of 16
Control
-
theoretic immune tradeoffs explain SARS
-
CoV
-
2 virulence and
1
transmission variation
2
Anish A. Sarma
1*
, Aartik Sarma, M.D.
2
, Marie Csete M.D., Ph.D.,
3
Peter P. Lee, M.D., John C. Doyle, Ph.D.
4
1
Computation and Neural Systems, California
Institute of Technology; Pasadena, CA 91105.
5
2
Division of Pulmonology, Critical Care, Allergy, and Sleep Medicine, Department of Medicine,
6
University of California
-
San Francisco; San Francisco, CA 94143.
7
3
Department of Immuno
-
Oncology, City of Hope Compr
ehensive Cancer Center; Duarte, CA
8
91010.
9
4
Control and Dynamical Systems, California Institute of Technology; Pasadena, CA 91105.
10
* Corresponding author. Email: aasarma@caltech.edu
11
12
.
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available under a
was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made
The copyright holder for this preprint (which
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;
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doi:
bioRxiv preprint
Abstract:
Dramatic variation in SARS
-
CoV
-
2 virulence and transmission b
etween hosts has
13
driven the COVID
-
19 pandemic. The complexity and dynamics of the immune response present
14
a challenge to understanding variation in SARS
-
CoV
-
2 infections. To address this challenge, we
15
apply control theory, a framework used to study complex
feedback systems, to establish rigorous
16
mathematical bounds on immune responses. Two mechanisms of SARS
-
CoV
-
2 biology are
17
sufficient to create extreme variation between hosts: (1) a sparsely expressed host receptor and
18
(2) potent, but not unique, suppress
ion of interferon. The resulting model unifies disparate and
19
unexplained features of the SARS
-
CoV
-
2 pandemic, predicts features of future viruses that
20
threaten to cause pandemics, and identifies potential interventions.
21
22
Main Text:
23
Variations in virulence
and transmission, shorthanded as the dual puzzles of
24
asymptomatic cases and superspreaders, have made SARS
-
CoV
-
2 infection and spread difficult
25
to predict and control
(
1
4
)
. The relationship between pathogen virulence and transmission has
26
been a subject of longstanding speculation and formal study
(
5
7
)
, and continues to be debated
27
in the context of variation in SAR
S
-
CoV
-
2 infection
(
3
,
8
10
)
. The complexity of the immune
28
response has impeded a un
ified mechanistic understanding of virulence, transmission, and
29
variation, relevant to SARS
-
CoV
-
2 and future emerging viruses (
Figure 1
). Here, we extend
30
techniques from control theory, a mathematical framework that has been used to analyze
31
complex feedback systems in both engineered and biological settings
(
11
14
)
, to immune
32
biology to analyze SARS
-
CoV
-
2 virulence and transmission.
33
34
Results
35
.
CC-BY-ND 4.0 International license
available under a
was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made
The copyright holder for this preprint (which
this version posted April 26, 2021.
;
https://doi.org/10.1101/2021.04.25.441372
doi:
bioRxiv preprint
A control
-
theoretic approach to immune dynamics
36
We use cont
rol theory to uncover mechanisms that lead to variation in virulence and
37
transmission. Informally, we compute the best
-
case immune response, consolidating unmodeled
38
immune dynamics into a control function
K
(
Figure 2A
). The best
-
case immune response
39
minimi
zes virulence and implicitly suppresses transmission. We implement mechanistic details
40
as constraints on the set of realizable control functions, and in this way identify mechanisms
41
(constraints) for which even a best
-
case
K
yields virulence and transmissi
on variation. This best
-
42
case
K
bounds any immune system model that we could have used, allowing us to pose rigorous
43
questions without a detailed model of immune dynamics. Formally, we consider the robust
44
control problem:
45
[
+
1
]
=
)
[
]
[
]
+
)
[
]
[
]
+
[
]
46
[
]
=
(
[
1
:
]
)
47
v
is a vector of viral loads,
u
is the immune action, and new virus enters the system as
δ
.
48
A
Δ
and
B
Δ
are sets of time
-
varying matrices describing uncertain linearized dynamics. We
49
leverage theorems guaranteeing that the best
-
case
K
alway
s corresponds to a convex set, so that
50
the best
-
case
K
computed from the set will be the best
K
over all realizable functions
(
15
)
.
51
52
The
open
-
loop problem
53
We first consider the open
-
loop dynamics of viral replication, or equivalently
K
= 0. We
54
model viral infection in the individual host as a three
-
step process: cell entry, replication in the
55
cell, and release of virus from the cell after an eclipse period
T
e
(
16
,
17
)
.
