Applied Ma hema ical Modelling 137 (2025) 115673
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Applied Ma hema ical Modelling
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Quasineu al mul is abili y in an epidemiological-like model o
de ec i e-helpe be aco ona i us in ec ion in cell cul u es
Juan C. Muñoz-Sáncheza,b,1, J. Tomás Láza o c,d,e, ,1, Julia Hillunga,
Ma ía J. Olmo-Ucedaa, Josep Sa danyése, , San iago F. Elena a, ,g,∗
aIns i u e o In eg a i e Sys ems Biology (I2SysBio), CSIC-Uni e si a de València, Pa e na, 46980 València, Spain
bDepa amen de Física Teò ica, Uni e si a de València, Bu jasso , 46100 València, Spain
cDepa amen de Ma emà iques, Uni e si a Poli ècnica de Ca alunya (UPC), 08028 Ba celona, Spain
dIns i u e o Ma hema ics, UPC-Ba celonaTech (IMTech), 08028 Ba celona, Spain
eCen e de Rece ca Ma emà ica (CRM), Ce danyola del Vallès, 08193 Ba celona, Spain
Dynamical Sys ems and Compu a ional Vi ology, CSIC Associa ed Uni CRM-I2SysBio, Spain
gSan a Fe Ins i u e, San a Fe, NM 87501, USA
A B S T R A C T
I is well known ha , du ing eplica ion, RNA i uses spon aneously gene a e de ec i e i al genomes (DVGs). DVGs a e unable o comple e an
in ec ious cycle au onomously and depend on coin ec ion wi h a wild- ype helpe i us (HV) o hei eplica ion and/o ansmission. The s udy
o he dynamics a ising om a HV and i s DVGs has been a longs anding ques ion in i ology. I has been shown ha DVGs can modula e HV
eplica ion and, depending on he s eng h o in e e ence, esul in HV ex inc ions o sel -sus ained pe sis en fluc ua ions. Ex ensi e expe imen al
wo k has p o ided mechanis ic explana ions o DVG gene a ion and compelling e idences o HV-DVGs i us coe olu ion. Some o hese obse a ions
ha e been cap u ed by ma hema ical models. He e, we de elop and in es iga e an epidemiological-like ma hema ical model specifically designed
o s udy he dynamics o be aco ona i us in cell cul u e expe imen s. The dynamics o he model is go e ned by se e al degene a e no mally
hype bolic in a ian mani olds gi en by quasineu al planes -i.e., filled by equilib ium poin s. Th ee diffe en quasineu al planes ha e been
iden ified depending on pa ame e s and in ol ing: (i) pe sis ence o HV and DVGs; (ii) pe sis ence o non-in ec ed cells and DVG-in ec ed cells;
and (iii) pe sis ence o DVG-in ec ed cells and DVGs. Key pa ame e s in ol ed in hese scena ios a e he maximum bu s size (𝐵), he ac ion o
DVGs p oduced du ing HV eplica ion (𝛽), and he eplica ion ad an age o DVGs (𝛿). Mo e p ecisely, in he case 0 <𝐵<1 +𝛽 he sys em displays
is abili y, whe e all h ee scena ios a e p esen . In he case 1 +𝛽<𝐵<1 +𝛽+𝛿 his is abili y pe sis s bu a ac ing scena io (ii) is educed o
a well-defined hal -plane. Fo 𝐵>1 +𝛽+𝛿, he scena io (i) becomes globally a ac o . Scena ios (ii) and (iii) a e compa ible wi h he so-called
sel -cu ing since he HV is emo ed om he popula ion. Sensi i i y analyses indica e ha model dynamics la gely depend on DVGs p oduc ion
a e (𝛽) and hei eplica i e ad an age (𝛿), and on bo h he in ec ion a es and i us-induced cell dea hs. Finally, he model has been fi ed o
single-passage expe imen al da a using an a ificial in elligence me hodology based on gene ic algo i hms and key i ological pa ame e s ha e been
es ima ed.
Con en s
1. In oduc ion.............................................................................. 2
2. Summa y o he expe imen al esul s .............................................................. 4
3. Ma hema ical model......................................................................... 4
* Co esponding au ho a : Ins i u e o In eg a i e Sys ems Biology (I2SysBio), CSIC-Uni e si a de València, Pa e na, 46980 València, Spain.
E-mail add ess: [email p o ec ed] (S.F. Elena).
1Equal con ibu ion.
h ps://doi.o g/10.1016/j.apm.2024.115673
Recei ed 13 Feb ua y 2024; Recei ed in e ised o m 24 July 2024; Accep ed 29 Augus 2024
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J.C. Muñoz-Sánchez, J.T. Láza o, J. Hillung e al.
4. Resul s ................................................................................. 5
4.1. Planes o equilib ia, s abili y and basins o a ac ion............................................... 6
4.2. Cell cul u e dynamics: es ima ion o he amoun o in ec i e pa icles .................................... 10
4.3. Case 𝑚 >1: all cells a e simul aneously in ec ed.................................................. 14
4.4. Sensi i i y analysis .................................................................... 16
4.5. Model fi ing o HCoV-OC43 da a: pa ame e s’ es ima ion............................................ 17
5. Discussion ............................................................................... 19
CRediT au ho ship con ibu ion s a emen ................................................................ 20
Decla a ion o compe ing in e es ..................................................................... 20
Da a a ailabili y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
Acknowledgemen s .............................................................................. 20
Appendix A. Time-se ies examples o he Π𝑉𝐷 , Π𝐶𝐷𝐷and Π𝐶𝐶
𝐷
................................................. 21
Appendix B. Fi s o de a ia ional equa ion wi h espec o ini ial condi ions........................................ 21
Appendix C. Addi ional fi ings esul s................................................................. 22
Appendix D. Model solu ions o some 𝜄∕𝛼 alues.......................................................... 23
Re e ences.................................................................................... 23
1. In oduc ion
RNA i uses can quickly adap and igge epidemics by c ossing species ba ie s due o hei high mu a ion a es, as eplica ion,
and la ge popula ion sizes [1–3]. Howe e , a high mu a ion a e is a double-edge swo d, as many mu a ions du ing in ec ion esul
in de ec i e i al genomes (DVGs). DVGs canno comple e he in ec ious cycle by hemsel es hus depending on he i al p o eins
syn hesized by a wild- ype helpe i us (HV). The gene ic e m DVG includes poin mu a ions, hype mu a ed genomes, dele ions,
inse ions, and genomic eo ganiza ions [4]. Huang and Bal imo e coined he e m de ec i e in e e ing pa icles (DIPs) o i al
pa icles con aining DVGs and no mal s uc u al p o eins encoded by he HV [5]. DIPs we e fi s iden ified in he la e 1940s by Von
Magnus and Ga d [6]based on he nega i e impac hey exe ed on i us accumula ion. DIPs ely on a HV o eplica ion, dis up ing
HV accumula ion and impac ing i al pa hogenesis [7,8]. Se e al s udies ha e shown ha i uses ich in DIPs educe i ulence [9],
induce high in e e on le els [10], aid i al pe sis ence [11,12], and modula e in ec ion capaci ies as shown o diffe en SARS-CoV-2
s ains [13].
Recen high- h oughpu sequencing echniques ha e unco e ed he eme gence o a ple ho a o DVGs wi hin a single in ec ed
hos [14–17]. No ably, hese s udies ha e shown ha dis inc subse s o p e alen DVGs ecu en ly appea along ime, sugges ing
in ica e dynamics wi hin he i al popula ion. These dynamics encompass compe i ion, and possibly compensa ion o coope a ion,
among a ious DVGs. Posi i e selec ion a ou s he mos compe i i e DVG a ian s, indica ing hei ela i e fi ness conce ning he
HV and o he DVGs. Despi e he gene a ion o hund eds o e en housands o dis inc DVGs du ing in ec ions, he majo i y a e los
due, among o he ac o s, o popula ion bo lenecks occu ing du ing in i o ansmissions among indi iduals [18]o du ing in i o
dilu ed se ial passages [19]. Howe e , DVGs migh pe sis o long pe iods o ime in immunosupp essed hos s o in hose wi h
como bidi ies [20], o i a high a io be ween i al pa icles and suscep ible cells (a pa ame e known as he mul iplici y o in ec ion,
MOI) is expe imen ally imposed [21–23]. Du ing in ec ions o in i o cell cul u es, as hose ha ha e mo i a ed his modelling wo k,
DVGs accumula e when i al popula ions a e epea edly passed a a high MOI, while hey do no accumula e i MOI is low o s ong
bo lenecks a e imposed a each ansmission e en [21,22,24]. The highe he MOI, he mo e likely DVGs and HV would coin ec he
same cell and hus pe sis in he popula ion [21,22,24]. While he molecula ecombina ion mechanisms by which DVGs a e gene a ed
a e well unde s ood, hei ole in modula ing he ou come o i al in ec ions is some imes unclea . Unde s anding he cha ac e is ics
and unc ions o DVGs is hus c ucial o comp ehending he complexi y o i al in ec ions and o de eloping s a egies o con ol
o mi iga e hei impac .
DIPs ac as ue hype pa asi es in e e ing wi h he HV eplica ion [23,25], compe ing o esou ces and educing i s accumu-
la ion and ansmission efficiency [5,22,26–28]. Expe imen al in i o s udies wi h cell cul u es ha e shown ha DIPs may engage
in an a ms ace wi h he HV [29–32]. Sho e genomes ea n an ad an age in e ms o eplica ion speed compa ed o he HV [33].
Addi ionally, he e is e idence o a s onge o m o in e e ence in which he DIPs compe e mo e effec i ely o he i al eplica ion
machine y [30,34]. Since hei disco e y, and gi en hei effec on he accumula ion o he HV, DIPs ha e a ac ed he a en ion
o esea che s as po en ial an i i al candida es [28,35,36], known as he apeu ic in e e ing pa icles (TIPs). Following his idea,
Xiao e al. c ea ed a TIP by dele ing he capsid-coding egion o polio i us [37]. Rema kably, he adminis a ion o his TIP o mice
igge ed a b oad an i i al esponse agains di e se espi a o y i uses, including en e o i uses, influenza A i us, and SARS-CoV-2.
The b oad-spec um an i i al effec s o his syn he ic TIP was a ibu ed o local and sys emic ype I in e e on esponses. No ably, a
single dose no only sa egua ded animals om SARS-CoV-2 in ec ion bu also s imula ed he p oduc ion o SARS-CoV-2 neu alizing
an ibodies, p o ec ing agains ein ec ion [37]. Ano he success ul applica ion o TIPs was epo ed by Cha u edi e al. [38]. Fol-
lowing a syn he ic biology app oach, hese au ho s gene a ed a ificial pa icles ha had significan an i SARS-CoV-2 effec in cell
cul u es and p ima y lung o ganoids, educing i al accumula ion 10- o 100- old. Fu he mo e, in anasal applica ion o hese TIPs
in in ec ed hams e s supp essed he i us by 100- old in he lungs, educed p o-inflamma o y cy okine exp ession, and p e en ed
se e e pulmona y edema [39]. In e es ingly, he p edic ion o SARS-CoV-2 inhibi ion by a single TIP adminis a ion was ob ained
om a wi hin-hos ma hema ical model based on diffe en ial equa ions [39].
