A Class of Solutions for the Hybrid Kinetic Model in the Tumor-Immune System Competition

In this paper, the hybrid kinetic models of tumor-immune system competition are studied under the assumption of pure competition.The solution of the coupled hybrid system depends on the symmetry of the state transition density which characterizes the probability of successful occurrences. Thus by defining a proper transition density function, the solutions of the hybrid system are explicitly computed and applied to a classical (realistic) model of competing populations.


Introduction
In this paper, the two-scale tumor immune-system competition hybrid model [1][2][3][4][5][6] is studied under the assumption that the transition density function is a symmetric and separable function.The competition between tumor and immune-system can be modeled at different scales.Cells of different populations are characterized by biological functions heterogeneously distributed, and they are represented by some probability distributions.The interacting system is characterized at a macroscopic scale by a density distribution function which describes the cells activity during the interaction proliferation.At this level, the distribution of cells fulfills some partial differential equations taken from the classical kinetic theory.In this case, the more general model consists in a nonlinear system of partial differential equations.From the solution of this system, one can define a parameter which defines the time evolving distance between the two distributions, and this parameter is the charactering coefficient of the microscopic equations, typically an ordinary differential system for the competition of two populations.
This parameter has been considered [4,5] as a random coefficient whose probability density distribution is modeled by the hiding-learning dynamics referred to biological events where tumor cells attempt to escape from immune cells which, conversely, attempt to learn about their presence.Therefore, when the coupling parameter is obtained by solving the kinetic equations for the distribution functions, then it will be included in the classical Lotka-Volterra competition equations.We will analyze on a concrete example the influence of this stochastic parameter on the evolution.This method can be easily extended to more realistic competition models (see, e.g., [7][8][9][10][11][12][13][14][15][16][17][18][19][20]).

The Hybrid Model for the Tumor-Immune System Competition
Let us consider a physical system of two interacting populations, each one constituted by a large number of active particles with sizes: for  = 1, 2 and R + def = [0, +∞).Particles are homogeneously distributed in space, while each population is characterized by a microscopic state, called activity, denoted by the variable .The physical meaning of the microscopic state may differ for each population.We assume that the competition model depends on the activity through a function of the overall distribution:  =  [  (, )] , ( [  (, )] : R + → R + ) . ( The description of the overall distribution over the microscopic state within each population is given by the probability density function: for  = 1, 2, such that   (, ) denotes the probability that the activity  of particles of the th population, at the time , is in the interval [,  + ]: Moreover, it is We consider, in this section, the competition between two cell populations.The first one with uncontrolled proliferating ability and with hiding ability; the second one with higher destructive ability, but with the need of learning about the presence of the first population.The analysis developed in what follows refers to a specific case where the second population attempts to learn about the first population which escapes by modifying its appearance.The hybrid evolution equations specifically can be formally written as follows [4,5]: where   , for  = 1, 2, is a function of  = { 1 ,  2 } and  acts over  = { 1 ,  2 }, while A  , for  = 1, 2, is a nonlinear operator acting on  and [] is a functional (0 ≤  ≤ 1) which describes the ability of the second population to identify the first one.Then, (6) denotes a hybrid system of a deterministic system coupled with a microscopic system statistically described by a kinetic theory approach.In the following the evolution of density distribution will be taken within the kinetic theory.The derivation of (6) 2 can be obtained starting from a detailed analysis of microscopic interactions.Consider binary interactions specifically between a test, or candidate, particle with state  * belonging to the ith population and field particle with state  * belonging to the jth population.The modelling of microscopic interactions is supposed to lead to the following quantities.
(i) The encounter rate, which depends for each pair of interacting populations on a suitable average of the relative velocity   , with ,  = 1, 2. (ii) The state transition follows from the mutual action of the field particle (F) of the th population on the test particle (T) of the th population and vice versa so that →  ⇐⇒  * () With respect to this mutual action, we can assume that this function depends on the biological model, as follows.
(1) Competition within the first group and with others: particles of the th population interact with any other particle both from its own th population and from the th population so that   ( * ,  * , ) ̸ = 0, ( fixed, ∀) .
In this case, each particle of the th population can change its state not only due to the competition with the th population but also by interacting with particles of its own population.Instead, the individuals of the th population change their state only due to the interaction with the other th populations.They do not interfere with each other within their th group.(2) Competition within the second group and with others: particles of the th population interact with any other particles both from its own th population and from the th population so that (3) Full competition within a group and with others: particles of each population interact with any other particles both from its own population and from the other population so that   ( * ,  * , ) ̸ = 0, (∀, ∀) .
(4) Competition of two groups: particles of each population interact only with particles from the other population so that   ( * ,  * , ) = 0, ( = ) .
We can assume that this kind of competition arises when the dynamics in each population are stable and each population behaves as a unique individual.
Then, by using the mathematical approach, developed in [1,2], it yields the following class of evolution equations: which can be formally written as (6) 2 .

Transition Density Function Based on Separable Functions
In this section, we give the solution of ( 15) under some simple assumptions on the form of the transition density (7).