56
+
3
4
7
57
7
{
9
:
}
<
=
훽푉
58
.
CC-BY-ND 4.0 International license
available under a
was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made
The copyright holder for this preprint (which
this version posted April 26, 2021.
;
https://doi.org/10.1101/2021.04.25.441372
doi:
bioRxiv preprint
?
59
7
?
A
60
3
?
A
61
We derive
A
in terms of
α
, the number of productively infected cells that result from a
62
single infected cell, where
(1
-
1/φ)
is the fraction of infec
ted cells that constitutively turn over in
63
a single eclipse period.
64
=
B
1
푘훼
G
H
I
0
K
65
=
푟훽
푘휙
[
3
]
66
α
s
c
a
l
e
s
l
i
ne
a
rl
y w
i
t
h [
C
s
].
A
s
m
a
l
l
fra
c
t
i
on of re
s
pi
ra
t
ory e
pi
t
he
l
i
a
l
c
e
l
l
s
a
re
s
us
c
e
pt
i
bl
e
67
t
o S
A
RS
-
CoV
-
2 w
he
n c
om
pa
re
d t
o rhi
novi
rus
, re
s
pi
ra
t
ory s
ync
yt
i
a
l
vi
rus
, a
nd i
nfl
ue
nz
a
(
18
68
22
)
. A
s
m
a
l
l
s
us
c
e
pt
i
bl
e
c
e
l
l
fra
c
t
i
on e
na
bl
e
s
l
a
rge
r
e
l
at
i
v
e
va
ri
a
t
i
on c
ons
i
s
t
e
nt
w
i
t
h re
port
e
d
69
s
i
ngl
e
-
c
e
l
l
da
t
a
(
20
,
21
)
(
F
i
gu
r
e
2B
-
C
).
70
71
T
he
c
l
os
e
d
-
l
oop pr
obl
e
m
72
W
e
ne
xt
c
ons
i
de
r t
he
e
ffe
c
t
s
of i
m
m
une
c
ont
rol
w
i
t
h i
nna
t
e
e
xt
ra
c
e
l
l
ul
a
r e
ffe
c
t
ors
.
73
H
i
ghe
r
α
re
qui
re
s
s
t
ronge
r i
m
m
une
re
s
pons
e
s
t
o a
c
hi
e
ve
a
c
om
pa
ra
bl
e
e
ffe
c
t
on vi
ra
l
l
oa
d
74
(
F
i
gu
r
e
2D
). T
he
i
m
m
une
re
s
pons
e
c
re
a
t
e
s
s
ym
pt
om
s
, w
hi
c
h e
na
bl
e
be
ha
vi
ora
l
m
e
a
s
ure
s
t
o
75
a
voi
d i
nfe
c
t
i
on
(
23
)
. W
e
us
e
a
hi
ghl
y s
i
m
pl
i
fi
e
d m
ode
l
of a
voi
da
nc
e
a
nd i
s
ol
a
t
i
on, e
m
pha
s
i
z
i
ng
76
th
e
c
ons
e
que
nc
e
s
of bi
ol
ogi
c
a
l
va
ri
a
t
i
on. W
e
de
fi
ne
t
ra
ns
m
i
s
s
i
on
R
CL
,
w
he
re
w(t
)
i
s
a
w
a
rni
ng
77
s
i
gna
l
a
nd
γ(t
)
=
e
x
p(
-
pw(t
))
. Ini
t
i
a
l
l
y, w
e
t
a
ke
w(t
)
t
o be
a
s
c
a
l
e
d norm
of t
he
i
m
m
une
re
s
pons
e
,
78
s
o t
ha
t
s
ym
pt
om
s
prom
ot
e
a
voi
da
nc
e
a
nd i
s
ol
a
t
i
on.
79
.
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available under a
was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made
The copyright holder for this preprint (which
this version posted April 26, 2021.
;
https://doi.org/10.1101/2021.04.25.441372
doi:
bioRxiv preprint
OP
=
4
R
(
)
(
)
푑푡
9
U
80
We extend this simple behavioral model to address a virus with a long presymptomatic
81
period followed by uniformly severe infection (
Figure 3A
-
B
). Advance warning and isolation
82
measures can contain such a virus. However, fully
asymptomatic cases make advance warning
83
more difficult. Fully asymptomatic cases need not be as contagious as presymptomatic
-
severe
84
cases to have this effect, and low rates of fully asymptomatic cases can be tolerated (
Figure 3C
).
85
Interferon
-
based control
varies less with
α
than extracellular responses, but interferon
-
86
suppressed control varies more. Early interventions with exogenous interferon can potentially
87
reduce the eventual symptom burden in what would otherwise be severe cases (
Figure 3D
-
E
) .