Applied Ma hema ical Modelling 137 (2025) 115673
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Fig. 1. (a) Expe imen al se up o co ona i uses HCoV-OC43 and MHV in cell 6-wells pla es [15]. (b) Passage dynamics o helpe i us (HV) and de ec i e i al
genomes (DVGs), wi h la ge i e fluc ua ions o he expe imen s pe o med in [15]. DVG abundance es ima ed using [63]. (c) Schema ic diag am o he dynamical
sys em modelling wi hin- and be ween-cell i us dynamics o he cell cul u e expe imen s pe o med in [64](see Sec ion 2and [15]). The model conside s in ec ion o
a suscep ible popula ion o hos cells, 𝐶, by HV ( ed) and/o by DVGs (yellow), p oducing HV only-in ec ed cells 𝐶𝑉, DVGs only-in ec ed cells 𝐶𝐷, and double-in ec ed
cells 𝐶𝐷𝑉 . Wi hin-cell eplica ion in ol es he amplifica ion o i al genomes, which p oduce DVGs a a a e 𝛽. In ec ed cells a e lysed eleasing HV and DVGs o he
medium. (d) Time se ies o HCoV-OC43 i al pa icles in ec ing BHK-21 cells wi h diffe en MOIs in cell cul u es. (e) S a e a iables o he model.
Ma hema ical models desc ibing ans in e ac ions be ween i al genomes a e ound in he li e a u e [40,41]. The dynamics o
DIPs ha e been ex ensi ely s udied as an ex eme case o complemen a ion [41–48]. Fo example, he ea ly wo k by Sza hmá y [43],
p esen ed s uc u ed deme models o p o ide a desc ip ion o he coexis ence o i us segmen s conside ing HV and DIPs, sensi i e and
esis an i uses oge he wi h DIPs, co i us pai s (i.e., i us ha exis as wo o mo e sepa a ed pa icles all o which mus be p esen
o he comple e eplica ion cycle o occu ), and i us–co i us sys ems. A deepe analysis o he model wi h HV and DIPs was la e
de eloped in [44]by conside ing cell popula ions in ec ed by pa icles diffe ing in numbe . Diffe en dynamics, such as s able fixed
poin s, pe iodic o bi s, and s ange chao ic a ac o s we e iden ified wi h his model. La e , Ki kwood and Bangham [49] de eloped a
diffe en ial equa ions model o analyze a sys em consis ing o well-mixed hos cells, HV, and DIPs unde se ial passage dynamics. The
model success ully explained a ious dynamic beha iou s obse ed in cell cul u e expe imen s: fluc ua ions in i us accumula ion
du ing successi e passages and sel -cu ing in ol ing he simul aneous ex inc ion o he HV and DIPs as shown expe imen ally [46,50].
Mo e ecen ly, passage expe imen s o baculo i uses in mo h la ae ha e also p o ided expe imen al e idence o chao ic dynamics
be ween HV and DIPs con aining long genomic dele ions [32].
Among co ona i uses, dele ions a e he mos common ype o DVGs [13,15], o med h ough ecombina ion due o conse ed
homology in specific egions and/o RNA s uc u es [51–54]. Indeed, SARS-CoV-2 dele ion DVGs ha e been pe asi ely ound bo h
in cell cul u es and in pa ien s [13,20]. In his ega d, asymp oma ic pa ien s ended o ha e lowe DVG loads han symp oma ic
ones, and highly di e se popula ions o DVGs we e obse ed a e long- e m COVID-19 in an immunosupp essed pa ien , sugges ing
a ela ionship be ween in e e on esponses and DVGs [20]. Gi en he pe asi eness o DVGs in be aco ona i us popula ions [20]and
he lack o e idences suppo ing hei sus ained in e e ence ac i i y (and hence hei po en ial de elopmen as TIPs), Hillung e al.
[15] pe o med long- e m in i o e olu ion expe imen s o explo e he ole o DVGs in be aco ona i uses dynamics o di e sifica ion
and e olu ion. Two i us models we e chosen o his s udy, he human co ona i us OC43 (HCoV-OC43) and he mu ine hepa i is
i us (MHV).
To be e unde s and he popula ion dynamics o his expe imen al sys em, he e we de elop and in es iga e a ma hema ical model
ga he ing key in e ac ions a he le el o wi hin-passage in ec ion dynamics (Fig. 1). The model shows he p esence o degene a e
no mally hype bolic in a ian mani olds (quasineu al in a ian mani olds). The exis ence o such mani olds implies ha he o bi s
Applied Ma hema ical Modelling 137 (2025) 115673
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J.C. Muñoz-Sánchez, J.T. Láza o, J. Hillung e al.
in he phase space each planes o equilib ia, depending on pa ame e s. Tha is, o bi s a e s ongly a ac ed o a gi en equilib ium
popula ion which is no a single poin a ac o bu a cu e o plane filled wi h equilib ia and diffe en ini ial condi ions each diffe en
equilib ium popula ion alues. Resea ch on quasineu al planes is a he limi ed. These neu al su aces ha e been cha ac e ised in
p eda o -p ey models wi h Holling ype III unc ional esponses [55]and in socio-economical models [56]. Mo eo e , quasineu al
s a es go e ned by lines o cu es o equilib ia ha e been iden ified in p ey-p eda o models gi en by diffe en ial [57]and pa ial [58]
diffe en ial equa ions, in Lo ka-Vol e a compe i ion models [59], in s ains’ compe i ion models o disease dynamics [60], and in
models o RNA genomes eplica ion [61]. Mo e ecen ly, a quasineu al cu e was ound in a ma hema ical model o an au oca aly ic
eplica o wi h an obliga e pa asi e [62]. Fo his la e case, a bis abili y mechanism de e mined whe he a gi en ini ial condi ion
achie ed he quasineu al cu e o co-ex inc ion. The model in es iga ed in he e exhibi s a a ie y o quasineu al objec s (mainly
planes) displaying, o some pa ame e combina ions, is abili y be ween diffe en scena ios as a unc ion o he ini ial condi ions.
2. Summa y o he expe imen al esul s
To in es iga e he dynamics o de ec i e i al genomes (DVGs) accumula ion, we pe o med se ial passages wi h wo be aco o-
na i uses: HCoV-OC43 in ei he baby hams e kidney cells (BHK-21) o human la ge in es ine ca cinoma cells (HCT-8), and MHV
in mu ine li e cells (CCL-9.1). Passages in ol ed s ochas ically a ying inocula size wi hin wo wide bu disjoin in e als ha can
b oadly be defined as low and high mul iplici y o in ec ion (MOI). A e e y passage, in ec ious i al i e (defined as concen a ion
o plaque o ming uni s, PFUs/mL) was measu ed as a p oxy o helpe i us (HV) accumula ion by plaque assays, while he amoun
and ype o he DVGs componen o he e ol ing popula ions was e alua ed a ou equidis an passages by high- h oughpu RNA
sequencing (RNA-seq). Fo illus a i e pu poses, he e we will ocus in he case o HCoV-OC43 in BHK-21 cells. In his pa icula
combina ion, 49 se ial passages we e pe o med, and he DVG componen e alua ed e e y 12 passages. Fig. 1c shows he dynamics
o he fi s 50 passages. Fo he high MOI ea men , he median MOI a he onse o each passage was 25 PFU/cell (IQR: 131.94),
while o he low MOI ea men , i was 3.75 ×10
−4 PFU/cell (IQR: 0.025). See Hillung e al. [15] o mo e expe imen al de ails.
Each passage in ol ed he eplica ion o he HV and DVGs in a cell cul u e dish wi h suscep ible hos cells. In o de o cha ac e ize
he in ec ion dynamics in he cell cul u e and es he alidi y o he model he e in es iga ed, a second se o expe imen s was pe -
o med by Hillung e al. [64]. Specifically, he dynamics o i us accumula ion was moni o ed wi hin a single in ec ious passage. To
do so, h ee independen confluen monolaye s o BHK-21 cells we e inocula ed a wo i al MOI in he o de o uni s pe cell wi h
HCoV-OC43. Then, HV accumula ion was e alua ed as abo e a he ime poin s indica ed in Fig. 1b.
3. Ma hema ical model
In his sec ion we in oduce he dynamical sys em modelling expe imen s by Hillung e al. [15,64] summa ised in Sec ion 2. The
model desc ibes he in ec ion dynamics aking place wi hin a single cell cul u e dish (Fig. 1a), conside ing a helpe i us (HV), which
in ec s and eplica es wi hin a popula ion o suscep ible cells, and p oduces de ec i e i al genomes (DVGs). By assump ion, a DVG
can only in ec a cell bu canno eplica e o lysa e cells on i s own, since i needs he p oduc s om he HV. I is also assumed ha
DVGs ha e a eplica ion ad an age in cells coin ec ed wi h he HV. The p esence o a DVG inside a cell, by con as , hinde s he
eplica ion o he i us, esul ing in a fi ness penal y. An impo an assump ion made is ha i al accumula ion domina es agains
i us’ pa icles decay, i.e., he a e o genomic RNA syn hesis is much g ea e han i s deg ada ion a e (p e ious esea ch on RNA
i uses suppo s his assump ion [65]). Fu he mo e, he model does no conside spon aneous cells’ decay and p esumes ha cell
dea h is d i en by i us-induced lysis. All hese assump ions a e consis en wi h he expe imen al ime scale (see below) and aim o
ocus on he dynamics ha a ise om he in ec ion p ocess.
The s a e a iables o he model a e gi en by in ec ing pa icles, he HV (𝑉) and DVGs (𝐷), and ou diffe en ypes o cells,
including suscep ible (𝐶) and in ec ed cells 𝐶𝑝, whe e he subsc ip indica es he pa icle (o pa icles) ha has (ha e) in ec ed he
hos cell, 𝑝 ∈{𝑉, 𝐷, 𝐷𝑉 }(see Fig. 1e). Wi hou affec ing any heo e ical conclusion and o a oid he use o la ge numbe s ( o
ins ance, he numbe o cells mo es a ound 106) in he p opaga ion o nume ical e o s, we ha e scaled all s a e a iables by di iding
hem by 𝐶(0), he ini ial numbe o cells. The esul ing sys em is dimensionless and i gi es ise o cell a iables anging in he in e al
[0, 1], wi h 𝐶(0) =1, and i uses and DVGs aking alues in an in e al ha depends on some pa ame e s o he sys em.
The in ec ion p ocess is modelled as ollows. When a 𝐶encoun e s a 𝑉o a 𝐷i becomes an in ec ed cell 𝐶𝑉o 𝐶𝐷, espec i ely.
This p ocess happens a an in ec ion a e 𝜄whose in e se could be conside ed as he a e age ime be ween in ec ions. Each ype
o hese in ec ed cells can be coin ec ed o supe in ec ed wi h he al e na i e pa icle ype, esul ing in a double-in ec ed cell 𝐶𝐷𝑉 .
