On a three-dimensional and two four-dimensional oncolytic viro-therapy models Rim Adenanea Eric Avila-Valesb Florin Avramc

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On a three-dimensional and two four-dimensional

oncolytic viro-therapy models

Rim Adenanea, Eric Avila-Valesb, Florin Avramc,

Andrei Halanayd, Angel G. C. P´erezb

October 4, 2022

aD´epartement des Math´ematiques, Universit´e Ibn-Tofail, Kenitra, 14000, Maroc

bFacultad de Matem´aticas, Universidad Aut´onoma de Yucat´an, Anillo Perif´erico Norte,

Tablaje Catastral 13615, C.P. 97119, M´erida, Yucat´an, Mexico

cLaboratoire de Math´ematiques Appliqu´ees, Universit´e de Pau, 64000, Pau, France

dDepartment of Mathematics and Informatics, Polytechnic University of Bucharest,

062203, Bucharest, Romania

Abstract

We revisit here and carry out further works on tumor-virotherapy compart-

mental models of [Tian, 2011, Wang et al., 2013, Phan and Tian, 2017, Guo

et al., 2019]. The results of these papers are only slightly pushed further. How-

ever, what is new is the fact that we make public our electronic notebooks,

since we believe that easy electronic reproducibility is crucial in an era in which

the role of the software becomes very important.

Keywords: Oncolytic viro-therapy, immune response, stability, compartmental

models, bifurcation analysis, electronic reproducibility.

Contents

1 Introduction 2

2 Warm-up: the 3 dimensional viral model [Tian, 2011, Kim et al.,

2020] 7

arXiv:2210.00401v1 [math.DS] 2 Oct 2022

3 The four-compartment viro-therapy and immunity model (1) 12

3.1 Boundedness ............................... 12

3.2 Boundary equilibria and their stability . . . . . . . . . . . . . . . . . 14

3.2.1 Stability of the boundary ﬁxed point EK............ 15

3.2.2 Stability of the boundary ﬁxed point E∗............ 16

4 The four-dimensional viro-therapy model with = 0 [Phan and Tian,

2017] 17

4.1 Interiorequilibria ............................ 17

4.2 The stability of E∗when =0...................... 19

4.3 Stability of the interior equilibria and bifurcation diagrams . . . . . . 19

4.4 Time and phase plots illustrating bi-stability and a limit cycle, with

=0.................................... 24

4.4.1 Bi-stability in the interval (b2, b2∗)................ 24

4.4.2 Limit cycle in the interval (bH, b∞)............... 25

5 The four-dimensional viro-therapy model of [Guo et al., 2019], with

logistic growth 26

5.1 Interiorequilibria............................. 26

5.2 Stability of interior equilibria and bifurcation diagrams . . . . . . . . 29

5.3 Time and phase plots illustrating diﬀerent behaviors, with = 1 . . . 32

5.3.1 Stability of E+in the interval (b0, bH).............. 32

5.3.2 Bi-stability and limit cycle in the interval (bH, b2∗)....... 34

5.3.3 Chaotic behavior in the interval (b2∗, b∞)............ 34

6 Conclusions 35

1 Introduction

Compartmental models became famous ﬁrst in mathematical epidemiology, fol-

lowing the pioneering work of Kermack and McKendrick [Kermack and McKendrick,

1927] on the SIR model; see [Haddad et al., 2010] for other domains of application,

and for some general theory. In the last thirty years, they have penetrated also in

mathematical virology [Perelson and Weisbuch, 1997, Nowak and May, 2000, Wodarz

and Komarova, 2005, Bocharov et al., 2018], and in mathematical oncolytic virother-

apy, i.e. in the modeling of the use of viruses for treating tumors [Santiago et al.,

2017, Rockne et al., 2019, Pooladvand, 2021].

We may distinguish between at least two main directions of work in these ﬁelds.

1. Part of the literature is dedicated to creating models to ﬁt speciﬁc viruses

and therapies – see for example [Perelson and Nelson, 1999, Perelson, 2002,

Antonio Chiocca, 2002, Smith and De Leenheer, 2003, Wodarz, 2003, Pillis

et al., 2006, Dalal et al., 2008, Tuckwell and Wan, 2000, Yuan and Allen,

2011, Yu and Wei, 2009, Huang et al., 2011, Chenar et al., 2018]. The models

proposed are high dimensional, and hence only analyzable numerically, for

particular instances of the parameters.

