profit or value, theoretically.(i)
Their ethical and viable plan of action in view of the rules.(ii)
The outcome, that players are getting, represent every move they could possibly make given the vast array of strategies at their disposal.Since acute or chronic leukemia are just different time scale cases theoretically, a case of acute leukemia is taken in consideration in this research (see [36]). Leukocytes, or white blood cells, are a part of the immune system that protect the body from diseases or external invaders (see Figure 3).Figure 3
Stem-cell proliferation showing the ability of self renewal and replication into mature cells. This is an infinite process and stem-cells are the only ones who can replicate themselves many times unlike other cells. Unusual proliferation or accelerated cell division can cause leukemia or tumours.While in leukemia, some of these healthy leukocytes get defected and works against the host body and do not let the healthy cells work either. That is the point when tumours are formed called neoplasm. Two types of replicating cells are investigated in this work i.e., cancerous leukocytes (c.leu) and normal leukocytes cells (leu). There will be lower but equal probabilities forleu to further multiply into large numbers, resulting in the ability to work as leu. In the process, the capacity to sustain neoplastic growth is acquired mainly by those cells which enjoy the capacity for self-renewal. This model concludes that only a limited and restricted number of cells inside the tumour are expected to relaunch tumour development. As per the stochastic theory, the tumour cells are relatively homogeneous, all of these tumour cells undergo active genetic modifications and progress towards malignancy growth. The replicator model suggests that the biologically and functionally explicit population of tumour-launching cells is limited. Therefore the focus of research is on those leu breeds that support the progression of targeted treatment for leukemia and this will strongly prevent disease recurrence. Simple and incomplex hypotheses have been made, for the initial discussion of the replicator dynamics model in terms of leukemia. But for more detailed analysis, more and more complex variables that can simulate the reality will be used in the future model.A set of pay off matrix in the evolutionary game theory, i.e., sums, costs, and benefits, of a specific strategy are calculated in terms of success and failure of leu and c.leu. To begin with the test, let the specific c.leu,leu phenotypes be set as external stimuli, in the general context of the payoff matrix.(1)A=a11a12a21a22. Here, aij is the player’s payoff using the i plan as opposed to the j plan player, since the matrix occupancy acquired in the form as seen in Table 1 is applied in this case, assuming that the interaction between the same type of cells stays neutral, which is why only diagonals are zeros. On the other hand, the leu-c.leu interplay capitulates into a negative payoff.Table 1
General form of the payoff matrix for cell to cell interaction.
c.lc.leuIleuc.lc.leu0a11−a21=−aIleua22−a12=b0In particular, for evolutionary game theory, this scenario exactly resembles the Hawk and Dove game, whereleu denotes the Doves and c.leu denotes the Hawks. Let the total number of cells be denoted by p in time t and this further can be classified into two types c.leu are PLt and leu are PIt. The cell masses selecting Si strategy i.e., Si=s1,s2,…,sk where i=1,2 at t. Let x1′t and x2′t denote the cells proportion of Hawk and Dove strategies, respectively.(2)x1′t=PLtpt,x2′t=PItpt=1−x1′t.The normalized symmetric matrix in Table1 gives the general replicator dynamics equation as follows:(3)x1′t=ax1−bx2x1x2,x2′t=bx2−ax1x1x2.Fitness is referred as the payoff in replicator dynamics.Hi is the fitness of the i phenotype, represented in terms of the A fitness and payoff matrix phenotypes, where the cell population’s fitness function is Hi=Axi and H¯ is the average fitness of the population of cells i.e., H¯=∑ixiHi=xTAx. Taking x1+x2=1 and with some algebra, the growth rate of c.leu and leu by selecting a strategy is Si:x1−xH1−H2 at any time t, here x is the frequency of cells. There are the three stationary points i.e., 0,1, and dfracH1H2=x∗. For H1<H2 it can be stated that x∗ is dimorphic. Also, for 1−H1/H2leu and H1/H2c.leu, population cells is the polymorphic and stationary point is asymptotically stable. If H1>H2, which means x∗=1 then the stationary point is asymptotically stable, which means there are only c.leu.In Figure4 the unique symmetric mixed Nash equilibrium is asymptotically stable i.e., H1/H2. Also, both pure Nash equilibrium of leu and c.leu are asymptotically stable. The risk-dominant one has the larger basin of attraction which is true in general because x∗>1/2 for games with an efficient equilibrium and a risk dominant one.Figure 4
Forn=2, the replicator dynamic phase plane showing the three equilibrium points as 0, 1 and a dimorphic equilibrium x∗.In this paper, the dynamical behaviour of the cells in mathematical models, is illustrated with the help of Mathematical software. These portraits are within the standard ranges ofleu and c.leu blood samples of patients recently diagnosed with leukemia, see [37]. In leukemia, leu counts differ rarely or sometimes at regular intervals ranging from 30 to 200×109 cells/liter. This variance was recorded during the 40 to 80 days time frame, which is a significantly long time relative to the c.leu life span and maturation acquisition.In these portraits, the white zigzag trajectory showsc.leu and pink dotted line shows leu. In Figure 5c.leu is greater in number than leu and if not treated soon c.leu will make leu extinct and the patient will die with the passage of time. Also, the trajectory shows an unstable node as the direction is moving away from the point. c.leu is much larger in amount than leu in Figure 6 and with the passing time, c.leu makes leu extinct and the patient dies. As the path shifts away from the point, the trajectory also shows unstable nodes. So this is the point where the cell population is monomorphic, which is that there will be just c.leu. It is studied by using fast-scale techniques that if not treated, how quickly a patient will die of acute leukemia.Figure 5
Diagnosis of leukemia is shown through the replicator dynamic phase plane.Figure 6
leu and c.leu phase plane of interaction. At the time of the spread of the disease when the patient is near death.