𝐶𝐷cells, as he DVG is no eplica ing i sel , will emain in he same s age un il a supe in ec ion wi h 𝑉occu s, i so. The nex
p ocess will be he eplica ion o HV and DVGs and he lysa e o he in ec ed cells. 𝐶𝑉will be lysed esul ing in o 𝜂new HV pa icles
and 𝛽𝜂 DVGs as a esul o e o s du ing he eplica ion p ocess. The o al numbe o pa icles eleased om he lysis, o bu s
size, is 𝐵=𝜂(1 +𝛽). I eplica ion occu s inside a 𝐶𝐷𝑉 cell, hen he DVG will use he eplica ion machine y o he HV o eplica e,
compe ing o common esou ces and esul ing in a dec ease o he HV accumula ion. The i al p oduc ion a e he 𝐶𝐷𝑉 lysis will
be educed o 𝜂∕𝜅wi h 𝜅>1. On he o he hand, as DVGs ha e a eplica ion ad an age deno ed by 𝛿, i he i us offsp ing a e
lysa e 𝐶𝐷𝑉 is 𝜂∕𝜅 hen he DVGs p oduced will be 𝜂(𝛿+𝛽)∕𝜅. This la e e m accoun s o he HV eplica ion e o s esul ing
in o addi ional DVGs. Thus, he o al offsp ing o he coin ec ed cells 𝐶𝐷𝑉 is 𝜂(1 +𝛿+𝛽)∕𝜅. Unde he assump ion ha he o al
bu s size o 𝐶𝑉and 𝐶𝐷𝑉 is he same, as bo h a e he same ype o cells, one could de i e he penal y coefficien as 𝜅=1 +𝛿∕(1 +
𝛽).
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Table 1
Model pa ame e s. All pa ame e s a e dimensionless excep he in ec ion and lysis a es wi h
in e se ime dimensions. HV: helpe i us; DVG: de ec i e i al genomes.
Pa ame e Desc ip ion Range
𝐵Numbe o pa icles eleased a e cell lysis (bu s -size) >0
𝜂Numbe o HVs p oduced pe cell >0
𝛽F ac ion o DVGs p oduced pe HV due o e oneous eplica ion [0,1]
𝜅Replica ion penal y o HV in cells coin ec ed wi h DVGs >1
𝛿Replica ion ad an age o DVGs >1
𝜄In ec ion a e >0
𝛼Vi us in ec ion-induced cell dea h a e >0
𝑚Mul iplici y o in ec ion (MOI) >0
Fo he sake o cla i y, he p e ious p ocesses a e ep esen ed s oichiome ically wi h he se o eac ions
𝐶+𝑉𝜄
⟶𝐶𝑉,(1)
𝐶+𝐷𝜄
⟶𝐶𝐷,(2)
𝐶𝑉+𝐷𝜄
⟶𝐶𝐷𝑉 ,(3)
𝐶𝐷+𝑉𝜄
⟶𝐶𝐷𝑉 ,(4)
𝐶𝑉
𝛼
⟶𝜂𝑉 +𝜂𝛽 𝐷, (5)
𝐶𝐷𝑉
𝛼
⟶
𝜂
𝜅𝑉+𝜂(𝛽+𝛿)
𝜅𝐷. (6)
F om hese eac ions, and using he law o mass ac ion, he ollowing se o au onomous o dina y diffe en ial equa ions (ODEs) is
de i ed:
𝐶=−𝜄𝐶(𝑉+𝐷),(7)
𝐶𝑉=𝜄𝐶𝑉 −𝐶𝑉(𝜄𝐷+𝛼),(8)
𝐶𝐷=𝜄(𝐶𝐷−𝐶𝐷𝑉),(9)
𝐶𝐷𝑉 =𝜄(𝐶𝐷𝑉+𝐶𝑉𝐷)−𝛼𝐶
𝐷𝑉 ,(10)
𝑉=𝛼𝜂(𝐶𝑉+
𝐶𝐷𝑉
𝜅)−𝜄𝑉(𝐶+𝐶𝐷),(11)
𝐷=𝛼𝛽𝜂(𝐶𝑉+
𝐶𝐷𝑉
𝜅)+𝛼𝛿𝜂𝐶𝐷𝑉
𝜅−𝜄𝐷(𝐶+𝐶𝑉).(12)
He e, 𝛼, 𝛽, 𝛿, and 𝜄a e independen pa ame e s, while 𝜂and 𝜅a e de i ed om he s oichiome ic ela ions h ough
𝜂=𝐵
1+𝛽and 𝜅=1+ 𝛿
1+𝛽.(13)
No ice ha he eplica ion ad an age o he DVG e sus i s HV, 𝛿, appea s in he e m 𝛼𝛿𝜂𝐶
𝐷𝑉 ∕𝜅. We assume 𝛽∈[0, 1], 𝛿>1, and
𝐵, 𝜄, 𝛼>0(see Table 1). The assump ion 𝛽∈[0, 1] elies on he biological implausibili y o 𝛽>1, since DVGs a ise due o dele ions
o he HV genomes. Hence, 𝛽can be in e p e ed as he ac ion o DVGs p oduced om HV du ing cell in ec ion. As we men ioned
abo e, he model assumes ha he amplifica ion o bo h HV and DVGs domina e o e hei deg ada ion wi hin he expe imen al
ime-scales. This assump ion is g ounded in he expe imen al measu es o accumula ion and deg ada ion a es o HCoV-OC43 done
in a single passage by Hillung e al. [64]. As i can be seen om his sys em, he scaling pe o med o he s a e a iables o mally
affec s only he exp ession o he pa ame e 𝜄. These da a show ha du ing he fi s 60 hou s pos -inocula ion (hpi), he effec o
deg ada ion was negligible compa ed o ha o p oduc ion. Only a e exhaus ion o p oduc i e cells, deg ada ion becomes ele an ,
hough he a e o deg ada ion was s ill 78.14% slowe han he a e o p oduc ion [64]. In any case, he possible consequences and
impac on dynamics o elaxing his assump ion a e discussed below.
4. Resul s
In he ollowing sec ions, we compu e he equilib ia o sys em Eqs. (7)-(12)and discuss hei s abili y. Phase diag ams o ele an
biological pa ame e s a e nume ically ob ained.2Then, he basins o a ac ion o equilib ia a e nume ically compu ed (Sec ion 4.1).
2Nume ical in eg a ions o he ODEs ha e been pe o med using he Runge-Ku a-Fehlbe g-Simó me hod o 7𝑡ℎ-8𝑡ℎ o de wi h au oma ic s ep size con ol and local
ela i e ole ance 10−13 .
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Sec ion 4.2 p o ides u he esul s, including hose cases wi h he deg ada ion o i al pa icles. Sec ion 4.3 explo es he pa icula
case o mul iplici y o in ec ion (MOI) >1whe e mos cells ge in ec ed simul aneously. Nex , Sec ion 4.4 con ains a sensi i i y
analysis o he solu ions wi h espec o pa ame e s and o ini ial condi ions. Las bu no leas , in Sec ion 4.5 he expe imen al ime
se ies o he HV dynamics in cell cul u es gene a ed by Hillung e al. [64]ha e been fi ed wi h he ma hema ical model using
a ificial in elligence.
4.1. Planes o equilib ia, s abili y and basins o a ac ion
As i is commonly done in Dynamical Sys ems Theo y, he s udy o he dynamics o sys em (7)-(12)is fi s based on he compu a ion
o i s equilib ium solu ions and he analysis o hei local s abili y.
Lemma 1 (The o igin and i s local s abili y). The o igin =(0, 0, 0, 0, 0, 0), he ull ex inc ion o all he popula ions, is always an equilib ium
poin o he sys em (7)-(12). Mo eo e , o any alue o he pa ame e s, he eigen alues o i s jacobian ma ix a he o igin a e 0(wi h
mul iplici y 4and semisimple) and −𝛼(double and semisimple as well).
Besides he o igin, o he equilib ia a e ound and discussed in he ollowing p oposi ion.
P oposi ion 1 (Planes o equilib ia). Sys em (7)–(12)has h ee planes o med by equilib ium poin s (i.e., all he poin s o ming he planes
a e fixed by he dynamics) ha we label as Π𝐶𝐶
𝐷
, Π𝐶𝐷𝐷, and Π𝑉𝐷. The o igin i ially belongs o all o hese planes o equilib ia, bu i
does no sha e hei biological in e p e a ion. Because o his, when we e e o hese planes he o igin will no be conside ed a pa o hem,
bu s udied aside. These planes in ol e diffe en biological equilib ium scena ios and a e defined as ollows:
(a) Pe sis ence only o non-in ec ed cells and de ec i e i al genomes (DVG)-in ec ed cells:
Π𝐶𝐶
𝐷
={(𝐶,𝐶
𝑉,𝐶
𝐷,𝐶
𝐷𝑉 ,𝑉,𝐷)=(𝐶,0,𝐶
𝐷,0,0,0) |||𝐶,𝐶
𝐷≥0,(𝐶,𝐶
𝐷)≠(0,0)}
The spec um o he jacobian ma ix a hese poin s inside he planes is
{0 (double, semisimple),−𝛼,−𝜄𝐶,𝜀±}(14)
whe e
𝜀±=−
𝜄(𝐶+𝐶𝐷)+𝛼
2±1
2√4𝛼𝐶𝜂𝜄 +(𝜄(𝐶+𝐶𝐷)−𝛼)2
+4𝛼𝐶𝐷𝜂𝜄
𝜅.(15)
Fo any alue o he pa ame e s, he disc iminan o (15)is always non-nega i e, and so all he eigen alues, o any poin in Π𝐶𝐶
𝐷
,
a e eal. This means ha Π𝐶𝐶
𝐷
is a so-called no mally hype bolic in a ian mani old (NHIM). This plane in ol es no popula ion o HV
(ei he ee o inside cells) o double-in ec ed cells, i.e., a si ua ion o sel -cu ing d i en by he quick and efficien ou compe i ion o he
HV by he DVGs. Non-in ec ed cells s ill emain in he sys em.
(b) Pe sis ence only o DVGs and DVG-in ec ed cells:
Π𝐶𝐷𝐷={(𝐶,𝐶
𝑉,𝐶
𝐷,𝐶
𝐷𝑉 ,𝑉,𝐷)=(0,0,𝐶
𝐷,0,0,𝐷)|||𝐶𝐷,𝐷≥0,(𝐶𝐷,𝐷)≠(0,0)}.
The spec um o he jacobian a any o i s poin s is
{0 (double, semisimple),−𝜄𝐷, −(𝜄𝐷 +𝛼),𝜆±},(16)
whe e
𝜆±=−
𝜄𝐶𝐷+𝛼
2±1
2√(𝜄𝐶
𝐷−𝛼)2+4𝛼𝐶𝐷𝜂𝜄
𝜅.(17)
As in he case abo e, he disc iminan is also non-nega i e o any choice o he pa ame e s. Hence, all he eigen alues a e eal and, hus,
Π𝐶𝐷𝐷is also a NHIM. This plane also in ol es sel -cu ing since, asymp o ically, no HV a e ound as DVGs ha e s eadily and efficien ly
displaced hem om he sys em.