2. Another part, which is our concern here, is in applying sophisticated math-

ematical tools, notably the theory of bifurcations for dynamical systems, to

“lower dimensional caricatures” of more complex models. This requires the

use of both symbolic software like Mathematica, Maple, or Sagemath, and also

of sophisticated numeric continuation and bifurcation packages like MatCont

(written in Matlab), PyDSTool (Python), XPPAuto (C) – see [Blyth et al.,

2020] for a recent review, and BifurcationsKit (written in Julia).

In our work below, we have combined the use of MatCont – see [P´erez, 2022] with that

of Mathematica – see [Adenane, 2022], and in particular the package EcoEvo. The

notebooks oﬀered on GitHub are an important part of our work, and we attempted

to achieve a roughly one to one correspondence between the equations numbered in

the text and those displayed in Mathematica.

The origins of the glioma viro-therapy four-compartment (x, y, v, z) model con-

sidered here, where untreated and tumor cells are denoted respectively by x, y, virus

cells by v, and innate immune cell by z, are in [O’Connell et al., 1999, Friedman

et al., 2006]. ¶Interestingly, these papers suggested a density dependent rate of im-

mune cells, linear up to a threshold z0, and quadratic afterwards. “The ﬁrst process

occurs when zis small and yields a linear clearance; the second process occurs when

zis large and yields a quadratic clearance” [Friedman et al., 2006, pg 2]. Subsequent

papers of Tian [Phan and Tian, 2017], [Guo et al., 2019] tackled symbolically the two

particular cases z0=∞,z0= 0. For further developments and further outstanding

questions in the ﬁeld, see [Vithanage et al., 2021, Phan and Tian, 2022a, Phan and

Tian, 2022b].

Since the quadraticity is hard to ascertain, we propose to study a uniﬁcation of

¶A four-dimensional model considerably more complex was proposed in [Senekal et al., 2021].

the four-compartment systems studied in [Phan and Tian, 2017, Guo et al., 2019]:

dt =λx 1−x+y

K−βxv

dt =βxv −γy −βyyz

dt =bγy −βxv −δv −βvvz

dt =z(ρβyy−cz),  ∈ {0,1},

(1)

where x,y,vand zrepresent the populations of uninfected (untreated) tumor cell

population, infected tumor cell population, free virus and innate immune cells, re-

spectively.

Remark 1.1. The invariance of the ﬁrst quadrant (also called “essential non-negativity”)

is immediate since each component fi(X)of the dynamics may be decomposed as

fi(X) = gi(X)−xihi(X),

where gi, hiare polynomials with nonnegative coeﬃcients, and xiis the variable whose

rate is given by fi(X). In fact, under this absence of “negative cross-efects”, even

more is true: the model admits a “mass-action representation” by the so-called “Hun-

garian lemma” [H´ars and T´oth, 1981, Haddad et al., 2010], [T´oth et al., 2018, Thm.

6.27] §

Remark 1.2. Scaling all the variables by x=Kex,y=Key, ... has the eﬀect

of multiplying all the quadratic terms by K, and one may ﬁnally assume K= 1,

at the price of renaming some other parameters. Also, scaling time by a constant

allows choosing another parameter as 1. Below, we will follow occasionally [Tian,

2011, Phan and Tian, 2017] in choosing K=γ= 1, which simpliﬁes a bit the results.

Figure 1 depicts a schematic diagram of this model. The interpretation of pa-

rameters can be seen in Table 1.

§The previous virology literature does not seem to be aware of this result, and oﬀers direct proofs

instead.



 



 

 







Figure 1: Schematic diagram of model (1). The compartments x,y,vand zdenote

uninfected tumor cells, infected tumor cells, free virus and innate immune cells,

respectively. Continuous lines represent transfer between compartments. Dashed

lines represent viral production or activation of immune cells.

Table 1: Interpretation of parameters for model (1).

Symbol Description

λintrinsic growth rate of uninfected tumor cells

K > 0 carrying capacity of uninfected tumor cells

β > 0 viral infection rate

βyrate at which immune system removes infected tumor cells

γ > 0 lysis rate of infected cells

b≥1 virus burst size

δclearance rate of viruses

βvrate at which immune system removes viruses

ρβy:= βz>0 proliferation rate of immune cells due to the interaction with infected tumor cells

crate of clearance of immune cells

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Onathree-dimensionalandtwofour-dimensionaloncolyticviro-therapymodelsRimAdenanea,EricAvila-Valesb,FlorinAvramc,AndreiHalanayd,AngelG.C.PerezbOctober4,2022aDepartementdesMathematiques,UniversiteIbn-Tofail,Kenitra,14000,MarocbFacultaddeMatematicas,UniversidadAutonomadeYucatan,AnilloPerifericoN...

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