## 3. Leukemia Dynamics by Leslie Predator Prey Model

The Leslie Predator-Prey Model has a significant impact on the field of mathematical biology and ecology. If the cancerous leukocytes (predator) density islt and the leukocytes (prey) density is it, then Leslie’s model can be represented by systems of nonlinear differential equations as follows:(4)didt=r1−c1l−bii,(5)dldt=r2−c2lil.In equation (5), the factor c2l/i is defined as Gower Term. The reason for adding the Gower word comes from the leukocyte growth rate factor, as countless i⟶∞ rise relative to the per capita growth of the cancerous leukocytes reaches their maximum growth i.e., 1/ldl/dt⟶r2. On the other hand, when leukocyte population declines i.e., i⟶0, then 1/ldl/dt⟶−∞ (i.e., cancerous leukocytes must vanish). Both cells have logistic growth. r1 is the growth rate of leu with carrying capacity r1/b1 and r2 is the growth rate of c.leu with carrying capacity c2/i in proportion to the population size of leukocytes. c1 and c2 provides the estimate of the quantity of transformation of c.leu and leu into each other respectively. The constant b signifies the effectiveness of c.leu in eliminating leu. In Leslie’s model it was suggested that the carrying capacity of a predator should be equal to prey. Also, it should be emphasized that the upper limits associated with predator prey together are met under the following conditions:(i)
For Predators: when there are more predators than prey in the population ratio(ii)
For Prey: when the ratio of Prey to Prey population is highTo estimate the level of success of the design function for achieving the set targets, take into consideration another type of function that acts as a fitness function and intends to represent the net outcome as a single merit value as follows:(6)H1=r1k1k1−i−bl,(7)H2=r21−lcbi,where H1 and H2 are the fitness functions of leu and c.leu, respectively. For a detailed derivation of equation (6) please see [36]. In contrast with c.leu, equation (7) defines the increased death rate of leu. c is the nutritional value constant of leu, which is proportional to c.leu. The densities of leu and c.leu at t time are i and l. Also, r1 & r2 are the intrinsic growth rate of leu and c.leu, k is the carrying capacity. The assumptions of this model are that a predator is intruding into each other’s behavior. This intervention has a negative effect on the predators themselves and increases the mechanism to bring the dramatic shifts in equilibrium, stability, and strength in the population of c.leu and leu. Consider the difference equation of equations (6) & (7) to be interpreted to understand this phenomenon more closely,(8)it+1=i1+r1k1k1−i−bl,lt+1=l1+r21−lcbi.Solving equation (5) the following equilibrium point is obtained E=r1c2/bc2+c1r2, r1r2/bc2+c1r2. The Jacobian matrix of equation (5) at this equilibrium point is given by the following equation:(9)Ji,l=−r1bc2bc2+c1r2−r1c1c2bc2+c1r2r22c2−r2.Therefore the characteristic equation is(10)λ2+r1bc2+bc2+c1r2r2λbc2+c1r2+r1br2c2+r1c1r22bc2+c1r2=0.Obviously,λ1λ2=r1br2c2+r1c1r22/bc2+c1r2>0 and λ1+λ2=−−r1bc2+bc2+c1r2/bc2+c1r2<0. Hence equilibrium E=r1c2/bc2+c1r2,r1r2/bc2+c1r2 is an asymptotically stable node if r1−r22≥r1c1r2/bc2+c1r2 or focus if r1−r22<r1c1r2/bc2+c1r2. Also, the positive equilibrium x∗,y∗ of system equation (5) is globally stable, where x∗=r1c2/bc2+c1r2 & y∗=r1r2/bc2+c1r2.Parameters, i.e., modest intraspecific competition, are shown on the left side of Figure7. Leukemia was the patient’s original diagnosis. As can be seen in the graph on the left, leu and c.leu coexist in the body. The stable node is thus present during this time interval.Figure 7
A phase plane of a two-dimensional Leslie predator-prey model that demonstrates the cohabitation ofleu & c.leu.An interspecific competition with high saddles. The parameters are displayed on the left. As an example, acute leukemialeuhas a typical range of 0to6; however, when acute leukemia flares up, the graph shows that c.leuclimbs to 6 or higher and leubecomes scarce. This node is unstable.Figure8 illustrates how chronic leukemia, or leu and c.leu, can coexist in a patient’s body for a very long time. The parameters are displayed on the left. While there is an equilibrium point between x=0 to 2 and y=0 to 4, it is a globally unstable node because the trajectory is moving away gradually. As mentioned above, there are three symptoms that a person with leukemia could encounter. They are all globally stable nodes since all of the estimated critical points are positive.Figure 8
Phase plane for a two-dimensional Leslie predator-prey model.Typically, a blood sample is taken when a patient has no symptoms yet the disease is present in his blood. Similar to the previous portraits, they are unstable nodes since both healthy and cancerous leukocytes are present in the blood, and in this model, the equilibrium stable point or node occurs when one cell population prevails or survives with a high rate. By utilizing a few algebraic procedures and the mathematical software Mathematica, two different scale models of various areas have been developed and studied. Although different results were found, the scenario remained the same. So let’s talk about leukemia treatment in the section afterwards.