(c) Pe sis ence only o ee HV and DVGs:
Π𝑉𝐷 ={(𝐶,𝐶
𝑉,𝐶
𝐷,𝐶
𝐷𝑉 ,𝑉,𝐷)=(0,0,0,0,𝑉,𝐷)|||𝑉,𝐷≥0,(𝑉,𝐷)≠(0,0)},(18)
in ol ing no cells and only HVs and DVGs in he medium. The spec um o he Jacobian a hese poin s is gi en by
{0,0,−𝜄𝑉 ,−𝛼,−𝜄(𝑉+𝐷),−(𝜄𝐷 +𝛼)} (19)
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The wo (semisimple) 0-eigen alues come om he ac ha , o any (𝑉, 𝐷) he poin (0, 0, 0, 0, 𝑉, 𝐷)is an equilib ium poin . All he
eigen alues a e non-posi i e (and semisimple) and so Π𝑉𝐷 is locally a ac ing o any (𝑉, 𝐷) ≠(0, 0). This plane ep esen s he mos
common ou come in which HV ends up killing all cells and DVGs a e una oidably p esen as byp oduc s o HV eplica ion.
I is no difficul o check ha he e a e no o he equilib ium solu ions ou om he o igin and hese h ee planes.
Le us discuss in mo e de ail he local s abili y o he equilib ium poin s con ained in he planes. Rega ding hose on Π𝐶𝐷𝐷, no ice
ha hei local s abili y depends on he sign o he eigen alue 𝜆+( he es a e all nega i e, excep hose 0coming om he ac ha
any poin on i wi h a bi a y 𝐶𝐷, 𝐷is also an equilib ium). Hence, he exp ession
𝜆+≥0⇔𝜄𝐶𝐷+𝛼≤√(𝜄𝐶𝐷−𝛼)2+4𝛼𝐶𝐷𝜂𝜄
𝜅
can be squa ed (no spu ious solu ions a e in oduced since he disc iminan is always non-nega i e), and i leads o
(𝜄𝐶𝐷+𝛼)2−(𝜄𝐶𝐷−𝛼)2≤4𝛼𝐶𝐷𝜂𝜄
𝜅⇔4𝜄𝐶𝐷𝛼≤4𝛼𝐶𝐷𝜂𝜄
𝜅⇔𝜅≤𝜂⇔1+ 𝛿
1+𝛽≤𝐵
1+𝛽.
Tha is,
𝜆+≥0⇔𝐵≥1+𝛽+𝛿. (20)
So, i 𝐵>1 +𝛽+𝛿all he poin s on Π𝐶𝐷𝐷a e uns able and i 𝐵<1 +𝛽+𝛿, hen hey a e all locally a ac ing.
Conce ning he equilib ium poin s o ming Π𝐶𝐶
𝐷
we ha e, like in he p e ious case, ha hei s abili y depends only on he sign
o he eigen alue 𝜀+. Thus, squa ing again, we ob ain
𝜀+≥0⇔4𝛼𝐶𝜂𝜄 +(𝜄(𝐶+𝐶𝐷)−𝛼)2
+4𝛼𝐶𝐷𝜂𝜄
𝜅≥(𝜄(𝐶+𝐶𝐷)+𝛼)2
⇔4𝛼𝐶𝜂𝜄 +4𝛼𝐶𝐷𝜂𝜄
𝜅≥(𝜄(𝐶+𝐶𝐷)+𝛼)2
−(𝜄(𝐶+𝐶𝐷)−𝛼)2
⇔4𝜄𝛼(𝐶+𝐶𝐷)≤4𝛼𝐶𝜂𝜄 +4𝛼𝐶𝐷𝜂𝜄
𝜅⇔(𝜂−1)𝐶≥(1− 𝜂
𝜅)𝐶𝐷.
Since
𝜂−1= 𝐵
1+𝛽−1,1− 𝜂
𝜅=1− 𝐵
1+𝛽+𝛿,
i ollows ha
𝜀+≥0⇔(𝐵
1+𝛽−1
)𝐶≥(1− 𝐵
1+𝛽+𝛿)𝐶𝐷.(21)
F om condi ions (20) and (21) h ee possible cases a ise (summa ized in Table 2): (i) 𝐵>1 +𝛽+𝛿; (ii) 1 +𝛽<𝐵<1 +𝛽+𝛿; and
(iii) 0 <𝐵<1 +𝛽. I is wo h ema king ha 𝐵is one o he pa ame e s ha can be be e in e ed expe imen ally by coun ing PFUs
as desc ibed in Sec ion 4.3. The plane o equilib ia Π𝑉𝐷 ( aking ou he o igin) is locally a ac ing in all h ee cases, so he ollowing
discussion will conce n only he planes o equilib ia Π𝐶𝐶
𝐷
and Π𝐶𝐷𝐷.
(i)Case 𝑩>𝟏+𝜷+𝜹. On one hand, his case in ol es 𝜆+>0, and so all he equilib ium poin s o ming Π𝐶𝐷𝐷a e, simul aneously,
uns able. On he o he , 𝐵>1 +𝛽+𝛿>1 +𝛽and so
𝐵
1+𝛽−1>0and 1− 𝐵
1+𝛽+𝛿<0.
This implies condi ion (21)and so all he equilib ia o ming Π𝐶𝐶
𝐷
become uns able. Biologically, his is he mos commonly
expec ed si ua ion: 𝐵con ains bo h HV and DVGs; he g ea e 𝛽+𝛿, he mo e DVGs a e p oduced a he expense o HV p oduc ion.
(ii)Case 𝟏+𝜷<𝑩<𝟏+𝜷+𝜹. F om (20)i ollows ha 𝜆+<0and, consequen ly, all he poin s cons i u ing Π𝐶𝐷𝐷a e locally
a ac ing. Mo eo e , one has ha
𝐵
1+𝛽−1>0and 1− 𝐵
1+𝛽+𝛿>0.
Thus, om exp ession (21), i de i es ha he poin s (𝐶, 0, 𝐶𝐷, 0, 0, 0) ∈Π
𝐶𝐶
𝐷
a e:
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Table 2
S abili y analyses: a ac ing planes in e ms o he alues o he pa ame e s 𝐵,
𝛽, and 𝛿.
Case Locally a ac ing equilib ium poin s
𝐵>1+𝛽+𝛿Π𝑉𝐷
1+𝛽<𝐵<1+𝛽+𝛿Π𝑉𝐷 ,poin sinΠ𝐶𝐶
𝐷
such ha (22)holdsandΠ𝐶𝐷𝐷
0<𝐵<1+𝛽Π𝑉𝐷 ,Π𝐶𝐶
𝐷
and Π𝐶𝐷𝐷
(iia) locally a ac ing i hey belong o he hal -plane
𝐶
𝐶𝐷
<
1− 𝐵
1+𝛽+𝛿
𝐵
1+𝛽−1
,(22)
(iib)o uns able (p ecisely, a saddle) i hey all in
𝐶
𝐶𝐷
>
1− 𝐵
1+𝛽+𝛿
𝐵
1+𝛽−1
.(23)
No ice ha condi ion (22) can be equi alen ly w i en as
𝐶<1− 𝜂
𝜅
𝜂−1𝐶𝐷,
wi h 1 <𝜂<𝜅. Since 𝐶, 𝐶𝐷a e posi i e, any choice (𝜂0, 𝜅0)inside he open sec o bounded by he lines 𝜂=1and 𝜂=𝜅(wi h
𝜂, 𝜅>1) gi es ise o a egion o (𝐶, 𝐶𝐷) ∈(0, +∞) ×(0, +∞) o a ac ing equilib ium poin s. Such egions a y om he oid
si ua ion ( o 𝜂=𝜅), wi h no poin inside, o he one whe e all he poin s a e a ac o s ( o 𝜂→1+). In he ex eme cases whe e
𝜂=𝜅, he p oduc ion o i us pe cell would be compensa ed by he penal y ha he HV ecei es by in ec ing a 𝐶𝐷cell.
(iii)Case 𝟎<𝑩<𝟏+𝜷. Clea ly, 𝐵<1 +𝛽+𝛿and he e o e all he equilib ium poin s o ming Π𝐶𝐷𝐷a e locally a ac ing. Besides,
one has ha 𝜀+in equa ion (15)is s ic ly nega i e and, consequen ly, all he poin s in Π𝐶𝐶
𝐷
a e also locally a ac ing.
Rema k 1. F om P oposi ion 1and i s local s abili y analysis abo e, he ollowing s a emen s should be highligh ed:
1. Recall ha he planes Π𝑉𝐷 , Π𝐶𝐶
𝐷
, and Π𝐶𝐷𝐷a e highly degene a e in he sense ha hey a e o med by equilib ium poin s. This
ac implies he exis ence o a couple o ze o eigen alues o he jacobian ma ix a any o he equilib ium poin s.
2. The equilib ium poin s (0, 0, 0, 0, 𝑉, 𝐷)cons i u ing he plane Π𝑉𝐷 a e all locally a ac ing o any alue o he pa ame e s and
o any alue o 𝑉and 𝐷. Π𝑉𝐷 is a NHIM (a ac ing in his case).
3. The local s abili y o he equilib ium poin s (𝐶, 0, 𝐶𝐷, 0, 0, 0) ∈Π
𝐶𝐶
𝐷
depends on he sign o i s eigen alue 𝜀+=𝜀+(𝐶, 𝐶𝐷). In
pa icula , as seen in (ii) abo e, he e exis s a line on Π𝐶𝐶
𝐷
, namely
𝐶
𝐶𝐷
=
1− 𝐵
1+𝛽+𝛿
𝐵
1+𝛽−1
,
sepa a ing hose poin s ha a e s able om hose ha a e uns able.
4. The local s abili y o he equilib ium poin s (0, 0, 𝐶𝐷, 0, 0, 𝐷) ∈Π
𝐶𝐷𝐷depends only on he alue o 𝐶𝐷. P ecisely, i is a ac ing
i and only i i s eigen alue 𝜆+=𝜆+(𝐶𝐷) <0 ⇔𝐵<1 +𝛽+𝛿, a condi ion ha elies only on he pa ame e s o he sys em.
Mo eo e , one has ha 𝜆+(𝐶𝐷) =𝜀+(0, 𝐶𝐷).
5. Since all he eigen alues o all he jacobian ma ices a ound he equilib ium poin s a e eal, he bi u ca ions (in s abili y) ha
hese poin s unde go a e always o ansc i ical ype, ha is, gi en by a change in he sign o a eal eigen alue.
The esul s shown in Table 2lead o some mul is abili y scena ios ha can be nume ically analysed in e ms o he pa ame e s.