On the Symmetries of the State Transition Density.
We assume that the integrability condition on   , holds true.As a consequence, if we write the transition density as a linear combination of separable functions, this definition implies some symmetries which will be useful for the following computations, in particular.

Theorem 1. If one defines the transition density as
with   ( * , ),   ( * , ) > 0 (,  = 1, 2), the following symmetry holds true: Proof.From ( 7), ( 17), we have There follows, with  = 1,  = 2 and  = 2,  = 1, so that by a comparison of to be valid for all  1 ,  2 , that is, as a consequence of the definition (17), In particular, to fulfill (20), we can assume from which, by taking into account (22), we get so that, by a difference, Thus, according to (25), the mutual action of the state transition given by the definition (7) can be summarized by (18).
In the following, we will consider a special choice for the transition density (17) as so that (18) is fulfilled.

Preliminary
is solved by where () is the solution of the second kind homogeneous Fredholm integral equation with  being the eigenvalue of the integral equation, and when  = 0, () is any arbitrary function fulfilling (32) 2 .
Proof.Let us first notice that in the trivial case of  = 0, there is no dependence on the function but this equation is also solved by (30) being In the more general case, (31) 2 , (32) 2 are direct consequence of the condition (5).By a simple computation, (29) can be transformed into the Fredholm integral equations (31), (32).
When  = 0, from the r.h.s, we have so that () cannot be univocally determined.
The proof of all cases above is followed by solving these two equations in (), () with respect to the initial condition (0, ).
For instance, for the first case (1), there follows that is However, if  ̸ = 0, the integral of the right side of the second equation is , while the integral of the first side must be zero.
With similar reasonings, we get the proof of the remaining cases.(15) In this section, we will give the explicit solution of the system (15) under some suitable hypotheses on both the encounter rate   and the transition density   ( * ,  * , ).Let us assume the symmetry of   so that

Solution of the System
Thanks to the previous theorems, and the symmetry of   ( * ,  * , ) as given by (18), system (15) simplifies, the following.
Theorem 4. Let the transition density   ( * ,  * , ) be defined as which fulfills (7) and the symmetries conditions (18) Thus, we obtain, by a variable change, so that (56) follows.
4.1.Pure Competition Model.We will consider the solution of (56) when, together with the hypotheses (53), (54) 2 , some more conditions are given on the parameters.According to (26), let us assume together with the symmetries conditions (18).If we define we will discuss only the following hypotheses: which seem to have some biological interpretations, being the pure encounter-competition model.This happens when the transition of state arises only when particles of one population interact only with an individual of the other population.In this case, individuals of one population do not interact with individuals of the same population.with   as given by (64), (65).This definition of the transition density fulfills (7) and the symmetries conditions (18).The density function (, ) is such that By assuming and for  0 ,  0 , the condition with   , Kronecker symbol.

Application to Lotka-Volterra Model
In this section, we will study a coupled system (6) where the macroscopic equations are the Lotka-Volterra equations (6) 1 .Concerning the coupling stochastic parameter [], we have to define the functional  in (2), (6) depending on the "distance" between distributions; that is, with where the maximum learning result is obtained when the second population is able to reproduce the distribution of the first one:  1 =  2 , while the minimum learning is achieved when one distribution is vanishing.In some recent papers; it has been assumed [4,5] that In this case, it is  = 1, when  1 =  2 ; otherwise  ̸ = 1 with  ↓ 0, depending on the time evolution of the distance between  1 and  2 .
Let us notice that  is the coupling term which links the macroscopic model (6) 1 to the microscopic model (6) 2 .There follows that the solution of the hybrid system (6) depends on the coupling parameter  (80) which follows from the solution of (15).System (15) is a system of two nonlinear integrodifferential equations constrained by the conditions (7), (5).Moreover, its solution depends also the constant encounter rate   , on the transition density function   , and the initial conditions   (0, ).In the following section, we will study the solution of ( 6), under some suitable, but not restrictive, hypotheses on   .
Under the hypotheses of Theorem 5 and the solution (72), we have In the last case we have the usual Lotka-Volterra system, therefore, we will investigate the first case.Thus, according to (6), we have the system with  ≥ 0,  ≥ 0,  ≥ 0.
The numerical solution of this system depends on both the parameters , , ,  and on the initial conditions  1 (0),  2 (0).We can see from Figures 1 and 2 that albeit the initial aggressive population  2 is greater than  1 , the first population can increase and keep nearly always over  2 in the quasilinear case in Figure 1(a) or always under  2 in presence of a strong nonlinearity in Figure 1(b).
If we invert the initial conditions so that the initial population of  1 is greater than  2 , we can see that in case of quasilinear conditions (see Figure 2(a)) the population  1 after some short time becomes lower than  2 .For a strongnonlinearity, instead after an initial growth  1 , it  tends to zero in a short time, while the second population grows very fast and becomes the prevalent population in Figure 2(b).

Conclusion
In this paper, the hybrid competition model has been solved under some assumptions on the transition density.In the simple case of Lotka-Volterra, the numerical solution gives some significant and realistic insights on the evolution of competing populations.