## 4. Hematopoietic Stem-Cell Transplantation (HSC)

In this section, a speculative patient who has already been given high chemotherapy measurements and has been transplanted withHSC. Since this is a mathematical study and a hypothetical patient is considered here (assumed parameter values are closer to the real patients data described in Section 5.1) so it is important to clarify here that the sole purpose of this study is to show the applicability of a powerful model such as “replicator dynamics” on a complex disease such as “leukemia.” Hence the complications occurring due to chemotherapy and its medicine have not been included in this study. Only the after effects of chemotherapy on blood count is considered in this work.To understand the interaction ofHSC, leu, and c.leu after the transplant, game theory have been used. Here are three players, namely, HSC, c.leu, and leu, with two execution strategies: kill or not kill. In such a case, there are two cell reaction possibilities when HSC enters the body of the patient. One is that, with them, leu will react and produce graft versus host disease that is not favourable for the patient and therefore can be said to be grouped with c.leu. And secondly, to boost the patient’s immune system, HSC creates a group of leu to help destroy the remaining c.leu to support him in recover.There is one and only one coalition that could conceivably occur in two-individual games, particularly the coalition between the two players. The amount of possibilities is, obviously greater in games with more than two players, calledn−individual games. As with this case, the following segments of the players in coalitions could occur in a game with three players:(a)
For any player on his own(b)
A coalition between Player 3 and Players 1 & 2(c)
Players 1 & 3 versus 2(d)
Players 2 & 3 versus 1(e)
the grand alliance, all three players included, see [36]So on the basis of theHSC decision, this case has two clear coalitions to decide if it applies to leu to fight against the remaining c.leu or to collate with c.leu against leu i.e.,(i)
FavourableHSC,leu unfavourable c.leu(ii)
Favourableleu unfavourable c.leu,HSCIt is presumed that interactions between cells of the same types are neutral, therefore, there are zeros on the diagonal, see Tables2 and 3). In the presence of the right stimuli that allow the cell to differentiate, while interacting with an unfavourable team, the favourable team receives a positive payoff, while the unfavourable team receives a negative payoff. This implies not only that it is believed that the favourable team gets a greater payoff under the favourable environment, but also that an unfavourable team will get negative feedback under this environment, see Tables 2 and 3. In Tables 2 and 3 normalized matrices, F1 stands for a favourable team and F2 for an unfavourable team.Table 2
General form of the payoff matrix for cell/cell interaction inHSC transplant (coalition i).