Fo example, le us conside he o bi s o sys em (7)-(12)wi h ini ial condi ions
(𝐶(0),𝐶
𝑉(0),𝐶
𝐷(0),𝐶
𝐷𝑉 (0),𝑉(0),𝐷(0)) = (1,0,0,0,𝑚𝑞𝑉0,𝑚(1 − 𝑞𝑉0)),(24)
and whe e
𝑚=𝑉(0) + 𝐷(0)
𝐶(0) =𝑉(0) + 𝐷(0)
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Fig. 2. Phase diag ams in he plane (𝑞𝑉0, 𝐵). The (app oxima e) 𝜔-limi s a e nume ically ob ained o an o bi wi h ini ial condi ions (1, 0, 0, 0, 𝑚𝑞𝑉0, 𝑚(1 −𝑞𝑉0)) in
e ms o 𝐵and he a io 𝑞𝑉0=𝑉(0)∕(𝑉(0) +𝐷(0). Diffe en colou s show he 𝜔-limi gi en by: Π𝑉𝐷 (blue), Π𝐶𝐷𝐷(oli e), and Π𝐶𝐶
𝐷
(black). An equilib ium is assumed
o be eached when ‖𝐹‖2<10−12 , whe e 𝐹is he ec o field o sys em (7)-(12). Dashed lines co espond o 𝐵=1 +𝛽and 𝐵=1 +𝛽+𝛿. The esul s a e displayed
o diffe en alues o 𝑚: om le o igh 𝑚 =0.01, 𝑚 =0.1, 𝑚 =1, and 𝑚 =10. In all he panels, we ha e used 𝛽=0.2, 𝛿=2, and 𝜄∕𝛼=10.
Fig. 3. (a) Same as in Fig. 2bu fixing 𝐵=1.5, 𝛽=0.2, 𝜄∕𝛼=10, and uning 𝛿. (b) Phase diag ams in he plane (𝑞𝑉0, 𝛿) o fixed 𝑚 =0.5and diffe en alues o 𝐵:
om le o igh and op o bo om 𝐵=1.1, 𝐵=1.5, 𝐵=2and 𝐵=3. The o he pa ame e s a e he same as in (a). (c) Examples o ajec o ies wi h a iables la ge
han ze o a he h ee planes o equilib ia. Nume ical alues o he diffe en g aphs: (i) (𝑞𝑉0, 𝑚) =(0.85, 0.1); (ii) (𝑞𝑉0, 𝑚) =(0.85, 0.5); (iii) (𝑞𝑉0, 𝑚) =(0.45, 0.1); (i )
(𝑞𝑉0, 𝑚) =(0.45, 0.5); ( ) (𝑞𝑉0, 𝑚) =(0.15, 0.1); ( i) (𝑞𝑉0, 𝑚) =(0.15, 0.5). The labels (i),...,( i) o he ime se ies (a he bo om) co espond o loca ions in he a ac ion
basin diag ams ( op). O he examples o ajec o ies o all he a iables can be ound a Fig. A1 in Appendix A.
is a fixed MOI. Obse e ha , in his sense, he pa ame e 𝑞𝑉0p o ides he i us p opo ion wi h his MOI, ha is, he a io 𝑉(0)∕(𝑉(0) +
𝐷(0)). Fo any choice
(𝑞𝑉0,𝐵), we compu e he app oxima e 𝜔-limi 3o i s co esponding o bi and assign i a colou depending on
he plane o equilib ia (Π𝑉𝐷, Π𝐶𝐷𝐷, o Π𝐶𝐶
𝐷
) i achie es. This s udy leads o diffe en diag ams in e ms o he alues o 𝑚. Some
o hem ha e been depic ed in Fig. 2. Diag ams compu ed unde changes in 𝛽o a ound a 10% −30% p o ide quali a i ely simila
esul s o he ones in Fig. 2. On he con a y, an inc ease in he alue o 𝛿( om 1.02 o 5in hese examples) exhibi s subs an ial
g ow h o he egion wi h final poin s on he plane Π𝐶𝐶
𝐷
( esul s no shown).
Simila explo a ions can be ca ied ou by se ing he alues o 𝛽and 𝐵and s udying he beha iou when 𝛿≥1 a ies. As a
sample, some o hese plo s and diag ams a e depic ed in Fig. 3 o fixed alues o he pa ame e s and a ying 𝛿.
I is no ewo hy o analyse he e olu ion o he basins o a ac ion o he planes Π𝐶𝐶
𝐷
, Π𝑉𝐷, and Π𝐶𝐷𝐷o Table 2and hei
biological in e p e a ion in e ms o 𝐵, 1 +𝛽, and 1 +𝛽+𝛿. Fo ins ance, he case 𝐵<1 +𝛽+𝛿exhibi s is abili y, in which sel -
3Recall ha , oughly speaking, he 𝜔-limi o an o bi 𝜓(𝑡)can be defined as 𝜔(𝜓) = lim𝑡→+∞ 𝜓(𝑡).
Applied Ma hema ical Modelling 137 (2025) 115673
16
J.C. Muñoz-Sánchez, J.T. Láza o, J. Hillung e al.
Thus, adding bo h con ibu ions, one ob ains
𝑉 =𝑉(𝑖)
+𝑉(𝑟)
=𝜂(𝐶0−𝐷0)+ 𝜂
𝜅𝐷0+(𝑉0−𝐶0)
𝐷 =𝐷(𝑖)
+𝐷(𝑟)
=𝛽𝜂(𝐶0−𝐷0)+ 𝜂
𝜅(𝛽+𝛿)𝐷0,
and i s sum
𝑉 +𝐷 =(𝑉0−𝐶0)+(1+𝛽)𝜂(𝐶0−𝐷0)+ 1+𝛽+𝛿
𝜅𝜂𝐷0=𝑉0+𝐶0(𝐵−1
),
whe e i has been aken in o accoun ha 𝐵=(1 +𝛽)𝜂and ha
1+𝛽+𝛿
𝜅𝜂=(1+𝛽)𝜂=𝐵.
4.4. Sensi i i y analysis
A c ucial piece o in o ma ion ha ma hema ical models p o ide, beyond p edic ions and quali a i e in e p e a ions, is o ge in-
sigh s in o he ele ance o hei pa ame e s de e mining o affec ing i s dynamics (sensi i i y). I is well known in Dynamical Sys ems
Theo y ha his in o ma ion can be ob ained om he so-called a ia ional equa ions. In i s gene al o m, hey supply in o ma ion
abou he dependence o any pa icula solu ion wi h espec o i s ini ial condi ions. Bu besides, a ia ional equa ions wi h espec
o pa ame e s p o ide knowledge on he sensi i i y o such solu ions wi h ega d o hese pa ame e s. These wo complemen a y
sou ces o in o ma ion can be bo h compu ed using a simila p ocedu e. Le us fi s in oduce he s udy ega ding pa ame e s and
show, a e wa ds, he one co esponding o he ini ial condi ions.
A p ac ical way o ge ing a ia ional equa ions in he fi s case is o diffe en ia e Eqs. (7)-(12)wi h espec o he chosen pa ame e ,
o commu e such de i a i e wi h he one wi h espec o he ime a iable 𝑡, and o sol e he esul ing diffe en ial equa ion. As an illus-
a i e example, we de ail he case o he a ia ional wi h espec o he pa ame e 𝛽. Le us deno e by (𝐶(𝑡; 𝛽0), 𝐶𝑉(𝑡; 𝛽0), … , 𝐷(𝑡; 𝛽0))
he solu ion o equa ions (7)-(12)wi h ini ial condi ion (𝐶0, … , 𝐷0). Le now (𝐶(𝑡; 𝛽), 𝐶𝑉(𝑡; 𝛽), … , 𝐷(𝑡; 𝛽)) be he solu ion o he same
Cauchy p oblem bu o a pa ame e 𝛽close o 𝛽0. We wonde how close he solu ion (𝐶(𝑡; 𝛽), … , 𝐷(𝑡; 𝛽)) will e ol e (in ime) wi h
espec o (𝐶(𝑡; 𝛽0), … , 𝐷(𝑡; 𝛽0)). To do i , we compu e he a ia ional equa ion wi h espec o 𝛽along (𝐶(𝑡; 𝛽0), … , 𝐷(𝑡; 𝛽0)), gi en
by he ollowing sys em o ODEs:
𝑑
𝑑𝑡 (𝜕𝐶
𝜕𝛽 )=𝜕
𝜕𝛽
𝐶=−𝜄(𝑉+𝐷)𝜕𝐶
𝜕𝛽 −𝜄𝐶 𝜕𝑉
𝜕𝛽 −𝜄𝐶 𝜕𝐷
𝜕𝛽 ,
𝑑
𝑑𝑡 (𝜕𝐶𝑉
𝜕𝛽 )=𝜕
𝜕𝛽
𝐶𝑉=𝜄𝑉 𝜕𝐶
𝜕𝛽 −(𝜄𝐷 +𝛼)
𝜕𝐶𝑉
𝜕𝛽 +𝜄𝐶 𝜕𝑉
𝜕𝛽 −𝜄𝐶𝑉
𝜕𝐷
𝜕𝛽 ,
𝑑
𝑑𝑡 (𝜕𝐶𝐷
𝜕𝛽 )=𝜕
𝜕𝛽
𝐶𝐷=𝜄𝐷 𝜕𝐶
𝜕𝛽 −𝜄𝑉 𝜕𝐶𝐷
𝜕𝛽 −𝜄𝐶𝐷
𝜕𝑉
𝜕𝛽 +𝜄𝐶 𝜕𝐷
𝜕𝛽 ,
𝑑
𝑑𝑡 (𝜕𝐶𝐷𝑉
𝜕𝛽 )=𝜕
𝜕𝛽
𝐶𝐷𝑉 =𝜄𝐷 𝜕𝐶𝑉
𝜕𝛽 +𝜄𝑉 𝜕𝐶𝐷
𝜕𝛽 −𝛼𝜕𝐶𝐷𝑉
𝜕𝛽 +𝜄𝐶𝐷
𝜕𝑉
𝜕𝛽 +𝜄𝐶𝑉
𝜕𝐷
𝜕𝛽 ,
𝑑
𝑑𝑡 (𝜕𝑉
𝜕𝛽 )=𝜕
𝜕𝛽
𝑉=(𝛼𝜕𝜂
𝜕𝛽 (𝐶𝑉+
𝐶𝐷𝑉
𝜅)−𝛼𝜂
𝜅2
𝜕𝜅
𝜕𝛽 𝐶𝐷𝑉 )
−𝜄𝑉 𝜕𝐶
𝜕𝛽 +𝛼𝜂𝜕𝐶𝑉
𝜕𝛽 −𝜄𝑉 𝜕𝐶𝐷
𝜕𝛽 +𝛼𝜂
𝜅
𝜕𝐶𝐷𝑉
𝜕𝛽 −𝜄(𝐶+𝐶𝐷)𝜕𝑉
𝜕𝛽 ,
𝑑
𝑑𝑡 (𝜕𝐷
𝜕𝛽 )=𝜕
𝜕𝛽
𝐷=(𝛼(𝜂+𝛽𝜕𝜂
𝜕𝛽 )(𝐶𝑉+
𝐶𝐷𝑉
𝜅)−𝛼𝛽𝜂
𝜅2𝐶𝐷𝑉
𝜕𝜅
𝜕𝛽 +𝛼𝛽 𝜕𝜂
𝜕𝛽
𝐶𝐷𝑉
𝜅−𝛼𝛽𝜂
𝜅2𝐶𝐷𝑉
𝜕𝜅
𝜕𝛽 )
−𝜄𝐷 𝜕𝐶
𝜕𝛽 +(𝛼𝛽𝜂 −𝜄𝐷)
𝜕𝐶𝑉
𝜕𝛽 +1
𝜅(𝛼𝛽𝜂 +𝛼𝛽𝜂)
𝜕𝐶𝐷𝑉
𝜕𝛽 −𝜄(𝐶+𝐶𝑉)𝜕𝐷
𝜕𝛽 ,
wi h
𝜕𝜂
𝜕𝛽 =− 𝐵
(1 + 𝛽)2,𝜕𝜅
𝜕𝛽 =− 𝛿
(1 + 𝛽)2,
and whe e he a iables 𝐶, 𝐶𝑉, … , 𝐷a e e alua ed a (𝑡; 𝛽0)and 𝛽=𝛽0. The ini ial condi ions o his sys em a e
𝜕𝐶
𝜕𝛽 (0) =
𝜕𝐶𝑉
𝜕𝛽 (0) =
𝜕𝐶𝐷
𝜕𝛽 (0) =
𝜕𝐶𝐷𝑉
𝜕𝛽 (0) = 𝜕𝑉
𝜕𝛽 (0) = 𝜕𝐷
𝜕𝛽 (0) = 0.
