key: cord-0537713-w5pxox0x
authors: Burini, Diletta; Chouhad, Nadia
title: Micro-Macro Derivation of Virus-Chemotaxis Models
date: 2021-12-14
journal: nan
DOI: nan
sha: 76e1b51c1efa5eb0906a9e60c70fce2255e178a1
doc_id: 537713
cord_uid: w5pxox0x

This paper deals with the micro-macro derivation of virus models coupled with a reaction diffusion models that generates the dynamics in space of the virus particles. The first part of the presentation focuses, starting from [5, 6] on a survey and a critical analysis of some phenomenological models known in the literature. The second part shows how methods of the kinetic theory can be used to model the dynamics of the system treated in our paper. The third part deals with the derivation of macroscopic models from the underlying description, delivered within a general framework of the kinetic theory.

The phenomenological derivation of models of biological tissues can be obtained, at the macroscopic scale, by conservation, or equilibrium, equations that involve locally averaged quantities deemed to describe the state of the system. These structures can be closed by means of heuristic models of the material behavior of the physical system under consideration. Different models correspond to each specific closure that are obtained by material models generally valid only for physical conditions closed to equilibrium, while dynamical models are required to describe physical reality far from equilibrium.

Alternative methods have been developed to tackle the aforementioned conceptual difficulty. Specifically, we refer to the method proposed in [9, 10] where the derivation of reaction-diffusion equations and of the celebrated Keller-Segel model [19, 20] was considered. This derivation is somehow inspired to the Hilbert's sixth problem, see [17] which suggests the search of a unified approach to physical theories at all representation scales, see also [16] .

In general, the dynamics at the low scale is modeled by the kinetic theory of active particles [3] . This equation is expanded in terms of a small parameter corresponding to the mean distance between pair of particles, which is closed by neglecting the contribution of higher order terms. The macro-scale model is obtained by taking low order terms of the expansion. This method is reviewed in the survey paper [1] mainly devoted to the micro-macro derivation Keller and Segel type models (in short KS model).

In more detail, our paper is devoted to the micro-macro derivation of reactiondiffusion and cross-diffusion models in which a virus model is coupled with a reaction diffusion models that regulate the spatial dynamics of viral particles. This class of models can be defined exotic by a term used to denote models in fields of behavioral sciences, for instance social and economical sciences, namely sciences where individual behaviors have an influence on the mechanical dynamics. Occasionally, the term model in complex environments is used to denote the interaction between a first model, essentially chemotaxis or cross diffusion, with a second (additional) model which describes the external dynamics. The survey [4] reports about a broad variety of this type of models.

In more details on the contents of our paper, Section 2 provides a description and the phenomenological derivation of the aforementioned class of models. Section 3 presents the methodological approach used for the micro-macro derivation. Section 4 shows how the approach can be applied to the derivation of the macroscopic description of the models reported in Section 2. Finally, a critical analysis is presented in the last section looking ahead to research perspectives.

The class of models presented in this section describes the dynamics of a virus model, where space dynamics is induced by a transport mechanism which is modeled by the action of a reaction-diffusion system. In more details, we consider a classic prototype model for virus dynamics in the spatially homogeneous case derived within a framework of population dynamics [7, 23] , which is also known by the acronym SIR [24, 25] . We briefly present the model which has been analytically and computationally studied in [5] , where a SIR type model is coupled with a Keller-Segel model. Let us consider a May-Nowak type model which describes, by a system of ODEs, the dynamics of three components, i.e. the densities of healthy uninfected immune cells u = u(t), infected immune cells v = v(t), and virus particles w = w(t) [24] . The model considers the following heuristic assumptions: Healthy cells are constantly produced by the body at rate r, die at rate d 1 u and become infected on contact with the virus, at rate βuw; Infected cells are produced at rate βuw and die at rate d 2 v; New virus particles are produced at rate kv and die at rate d 3 w. These assumptions lead to the following system of ODEs:

This model has been quite comprehensively understood via a thorough qualitative analysis of corresponding initial value problems (for instance cf. [7, 24] ). As it is known, in addition to the infection-free equilibrium Q 0 := ( r d1 , 0, 0), if the so-called basic reproduction number R 0 is greater than 1, namely R 0 > 1 with