F1F2F10a−c=−a1F2d−b=a20Table 3
General form of the payoff matrix for cell/cell interaction inHSC transplant (coalition ii).
F1F2F10e−g=−b1F2h−f=b20


## 5. Replicator Dynamics and Nash Equilibrium ofHSC Transplantation

If there aren distinct behaviour patterns called “Pure Strategies,” E1,E2,…,En where n=1,2. The state of a population of cells concerning this conflict is fully described by a vector xi=x1,x2,…,xn∈Rn, where xi represents the frequency of leu and c.leu with behaviour Ei,i=1,2,…,n. Consequently, xi≥0 for all i and ∑ixi=1 the set of all such x′s is denoted by Sn. If an Ei individual cell contends with an Ej individual cell then the payoff will be aij. Let A be the matrix of all those aij’s. Then, assuming random encounters of cells as,(11)ei.Ax=∑iaijxj,is the average payoff for an Ei-cell in a cell population x and(12)x.Ax=∑ixiei.Ax=∑i∑jaijxixj,is the mean average payoff within cell population x. Taylor and Jonker [38] assumed that the growth rates x1./xi=dxi/dt/xi of the frequency of strategy Ei are equal to the payoff difference ei.Ax−x.Ax, This yields the replicator equation(13)x˙1=xiei.Ax−x.Ax;i=1,2,…,n,on the invariant set Sn. If the game begins with xit°=0, so for all t, the strategy is xit=0. And if there are no favourable cells or unfavourable cells that play strategy from the beginning, then there are no cells that can replicate that approach. The general replicator dynamics equation for normalized symmetric matrix from Tables 2 and 3 is(14)x1′=a1x1−a2x2x1x2,x2′=a2x2−a1x1x1x2.After some algebra, these equations become(15)x1′=x11−x1a1−a1+a2x2.Also,(16)x1′=b1x1−b2x2x1x2,x2′=b2x2−b1x1x1x2.After some algebra these equations become(17)x2′=x21−x2b1−b1+b2x1. Which gives five solutions for equations (15) and (17) i.e., the pure strategy unstable solutions are 0,0, 1,1, 1,0, 0,1 or a mixed strategy solution a1/a1+a2,b1/b1+b2. Clearly, system has positive equilibrium hence the Jacobian matrix at equilibrium is given by the following equation:(18)J∗=1−2x1∗×a1−a1+a2x2∗−x1∗1−x1∗a1+a2−x2∗1−x2∗b1+b21−2x2∗−b1−b1+b2x1∗.For easier calculation leta1=a,b1=b,a1+a2=m,b1+b2=n then a mixed strategy equilibrium occurs when 0<a/m,b/n<1 where x1∗=b/n,x2∗=a/m. From this, it can be deduced that this system has a positive equilibrium. Therefore, its Jacobian matrix at this equilibrium point becomes(19)J∗=0−bmn1−bn−anm1−am0.Now, further this matrix describes two cases:(i)
ifa1 & b1 have different sign, then after solving equations (15) & (17), it is observed that the function shows closed spirals and is monotonic if the following conditions are satisfied:(a)
a1≤a1+a2 & b1>b1+b2(b)
a1>a1+a2(ii)
ifa1 and b1 have same sign than it is a saddle point which means an unstable point. Also, its equilibrium is hyperbolic i.e., x1∗,x2∗, so from this, it can be observed that one monomorphic fixed point is asymptotically stable whereas the fixed points under replicator dynamics is a saddle.