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Fig. 7. Time solu ions o he a ia ional equa ions wi h espec o pa ame e s 𝛽, 𝛿, and 𝜄∕𝛼( op) and ini ial condi ions 𝐶(0), 𝑉(0), 𝐷(0) (bo om) o high mul iplici y
o in ec ion (MOI) (𝑚 = 100) a ound he solu ion plo ed in Fig. 6wi h no deg ada ion and alues 𝑞𝑉0=0.75, 𝐵= 500, 𝛽=10
−6, 𝛿=10, and 𝜄∕𝛼=0.1.
The solu ions 𝜕𝐶(𝑡)∕𝜕𝛽, 𝜕𝐶𝐷(𝑡)∕𝜕𝛽, ... 𝜕𝐷(𝑡)∕𝜕𝛽 o his sys em p o ide he (fi s o de ) ime e olu ion o he a ia ion o he a iables
𝐶, 𝐶𝑉, … , 𝐷 o 𝛽close o 𝛽0. Thus, o ins ance, in he case o 𝑉(𝑡; 𝛽)we ha e
𝑉(𝑡;𝛽)=𝑉(𝑡;𝛽0)+ 𝜕𝑉 (𝑡;𝛽0)
𝜕𝛽 (𝛽−𝛽0)+((𝛽−𝛽0)2),
and he e o e, o small alues o |𝛽−𝛽0|, we ge ha
𝑉(𝑡;𝛽)∼𝑉(𝑡;𝛽0)+ 𝜕𝑉 (𝑡;𝛽0)
𝜕𝛽 (𝛽−𝛽0),
whe e
𝜕𝑉
𝜕𝛽 (𝑡; 𝛽0)has been ob ained sol ing he a ia ional equa ion abo e.
La ge alues (posi i e o nega i e) o 𝜕𝑉 (𝑡; 𝛽0)∕𝜕𝛽 a a ime, say, 𝑡 =𝑡∗would co espond o significan a ia ion (g ow h o
decay) in he alue o 𝑉(𝑡; 𝛽)wi h espec o 𝑉(𝑡; 𝛽0) o 𝛽∼𝛽0. A simila sensi i i y analysis can be pe o med o he es o he
s a e a iables and wi h espec o he pa ame e s 𝐵, 𝛿, and 𝜄∕𝛼. Fu he mo e, since he expe imen al da a measu emen s a e educed
o alues o 𝐶, 𝑉, and 𝐷, we ocus ou a en ion only on he a ia ion o hese a iables wi h espec o he selec ed pa ame e s. An
illus a i e example o he solu ion o hese a ia ional equa ions (wi h espec o he pa ame e s 𝐵, 𝛽, 𝛿, and 𝜄∕𝛼) can be ound in
Fig. 7. On one hand, he a ia ionals o 𝑉a e always in he nega i e domain, sugges ing ha inc eases in h ee pa ame e s always
ansla e in educ ions o HV accumula ion. On he o he , he a ia ionals o 𝐷a e always in he posi i e domain, indica ing ha
inc eases in he magni ude o any o he h ee pa ame e s esul in mo e accumula ion o DVGs a he cos o he HV. In e es ingly,
he a ia ional equa ions wi h espec o 𝛿show a small ange o alues, hus sugges ing a weak dependence o he dynamics on
he ad an age DVGs ha e on HV in e ms o eplica ion. In sha p con as , he dependence on 𝛽is s ong and e en s onge in
he case o 𝜄∕𝛼. Toge he , hese obse a ions suppo he idea ha DVGs mos ly gain ad an age by in e e ing wi h he i us ( ia
𝛿) and, in e es ingly, educing he efficiency by which ee i us esul s in i us-p oducing in ec ed cells (see (34)and discussion
he ein).
Va ia ional analysis can also be ex ended o he ini ial condi ions o he sys em. The de i a ion o he a ia ional equa ions o a
solu ion wi h espec o i s ini ial condi ions is de e ed o a b ie explana ion in Sec ion Bin he Appendix. As in he e, ou analysis
only ocused on he a iables 𝐶, 𝑉, and 𝐷. Fig. 7illus a es he effec o 𝐶(0), 𝑉(0) and 𝐷(0) on he esul o he p ocess. He e, he
si ua ion is mo e complex han in he case o pa ame e s. The numbe o a ailable cells a he beginning o he in ec ion has a s ong
posi i e effec bo h in he 𝑉and 𝐷, al hough he accumula ion o DVGs is ema kable mo e sensi i e o 𝐶(0) han he accumula ion
o HVs. In con as , he effec o he s a ing numbe o HVs and o DVGs has mino effec s on he ou come o he p ocess. As expec ed,
s a ing wi h mo e 𝑉pa icles sligh ly benefi s he HV and weakly penalizes he DVGs. This has o be unde s ood in a si ua ion in
which 𝑚 = 100 and 𝑞𝑉0=0.75. Likewise, inc easing he numbe o 𝐷pa icles in he inoculum sligh ly penalizes he HV in he same
small magni ude ha a ou s DVGs accumula ion.
4.5. Model fi ing o HCoV-OC43 da a: pa ame e s’ es ima ion
To assess he accu acy o he model in ep oducing eal ime-se ies da a and o es ima e model pa ame e s, we ha e fi ed he
model using a ificial in elligence o he da a epo ed in Hillung e al. [15], shown in Fig. 1b. The aim o such wo k was o compa e
Applied Ma hema ical Modelling 137 (2025) 115673
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J.C. Muñoz-Sánchez, J.T. Láza o, J. Hillung e al.
Table 3
Ranges o he pa ame e s o he model explo ed by he gene ic al-
go i hm.
Pa ame e
𝐵∈[10
1,104]
𝛽∈[10
−8,100]
𝛿∈[1,200]
𝛼∈[10
−3,101]h−1
𝜄∈[10
−5,100]h−1
𝛾∈[0.01,0.10] h−1
𝑉0∈{[10
5,107] o 𝑚𝑉=1.8∥[10
6,108] o 𝑚𝑉=3.8}
𝐷0∈{[1,107] o 𝑚𝑉=1.8∥[1,108] o 𝑚𝑉=3.8}
he decay o he HCoV-OC43 in cell cul u e lysa es and in esh media. Fo his eason he expe imen was conduc ed o se e al
hou s a e cellula ex inc ion, including deg ada ion p ocesses in ou model. A gene ic algo i hm (GA) has been used o es ima e he
ec o o pa ame e s (𝐵, 𝛽, 𝛿, 𝛼, 𝜄, 𝛾), and he ini ial condi ions 𝑉0, 𝐷0be e fi ing he expe imen al da a. The size o he popula ion
o ec o s o pa ame e s was fixed a 600, and he numbe o gene a ions was se o 104. The explo a ion o he pa ame e space was
efined h ough successi e sea ches, and he pa ame e ’s ange cons ained by biological and expe imen al conside a ions. Wi hin
each gene a ion, he 5% o he pa ame e s wi h he lowes alues (see below) we e designa ed as he eli e popula ion, emaining
cons an o he nex gene a ion. The nex -bes 80% unde wen pa ame e c osso e , while he emaining 15%, ep esen ing he leas
a ou able pa ame e s, expe ienced andom mu a ions. The pa ame e space was subdi ided in o a loga i hmic scale o ensu e a
mo e balanced explo a ion, gi en ha he in e als spanned se e al o de s o magni ude. Excep ions we e made o 𝛿and 𝛾, whe e
linea scale sea ches we e pe o med due o he na u e o hei in e als. Conce ning he deg ada ion a e, 𝛾, a na ow in e al was
chosen a ound he epo ed alues a [15]. The explo ed pa ame e anges a e lis ed in he Table 3.
The cos unc ion used o compu e he diffe ences be ween he expe imen al and he simula ed da a is defined as ollows:
=log(𝑁
∑
𝑖=1 (log(𝑉𝑖⋅
𝑉−1
𝑖))2
+𝜎(log
4
∑
𝑗=1
𝐶𝑗(𝑡= 65);𝑘, 𝑥∗,𝑝))(39)
wi h
𝜎(𝑥;𝑘,𝑥∗,𝑝)= 𝑝
1+𝑒−𝑘(𝑥−𝑥∗).(40)
He e, {𝑡𝑖, 𝑉𝑖}𝑁
𝑖=1 ep esen he expe imen al da a and
𝑉𝑖deno es he alues o he i al popula ion ob ained wi h he ma hema ical
model o a gi en se o pa ame e s. Addi ionally,
∑4
𝑗
𝐶𝑗(𝑡 = 65) co esponds o he o al numbe o cells a 𝑡 =65hpi, a ime poin
es ima ed o ha e a low emaining cell coun . The unc ion is o mula ed as he loga i hm o a 𝜒2- unc ion o he loga i hm o he
expe imen al da a, augmen ed by a penal y e m. The penal y e m akes he o m o a sigmoidal unc ion wi h adjus able pa ame e s
and is inco po a ed o accoun o he app oxima e ime o cell dea h on he cul u e, enhancing alignmen wi h he expe imen al
da ase . The pa ame e s o he penal y e m ha e been empi ically adjus ed o guide he GA in selec ing pa ame e amilies whe e
he nume ical in eg a ion o he model significan ly educes he cell coun a 𝑡 =65hpi (wi h 𝑘 =5, 𝑥∗=−1, 𝑝 =10). Ex e nal
es ima es o he cell coun o e ime would enable he emo al o his penal y e m. To assess he obus ness o he final esul , i
was decided o conduc fi e iden ical ba ches, he eby ob aining fi e dis inc popula ions o op imized pa ame e s. While pe o ming
his p ocedu e, we obse ed ha a e 104gene a ions, he pa ame e s o he eli e popula ion we e nea ly iden ical. Consequen ly,
we op ed o ex ac he pa ame e ec o s om each ba ch and compu e he mean and s anda d de ia ion ac oss ba ches.