, then, the system shows an additional equilibrium Q * := (u * , v * , w * ), where

This equilibrium is globally asymptotically stable and positive defined, whereas if R 0 ≤ 1 then the infection-free equilibrium Q 0 enjoys this property [22] . The space dynamics in the model studied in [5, 6, 13] is modeled by a deterministic reaction diffusion dynamics acting on u = u(t, x), v = v(t, x) and w = w(t, x) which now include space dependence:

where D u , D v and D w denote the respective, positive defined, diffusion coefficients and where χ represents strength and direction of the cross-diffusive interaction, while the parameters β, k, d 1 , d 2 , d 3 have been already defined above. The reaction-diffusion action term corresponds to a simplified Keller-Segel chemotaxis system [5] . This model can be viewed as a specific example of interaction between a dynamical system modeled by ODEs and a reaction-diffusion system which creates pattern formation. Further developments may focus on the modeling of the virus dynamics, that might go beyond the limited validity of SIR models, as well as on the modeling space dynamics by selecting cross diffusion-reaction models consistent with the specific biological and physical environment where the dynamics develops.

This section presents the micro-macro derivation by an asymptotic expansion somehow inspired to the Hilbert [10] . Firstly, we derive a general kinetic model for three interacting population corresponding to the specific case of the SIR model. The derivation of macroscopic equations is treated in the following two subsections. Firstly, we derive a general macroscopic model and then, we show how it can be specifically referred to the virus model under consideration.

This subsection presents the derivation of macroscopic models, by micro-macro decomposition, of linear transport models a binary mixture of self-propelled particles whose state, called microscopic state, is denoted by the variable (x, v), where x and v are, respectively, position and velocity. The collective description of a mixture of particles can be encoded in the statistical distribution functions

Weighted moments provide, under suitable integrability properties, the calculation of macroscopic variables.

Let us now consider the following class of equations:

where G 1 , G 2 , G 3 are interactions terms assumed depending on the quantities f 1 , f 2 , f 3 , while the operator T i (f ) models the dynamics of biological organisms by a velocity-jump process:

is the probability kernel for the new velocity v ∈ V assuming that the previous velocity was v * . The derivation of macroscopic models from the kinetic model (4), can be obtained in the regime ν 1 , ν 2 , ν 3 , → +∞ corresponding to the distance between particles tending to zero. After a dimensionless of the system is obtained, see [1] , a small parameter ε can be chosen, for the parabolic scaling, such that

Then, the model (4) can be rewritten as follows:

Assumption 3.1. The turning operators T 1 , T 2 , T 3 are supposed to be decomposable as follows:

where T j 1 for j = 0, 1, is given by

with T j

where the dependence on f 2 , f 3 , of the operator T 1 stems from T 1 1 , while we suppose that T 0 1 is independent of f 2 , f 3 , and T l for (l = 2, 3), is given by

Assumption 3.2. We assume that the turning operators T i (i = 1, 2, 3) satisfy the following equality:

Assumption 3.3 There exists a bounded velocity distribution M j (v) > 0(j=2, 3) and M 1 (v) > 0 , independent of t, x, such that the detailed balance

and

hold true. Moreover, the flow produced by these equilibrium distributions vanishes, and M i are normalized

In addition, we assume that the kernels T j (v, v * ) and T 0 1 (v, v * ) are bounded and that there exist constants σ j > 0 and σ 1 > 0, j = 2, 3, such that

for all (v, v * ) ∈ V × V , x ∈ Ω and t > 0.

Given that L j = T j (j = 2, 3) and L 1 = T 0 1 . Technical calculations yields the following Lemma:

Lemma 1 Suppose that Assumptions 3.3 holds. Then, for i = 1, 2, 3, the following properties of the operators L 1 , L 2 and L 3 hold:

A system coupling a hydrodynamic part with a kinetic part of the distribution functions, is derived in this subsection. Then it is proved that such a system is equivalent to the two scale kinetic equation (6). This new formulation provides the basis for the derivation of the general model we are looking for. In the remainder, the integral with respect to the variable v will be denoted by · . This notation is used also for any argument within . In addition, let us denote by f = (f 1 , f 2 , f 3 ) the solution of (6), where f is decomposed as follows:

and

In order to develop asymptotic analysis of Eq. (6), additional assumptions on the operator T 1 1 and the interaction terms Gî(î = 1, 2, 3) are needed.