### 5.1. Complete Description of Leukocytes (NK Cells, Peripheral CD34+cells):

The number ofNk cells after chemotherapy is given in parameters see [37] where 2.08×10−7Nk cells and 5.00×108leu are remaining cells in the body. The mean value of the peripheral CD34+ cells infused was 9.19×106cells/kg recipient’s body weight and the mean CD34+ cell number was 3.47×104cells/kg. Whereas Pillis et.al [37] shows remaining c.leu i.e., 4.31×10−1 and 1.42×10−6cells/kg.Figure9 shows how badly this coalition can affect the patient’s life as at the time of chemotherapy the number of c.leu was minor but after transplantation if graft versus host disease occurs, it can kill the patient. The rise can be observed in an unfavourable coalition of cells. The payoff matrix for this simulation becomes(20)J∗=05×1089.2247×1060.Figure 9
Phase portrait of replicator dynamics after stem-cell transplantation forleu and (HSC, c.leu).Interaction of favourable and unfavourable coalition ofleu and c.leu with HSC is described in Figure 10. As it can be seen after a small oscillation the unfavourable group has stopped rising, whereas leu is approaching infinity. Payoff matrix for this simulation becomes(21)J∗=010092247000.4310010.Figure 10
Phase portrait of replicator dynamics after stem-cell transplantation for favourableHSC,leu and unfavourable c.leu.Equations (20). And (21). Gives real and distinct eigen values as they have opposite signs. Equation (20). Have detJ∗=−4.61235×1015<0, τ2−4detJ∗>0 shows a saddle point. Equation (21). Also, have distinct and real eigen values and detJ∗=−434976854.9<0, τ2−4detJ∗=83424.39498>0 which means its also a saddle point.


## 5.1. Complete Description of Leukocytes (NK Cells, Peripheral CD34+cells):

The number ofNk cells after chemotherapy is given in parameters see [37] where 2.08×10−7Nk cells and 5.00×108leu are remaining cells in the body. The mean value of the peripheral CD34+ cells infused was 9.19×106cells/kg recipient’s body weight and the mean CD34+ cell number was 3.47×104cells/kg. Whereas Pillis et.al [37] shows remaining c.leu i.e., 4.31×10−1 and 1.42×10−6cells/kg.Figure9 shows how badly this coalition can affect the patient’s life as at the time of chemotherapy the number of c.leu was minor but after transplantation if graft versus host disease occurs, it can kill the patient. The rise can be observed in an unfavourable coalition of cells. The payoff matrix for this simulation becomes(20)J∗=05×1089.2247×1060.Figure 9
Phase portrait of replicator dynamics after stem-cell transplantation forleu and (HSC, c.leu).Interaction of favourable and unfavourable coalition ofleu and c.leu with HSC is described in Figure 10. As it can be seen after a small oscillation the unfavourable group has stopped rising, whereas leu is approaching infinity. Payoff matrix for this simulation becomes(21)J∗=010092247000.4310010.Figure 10
Phase portrait of replicator dynamics after stem-cell transplantation for favourableHSC,leu and unfavourable c.leu.Equations (20). And (21). Gives real and distinct eigen values as they have opposite signs. Equation (20). Have detJ∗=−4.61235×1015<0, τ2−4detJ∗>0 shows a saddle point. Equation (21). Also, have distinct and real eigen values and detJ∗=−434976854.9<0, τ2−4detJ∗=83424.39498>0 which means its also a saddle point.