Fo he da ase ob ained om inocula ion a i al MOI = 1.8, he es ima ed pa ame e s ha bes fi ed he expe imen al da a
we e ound o be 𝐵=78, 𝛽=10
−8, 𝛿=1, 𝛼=0.1827 h−1, 𝜄 =0.0027 h−1 and 𝛾=0.0154 h−1, wi h co esponding ini ial condi ions
𝑉0=1, 516, 364 and 𝐷0=1. Simila ly, o he ones ob ained a i al MOI = 3.8, he es ima ed pa ame e s yielding he op imal
ep oduc ion o expe imen al da a we e iden ified as 𝐵= 523, 𝛽=10
−8, 𝛿= 200, 𝛼=0.1898 h−1, 𝜄 =0.0006 h−1 and 𝛾=0.0361 h−1,
wi h co esponding ini ial condi ions 𝑉0=3, 676, 350 and 𝐷0=1. The s anda d de ia ion o he op imal pa ame e s in bo h cases
was less han 0.005% in all cases. The fi ings o he ma hema ical model o he expe imen al da a using he pa ame e s ob ained
wi h he GA a e displayed in Fig. 8.
The op imal combina ion o bo h da ase s, as e iden om he esul ing op imal combina ion, shows ha 𝛽and 𝛿do no find an
op imal alue wi hin he sea ch ange. Howe e , examining he alue o 𝛿, i is no ewo hy ha unde such simila expe imen al
condi ions, hei esul s a e ema kably diffe en . Consequen ly, we decided o analyze how he unc ion beha ed while keeping he
emaining pa ame e s fixed and a ying bo h 𝛽and 𝛿(Appendix Fig. C1). As a esul , we ound ha he unc ion ha dly changed i s
alue wi hin he pa ame e sea ch ange o 𝛽and 𝛿. This obse a ion a ises because one o he pa ame e combina ions op imizing
is he absence DVGs (no e ha 𝐷0does no also find an op imal alue wi hin he sea ch ange). In he nea ly DVGs- ee scena io
(small 𝛽and 𝐷0), he alue o 𝛿becomes i ele an .
These esul s unde sco e he flexibili y o he model and he necessi y o acqui ing addi ional da a o he fi ing p ocess o a
igo ous in e p e a ion o he pa ame e s. Measu es ha could p o e help ul would include, o ins ance, coun ing o al cell numbe s
o e ime using li e imaging echniques and easing apa in ec ed om non-in ec ed cells by using i uses agged wi h fluo escen
Applied Ma hema ical Modelling 137 (2025) 115673
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J.C. Muñoz-Sánchez, J.T. Láza o, J. Hillung e al.
Fig. 8. Expe imen al da a (ci cles) and he co esponding fi ing o he ma hema ical model (solid lines) using he pa ame e s alues op imised by he gene ic algo i hm.
We show he cases o inocula ion conduc ed a i al mul iplici y o in ec ion (MOI) 1.8(blue) and 3.8( ed).
p o eins and li ing om dead cells by using dead-specific dyes such as ypan blue o p opidium iodide. Las ly, o p ecisely es ima e
he amoun o DVGs, i would be essen ial o ex ac o al RNA om he cul u es a a ious ime poin s, subjec ing hem o RNA-seq
analysis ollowed by bioin o ma ic analyses using ools such as DVGfinde [67] o iden i y diffe en DVGs and es ima o s as desc ibed
in [63] o quan i y hei abundance.
5. Discussion
De ec i e i al genomes (DVGs) ha e been o en iewed as eplica ion byp oduc s, s emming om cell cul u e passaging con-
di ions. Howe e , a enewed in e es in his ac ion o he o al i al popula ion sugges s ha hei exis ence migh ha e mo e
unc ional o biological significance han p e iously hough [4]. Gi en he no ably high ecombina ion a es seen in some i uses,
including be aco ona i uses, i is plausible ha DVGs could influence he adap i e e olu ion o he iable i us popula ion [13]. The
ques ion o whe he DVGs con e o no a selec i e ad an age o he i al popula ion and i he condi ions o high mul iplici y o
in ec ion (MOI) o localized co-in ec ion hold biological ele ance beyond cell cul u e emains o be explo ed.
Se ial passage expe imen s se e o many diffe en applica ions in Vi ology, no only s udying in i o e olu ion [68]. The wo
mos common applica ions a e (i) he main enance and amplifica ion o i al s ocks in he labo a o y and (ii) he classic way o
p oducing a enua ed li e accines by adap ing he a ge i us o cells om a hos as diffe en as possible om he one o be
immunised. DVGs ha e been ound in li e-a enua ed accines o polio, measles and influenza i uses ( e iewed in [4]). Howe e ,
hei impac on he de elopmen o p o ec i e immuni y and accine efficacy has no been o mally e alua ed. Gi en hei po en ial
o in e e e wi h and s imula e he immune sys em, he e is specula ion ha DVGs could imp o e accine efficiency while ensu ing
i us sa e y by limi ing i s eplica ion and sp ead. I his hypo hesis holds ue, i becomes c ucial o ca e ully con ol he amoun
o DVGs in accine p epa a ions o p e en comple e in e e ence and a significan educ ion in he i us’ effec i eness. Ou esul s
sugges ha he numbe o DVGs gene a ed om helpe i us (HV) eplica ion (𝛽) and he eplica i e ad an age o DVGs (𝛿) a e he
wo mo e ele an pa ame e s o be manipula ed in o de o op imize he a io HV:DVGs in e ms o accine efficiency.
Mul i ude o ma hema ical models o HV-DVGs ha e been in es iga ed a diffe en scales [32,38,40–48]. In e es ingly, hese
models ha e ep oduced expe imen ally obse ed dynamics such as de e minis ic chaos [32]and sel -cu ing [46,49], which in ol es
he simul aneous ex inc ion o he HV and de ec i e in e e ing pa icles (DIPs), esul ha was obse ed in i o by Jacobson e al.
[50]and S auffe Thompson and Yin [46]. Ou model also iden ifies combina ions o pa ame e s in which sel -cu ing a ises: la ge 𝛿
alues would in u n allow o he ela ionship 1 +𝛽+𝛿>𝐵 o be ulfilled, shi ing om he scena io o a single a ac ing plane Π𝑉𝐷
o a mo e complex scena io wi h h ee a ac ing planes and is abili y, including he planes Π𝐶𝐶
𝐷
and Π𝐶𝐷𝐷(being hese planes
degene a e NHIMs) ep esen ing i us- ee sel -cu ing solu ions.
Model p edic ions a e as alid and ealis ic as he assump ions om which hey build up. Ou fi s s ong assump ion, al eady
discussed a la ge, was ha a he ime scales o he e olu ion expe imen s, i us p oduc ion o e -weigh s i us deg ada ion. This
assump ion is well suppo ed by expe imen al da a (see [64]and e e ences he ein as well as [65]). Ou model was designed o
shed ligh on he dynamics o be aco ona i us popula ions du ing se ial passages and hus could no be applied o a much mo e
complex in i o si ua ion. Fo example, we a e collapsing all possible DVGs in o a single ca ego y, while in eali y, i al popula ions
con ain a la ge ac ion o e y di e se DVGs ha a e in a dynamic equilib ium, wi h some appea ing and disappea ing a e e y
ansmission e en and o he s pe sis ing o long pe iods o ime [15–17,20]. Ou app oach aligns wi h he cu en ma hema ical
models men ioned in he p e ious pa ag aph ha ypically ocus on a single dominan DVG. Fu he de elopmen s o inco po a e he
po en ial coope a ion and compe i ion among mul iple ypes o DVGs a e needed.
Ou model is simila o hose p oposed by F ank [45]and by Liang e al. [48]in he sense ha hey inco po a e 𝑉, 𝐷, 𝐶, 𝐶𝐷,
𝐶𝐷𝑉 , and 𝐶𝑉as s a e a iables. Howe e , he e a e wo majo diffe ences. Fi s ly, hei models conside a spa ial dis ibu ion o cells
and hus use eac ion-diffusion sys ems. Secondly, hey inco po a e an addi ional ca ego y o cells, namely 𝐶∗
𝑉, conside ed as cells
ha we e in ec ed by 𝑉some ime ago, becoming 𝐶𝑉, and a e al eady p oducing HV bu canno be supe in ec ed by DVGs. In his
case, he a e o 𝐶∗
𝑉is o die ou a e some ime p oducing only 𝑉. This makes a no able diffe ence wi h ou model, as we op ed o
Applied Ma hema ical Modelling 137 (2025) 115673
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J.C. Muñoz-Sánchez, J.T. Láza o, J. Hillung e al.
simplici y by supp essing his ca ego y as he modula ion o he quo ien 𝜄∕𝛼embeds he scena io in which 𝐶𝑉lysa es p e ious o
supe in ec ion wi h 𝐷(Appendix Fig. D2). Al hough we ecognize ha supe in ec ion exclusion [69]migh be a ele an p ocess o
many i uses, i is no a uni e sal one, and no e idence o such p ocess ha e been epo ed o co ona i uses.
The model p esen ed he e p o ides wo new esul s in heo e ical Vi ology: (i) HV-DVGs eplica ion and in ec ion dynamics
go e ned by quasineu al planes (degene a e 2D NHIMs o med by equilib ium poin s in his case); (ii) scena io o is abili y o med
by hese degene a e NHIMs. Conce ning poin (i) and, as a as we know, quasineu al dynamics in RNA i us dynamics ha e been up
o now ound in degene a e one-dimensional objec s [61]. Hence, ou findings ex end his esul on degene a e mani olds o planes.
Conce ning poin (ii), hese men ioned one-dimensional mani olds a e usually ound in scena ios o monos abili y and bis abili y. We
he e p o ide an example whe e h ee diffe en quasineu al planes can be achie ed depending on ini ial condi ions. The exis ence o
hese degene a e planes is subjec o non spon aneous deg ada ion o i al pa icles and cells (𝛾=0), scena io ha may seem unlikely
in a eal expe imen . Howe e , expe imen al es ima es ha e indica ed ha deg ada ion emains e y low and ha he dynamics
wi hin he ime scale used in he expe imen s is domina ed by i us eplica ion o e deg ada ion [64]. Hence, we conjec u e ha he
long ime delays a ising wi h pa ame e alues close o he ones a which hese degene a e NHIMs a e ound [62]could be obse ed
in eal expe imen s. Mo eo e , he passage expe imen s showed ex emely la ge fluc ua ions be ween passages (some o hem o
abou 1-2 o de s o magni ude), as we show in Fig. 1b (see [15] o u he de ails). These ex emely la ge fluc ua ions could be due
o he combined effec o s ochas ic sampling be ween passages and he dynamics ied o degene a e NHIMs. Unde his scena io,
diffe en ini ial condi ions (s ochas ically a ying a each passage) may gi e place o diffe en s a iona y alues o HV and DVGs due
o influence o NHIMs o o hei emnan s o hose cases wi h some (and small) con ibu ion o deg ada ion.