Assumption 3.4. We assume that the turning operator T 1 1 and the interaction terms Gî(i = 1, 2, 3) satisfy the following asymptotic behavior as:

for j = 0, 1, ..., q 2 + 1 and l = 0, 1, ..., q 3 + 1.

Then, (10) rapidly yields:

and

Then, the first terms of Hilbert expansion of equal order in ε i , ε j and ε l for i = 0, 1, ..., q 1 + 1, j = 0, 1, ..., q 2 + 1 and l = 0, 1, ..., q 3 + 1 are: 

∀p ≤ q 1 ,

where δ a,b stands for the Kronecker delta. The first equation of (21) implies that

Therefore ∃c(t, x), ∃s(t, x), ∃u(t, x) such that

Using (10), (13) and (26)-(27), we conclude that Eq. (22) satisfies the solvability condition, therefore g 1 and h 1 are given by

The calculations of g q1+1 , h q2+1 , and k q3+1 are obtained from the solvability conditions at O(ε q1+1 ), O(ε q2+1 ) and O(ε q3+1 ), which are given by the following:

and

Using (13), (10), (19) and (26)-(28), denoting by < · > the integral with respect to the variables v, shows that the system (29)-(30) can be rewritten as follows:

As T 0 1 , T 2 and T 3 are self-adjoint operators in

and L 2 D v , dv M3(v) , one has the following computations:

where θ 1 and θ 2 are given in Lemma 2. Therefore, the macroscopic model (31) can be written as follows:

where D c , D s , D u , α are given, respectively, by

and

while H i (c, s, u), (i=1,2,3) are given by:

More in detail, let us consider the following kernels:

with σ 1 , σ 2 , σ 3 > 0. Hence, the leading turning operators T j (j=2,3) and T 0 1 can be viewed as relaxation operators:

and

Moreover, θ 1 , θ 2 , and θ 3 are given by

while α, are defined by (34)-(35), and are computed as follows:

The diffusion tensors D c , D s , and D u are given by

while H 1 , H 2 , H 3 are still given by (35).

• Let us now consider that q 1 = q 2 = q 3 = 2 and p = 2, then from (32) one has the following macro-scale type models up to ε:

The role of the terms H 1 (c, s, u), H 2 (c, s, u), and H 3 (c, s, u) in (35) consists in modeling the interaction between the for quantities of the mixture. For example, by choosing:

Therefore, the macroscopic model (44) writes:

• Let us also consider that q 1 = q 2 = q 3 = 1, p = 1, and the following choice:

is a vector valued function satisfying the following:

Then T 1 1 satisfies (13), and leads to the following:

Finally, α(s), defined in (41), is given by α(s) = χ(s) · ∇ x s, where the chemotactic sensitivity χ(s) is given by the matrix

Therefore, the macroscopic model (32) can be written as follows:

∂ t c + div x c χ(s) · ∇ x s − (D c · ∇ x c) = H 1 (c, s, u), ∂ t s − div x (D s · ∇ x s) = H 2 (c, s, u),

Supposing that the G i (i = 1, 2, 3), are given by (45)-(47), yields the following macroscopic model (53):

where the D c , D s , D u , and χ(s) are given by (42)-(43) and (52).

We trust that the micro-macro derivation developed in our paper, can be further extended to a variety of exotic models including models of the dynamics of different types of virus. This subsection simply introduces this topic which will be developed in a well defined research program. Indeed, the complexity of the dynamics may substantially increase thus requiring nontrivial developments of the approach. In more details on virus dynamics, possible developments might account for delay-distributed dynamics [11] , infection model with multitarget cells [12] , stochastic models which include selective progression [15] . In addition, the recent Covid-19 virus pandemic has generated a huge number of models that can be viewed as a technical development of SIR type models. The research article [14] is one of the first contributions to this topic. Subsequently, it has been followed by various papers which have gone beyond the framework of compartmental models accounting for multiscale features and of the immune competition inside the lung [2] , transport dynamics [8] , contagion in crowds [21] , and various others.

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