## 6. Discussion of Results

According to the calculations, it showed that replicator dynamics has three stationary points,0,1 and H1/H2=x∗. Also, it showed that x∗ is dimorphic for H1≤H2 which means that in blood, both cell populations (i.e., leuand c.leu) exist.For asymptotically stable stationary point, it showed that the cell population is polymorphic i.e.,1−H1/H2c.leuand H1/H2 leu.For monomorphic cell population i.e., onlyc.leuexists, where x∗=1, for H1≥H2 which implies that x∗ is asymptotically stable.By considering the data of a patient who has been diagnosed with leukemia, the two-dimensional Leslie predator prey model of equation (10). Shows in Figure 7 a stable node, that is, r1−r22=4.5548×10−3 and r1c1r2/bc2+c1r2=8.519876×10−6.The coexistence equilibrium point with parameters given in Table4 is obtained as x∗=4.99 and y∗=5.99. At this point the patient can survive for some time period.Table 4
Description of parameters for leukemia patient for different stages of disease.
DescriptionEarly prognosis valueAcute leukemiaChronic leukemiar10.50.70.6c11.42×10−60.70.6b0.10.250.6r20.60.750.6c20.50.40.6Also, from Figure11 if r1−r22≥r1c1r2/bc2+c1r2 with parameters in Table 4 it gives r1−r22=8.623253×10−4, r1c1r2/bc2+c1r2=0.588 then both the eigenvalues are positive i.e., λ1+λ2=0.638,λ1λ2=0.525 so it is unstable knot. This situation describes an acute leukemia i.e., variation of parameters shows a drastic change in patients condition.Figure 11
Leslie’s predator-prey phase plane depicts the onset of the disease or the decrease in leukocytes following an increase in cancerous leukocytes.For a chronic leukemia patient, the parameters are given in Table4 and the critical point is x∗=0.5 and y∗=0.5. As it can be observed in Figure 8, in this case, the patient condition does not change drastically but still, it is an unstable node due to the positive eigen values i.e., λ1+λ2=0.3,λ1λ2=0.36In the last section, a three player game interaction is described after theHSC transplantation between human normal leu, c.leu, and HSC. It is shown by taking two coalitions i.e., favourable and unfavourable. Different results have been obtained to show the accuracy and applicability of replicator dynamics to this disease. The phase portrait Figure 10 for the first coalition between favourable and unfavourable cells depicts the rise in leu. Which means that chemotherapy is being successful in eliminating the c.leu, the transplant have been successful and patient will surely survive.The portrait in Figure 9 indicates the increase in unfavourable cell coalitions. The second coalition indicates how badly it can affect the life of the patient because the amount of cancerous leukocytes are limited at the time of chemotherapy, but if graft versus host disease happens after transplantation, it can destroy all the remaining leukocytes that cause the patient to die.


## 7. Conclusion & Future Work

This paper concludes that even a complex disease such as leukemia can be described using population dynamics and evolutionary game theory. These techniques have been implemented and their results provided. These models successfully deduce steady states and their stability. These models individually permit the existence of two types of stationary states:(1)
State of no disease, with no cancerous leukocytes(2)
State of coexistence where a cancerous leukocyte persists against the background of the immune responseThe state of no disease is asymptotically stable and a state of leukemia is unstable. Both models are in complete agreement about stable and unstable situations. It was found from the study, that the state of no disease represents the immune state.In this research, a two person zero-sum game of evolutionary game theory is used to describe the patient’s condition having leukemia. Different results have been obtained for this situation by incorporating a mathematical software Mathematica. Then leukemia dynamics is studied by an ecological model Lesli predator prey model. Different results have also been obtained for this situation as well by incorporating a mathematical software Mathematica. The treatment of leukemia i.e., HSC is then studied by using a three player zero sum game as it was observed that evolutionary game theory gives a better understanding of leukemia than the ecological model. This study is a sincere effort to show another perspective of game theory that it can also define the complex dynamics of leukemia. By adding more complex factors, these calculations can give more realistic and accurate results.The game is based on the number of leukocytes in this paper, but in future work, we may investigate evolutionary game theory for leukemia into specific cell types and their count to better determine which cell types should specifically be targeted and can result in eradicating the disease. Because current treatment advancements remove both healthy and malignant cells from the body, patients suffer greatly and are more susceptible to getting infected by a virus.



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