In conclusion, he ma hema ical model s udied in his manusc ip fi s well he dynamics o i us accumula ion wi hin single-
passages i.e., wi hin- and be ween-cell dynamics in a cell cul u e, in cell cul u es o be aco ona i uses. Ou pa ame e sensi i i y
analysis also sugges s ha he mos ele an pa ame e s o explain he obse ed dynamical pa e ns a e he p oduc ion o DVGs
om he HV (𝛽) and he in ec ion o lysa e a es (𝜄∕𝛼). The combina ion o expe imen s and biologically inspi ed modelling p o ides
a powe ul ool o iden i y pa ame e s o be op imized o u u e de elopmen s o DVGs as he apeu ic in e e ing pa icles o as
adju an s in li e a enua ed accines.
CRediT au ho ship con ibu ion s a emen
Juan C. Muñoz-Sánchez: Concep ualiza ion, Da a cu a ion, Fo mal analysis, In es iga ion, So wa e, Visualiza ion, W i ing –
o iginal d a , W i ing – e iew & edi ing. J. Tomás Láza o: Concep ualiza ion, Da a cu a ion, Fo mal analysis, In es iga ion, Me hod-
ology, So wa e, Visualiza ion, W i ing – o iginal d a , W i ing – e iew & edi ing. Julia Hillung: Da a cu a ion, Me hodology,
Resou ces, W i ing – e iew & edi ing. Ma ía J. Olmo-Uceda: Concep ualiza ion, Da a cu a ion, So wa e, W i ing – e iew & edi -
ing. Josep Sa danyés: Concep ualiza ion, Da a cu a ion, Fo mal analysis, Me hodology, So wa e, Visualiza ion, W i ing – o iginal
d a , W i ing – e iew & edi ing. San iago F. Elena: Concep ualiza ion, Da a cu a ion, Fo mal analysis, Funding acquisi ion, Me hod-
ology, P ojec adminis a ion, Resou ces, Supe ision, W i ing – o iginal d a , W i ing – e iew & edi ing.
Decla a ion o compe ing in e es
The au ho s decla e ha hey ha e no known compe ing financial in e es s o pe sonal ela ionships ha could ha e appea ed o
influence he wo k epo ed in his pape .
Da a a ailabili y
Da a will be made a ailable on eques .
Acknowledgemen s
We hank José A. O eo o aluable sugges ions and c i ical eading o he manusc ip . JCM has been unded by g an
ACIF/2021/296 (Gene ali a Valenciana). MJO was unded by con ac FPU19/05246 by MCIU/AEI/ 10.13039/501100011033 and
“ESF in es s in you u u e”. JTL has been unded by he p ojec s PGC2018-098676-B-100 and PID2021-122954NB-I00 unded by
MCIU/AEI/10.13039/501100011033/ and “ERDF a way o making Eu ope”, and by he g an “Ayudas pa a la Recualificación del
Sis ema Uni e si a io Español 2021-2023”. JTL also hanks he Labo a o io Sub e áneo de Can anc, he I2SysBio and he Ins i u de
Ma héma iques de Jussieu-Pa is Ri e Gauche (So bonne Uni e si é) o hei hospi ali y as hos ing ins i u ions o his g an . We also
hank he MCIU/AEI/10.13039/501100011033/, h ough he Ma ía de Maez u P og am o Uni s o Excellence in R&D (CEX2020-
001084-M) and CERCA P og amme/Gene ali a de Ca alunya o ins i u ional suppo . JS has been also suppo ed by he Ramón y
Cajal g an RYC-2017-22243 unded by MCIU/AEI/10.13039/501100011033 and “ESF in es s in you u u e”. SFE was suppo ed by
CSIC PTI Salud Global g an 202020E153 and by g an s SGL2021-03-009 and SGL2021-03-052 om Eu opean Union Nex Gene a-
ion EU/PRTR h ough he CSIC Global Heal h Pla o m es ablished by EU Council Regula ion 2020/2094. Many compu a ions we e
pe o med on he HPC clus e Ga na xa a I2SysBio (CSIC-UV). JCMS, JTL, MJOU, and SFE acknowledge he suppo o he San a Fe
Ins i u e, whe e pa o his esea ch was de eloped.
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J.C. Muñoz-Sánchez, J.T. Láza o, J. Hillung e al.
Appendix A. Time-se ies examples o he 𝚷𝑽𝑫 , 𝚷𝑪
𝑫𝑫and 𝚷𝑪𝑪
𝑫
Fig. A1. Time se ies o ex inc ions o cells (end s a e a Π𝑉𝐷 ) and o i al in ec i e pa icles (IP) (end s a e a Π𝐶𝐷𝐷) and double ex inc ion (end s a e Π𝐶𝐶
𝐷
) Pa ame e s:
𝐵=10, 𝛽=0.5and 𝛿=20and 𝜄∕𝛼= 100. Ini ial condi ions: (a) 𝑉0=10
−3 and 𝐷0=10
−3; (b) 𝑉0=10
−3 and 𝐷0=0.9; and (c) 𝑉0=10
−3 and 𝐷0=0.5. 𝐶0=1.
Appendix B. Fi s o de a ia ional equa ion wi h espec o ini ial condi ions
In his sec ion we p esen a well-known de i a ion o he so-called fi s a ia ional equa ion wi h espec o ini ial condi ions. To
his end, le us conside he ollowing Cauchy p oblem:
𝑥 =𝑓(𝑥),𝑥(0) = 𝑥0,𝑥∈ℝ𝑛,=𝑑
𝑑𝑡,(B.1)
which, wi hou loss o gene ali y, we can assume au onomous and sa is ying all he equi ed condi ions o smoo hness and de i abili y,
ensu ing he exis ence and uniqueness o solu ions. We deno e by 𝑥(𝑡; 𝑥0)i s unique solu ion. We a e in e es ed in he compu a ion
o he fi s -o de (in 𝜀) a ia ion o he solu ion 𝑥(𝑡; 𝑥0+𝜀)o he Cauchy p oblem
𝑥 =𝑓(𝑥),𝑥(0) = 𝑥0+𝜀, (B.2)
whe e, abusing o no a ion, 𝑥0+𝜀 =𝑥0+𝜀Id, Id being he 𝑛-dimensional iden i y ma ix. I 𝜀is small enough, we can w i e, using
Taylo ,
𝑥(𝑡;𝑥0+𝜀)=𝑥(𝑡;𝑥0)+𝜀𝜕𝑥(𝑡;𝑥0)
𝜕𝑥0
+(𝜀2).(B.3)
Diffe en ia ing his exp ession wi h espec o 𝑡we ge
𝑥(𝑡;𝑥0+𝜀)= 𝑥(𝑡;𝑥0)+𝜀𝑑
𝑑𝑡 (𝜕𝑥(𝑡;𝑥0)
𝜕𝑥0)+(𝜀2).(B.4)
On he o he side, using ha 𝑥(𝑡; 𝑥0+𝜀)is a solu ion o (B.2)and using Taylo again, we ha e ha
𝑥(𝑡;𝑥0+𝜀)=𝑓(𝑥(𝑡;𝑥0+𝜀)) = 𝑓(𝑥(𝑡;𝑥0)) + 𝜀𝜕𝑓(𝑥(𝑡;𝑥0))
𝜕𝑥0
+(𝜀2)
=𝑓(𝑥(𝑡;𝑥0)) + 𝜀𝐷𝑓(𝑥(𝑡;𝑥0)) 𝜕𝑥(𝑡;𝑥0)
𝜕𝑥0
+(𝜀2),
whe e 𝐷𝑓 deno es he diffe en ial ma ix o 𝑓wi h espec o 𝑥 =(𝑥1, 𝑥2, … , 𝑥𝑛). Equa ing e ms o o de 𝜀1in he la e exp ession
wi h hose in (B.4)i u ns ou ha 𝜕𝑥(𝑡; 𝑥0)∕𝜕𝑥0sa isfies he ODE
𝑑
𝑑𝑡 (𝜕𝑥(𝑡;𝑥0)
𝜕𝑥0)=𝐷𝑓(𝑥(𝑡;𝑥0)) 𝜕𝑥(𝑡;𝑥0)
𝜕𝑥0
.(B.5)
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Rega ding i s ini ial condi ion, we know ha (subs i u ing 𝑡 =0in exp ession (B.3))
𝑥(0;𝑥0+𝜀)=𝑥(0; 𝑥0)+𝜀𝜕𝑥(0; 𝑥0)
𝜕𝑥0
+(𝜀2)⇔𝑥0+𝜀=𝑥0+𝜀𝜕𝑥(0;𝑥0)
𝜕𝑥0
+(𝜀2),
om whe e, equa ing again powe s in 𝜀, i ollows ha
𝜕𝑥(0;𝑥0)
𝜕𝑥0
=Id,(B.6)
whe e Id is he iden i y ma ix. Equa ions (B.5) and (B.6)a e usually called fi s a ia ional equa ions a ound a solu ion 𝑥(𝑡; 𝑥0).
They p o ide in o ma ion abou he local dynamics ( angen ial and no mal) a ound a solu ion 𝑥(𝑡; 𝑥0)o (B.1). This fi s a ia ional
equa ion is linea and homogeneous. Recu en ly, one can compu e he a ia ional equa ions o highe o de (in 𝜀). All o hem a e
also linea bu non-homogeneous.
Appendix C. Addi ional fi ings esul s
Fig. C1. (a) Mean (solid line) and minimum (do ed line) alue o a each gene a ion o da a om inocula ion a i al mul iplici y o in ec ion (MOI) 1.8(blue) and
3.8( ed). (b) Value o o diffe en (𝛽, 𝛿) combina ions being he es o he pa ame e s selec ed as op imum a he gene ic algo i hm. The same s udy has been
pe o med wi h 𝐷0=0 o bo h da ase s ob aining he same esul s.
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J.C. Muñoz-Sánchez, J.T. Láza o, J. Hillung e al.
Appendix D. Model solu ions o some 𝜾∕𝜶 alues
Fig. D2. Some nume ical in eg a ions o he sys ems o 𝐵= 100, 𝛽=0.01 and 𝛿=10. F om le o igh he 𝜄∕𝛼quo ien is 0.01, 0.1, 1 and 10. This figu e aims o
illus a e how he esul depends on hese pa ame e s, bo h in e ms o cellula ype abundance and he helpe i us (HV) - de ec i e i al genomes (DVGs) final a io
(IP: in ec i e pa icles).
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