88
Taking
these control layers together, we consider virulence and transmission as
α
varies.
89
Presymptomatic
-
severe high
-
α
cases take a dominant role in spreading the pathogen, especially
90
where they interact with other high
-
α
individuals (
Figure 4
).
91
92
Discussion
93
Other
viruses
94
HCoV
-
NL63 and SARS
-
CoV
-
1 also bind to ACE2. HCoV
-
NL63 infection is
95
asymptomatic or cold
-
like
(
24
)
, while SARS
-
CoV
-
1 infection is typically severe, with some
96
reported asymptomatic cases
(
4
,
25
)
. Viral infection in these cases could be biologically variable
97
with median effects that are too mild or too severe to be evident in clinical outcomes. SARS
-
98
CoV
-
1 exhibits variable transmission, consistent with this interpretation
(
26
)
. HCoV
-
NL63’s
99
reduced virulence may result from a spike glycoprotein structure that decreases
α
(
27
)
.
100
101
Interferon signaling
102
.
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was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made
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;
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doi:
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Clinical and experi
mental studies have shown that early exogenous interferon
103
administration can reduce coronavirus infection severity
(
28
32
)
. Our results suggest that
104
presymptomatic interferon could be partic
ularly beneficial in averting severe outcomes.
105
Conversely, our results suggest a mechanism for harm from early immunosuppression in patients
106
with SARS
-
CoV
-
2, consistent with clinical trial evidence
(
33
)
. Because of the ubiquity of
107
interferon suppression strategies in respiratory viruses, studies of contro
l
-
guided interventions
108
could facilitate responses to future emerging viruses.
109
110
111
.
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was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made
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;
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doi:
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211
212
Fig. 1. A control theory framework to analyze virulence and transmission.
213
(A) Schematic time
-
series representations of symptoms for three viruses. The red and blue cases
214
have relatively low variation across hosts and acro
ss time. The purple case, schematically like
215
SARS
-
CoV
-
2, varies across hosts and across time, suggesting variation in host immune
216
responses.
217
.
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(B) An apparent tradeoff between virulence and transmission can result from host immune
218
responses. However, this tr
adeoff depends on host control mechanisms, and can be made more
219
favorable to the host population or more favorable to the virus.
220
(C) A block diagram shows the relationships between virulence and transmission control in two
221
hosts, one infected with SARS
-
Co
V
-
2 and one susceptible. The dynamics without control are
222
shown in the lower half of the diagram, and the control responses in the upper half. Immune
223
responses suppress shedding, create symptoms, and allow behavioral responses.
224
225
226
227
228
.
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229
Fig. 2. Host dynamics
shape virulence and transmission.
230
231
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(A) Within each host, well
-
characterized kinetics govern viral replication. Viruses enter cells,
232
replicate, exit after a delay, and degrade in the extracellular space. These kinetics are coupled to
233
immune responses, for
which we compute best
-
case bounds with control theory.
234
(B) A low susceptible cell percentage (SCP) in the host enables variation. The maximum fold
-
235
change deviation from the median and the effect of a small fluctuation both grow as the median
236
SCP approache
s 0%.
237
(C) An open
-
loop model removes all control elements and considers the underlying dynamics.
238
Open
-
loop variation in viral shedding varies dramatically on relevant time
-
scales, amplifying
239
variations in SCP.
240
(D) Ideal extracellular immune control can c
reate similar, low
-
variation viral load trajectories
241
between hosts, but these similar trajectories require differing immune effort. The underlying
242
open
-
loop dynamics directly shape virulence.
243
.
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244
Fig. 3. Layered control of virulence,
transmission, and variation.
245
(A) When all infected hosts are eventually symptomatic, advance warning allows isolation and
246
potentially treatment measures in pre
-
symptomatic individuals.
247
(B) Interferon
-
suppressed responses allow an extended period of
viral replication and shedding
248
during which avoidance behaviors are not possible (without advance warning). Transmission can
249
be computed from the viral and immune trajectories.
250
(C) Advance warning can reduce the effective transmission rate of presymptomati
c individuals,
251
but asymptomatic cases facilitate escape. As the rate of escape increases, the effective
252
transmission from presymptomatic
-
severe individuals increases sharply.
253
.
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(D) Timing is a crucial determinant of interferon efficacy. Presymptomatic exoge
nous interferon
254
administration can potentially reduce the eventual symptom burden in an individual who would
255
otherwise experience severe disease.
256
(E) Extracellular immune responses vary more with
α
than interferon
-
based immune responses
,
257
but interferon
-
su
ppressed responses vary most.
258
.
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259
260
Fig. 4. Virulence and transmission depend on host control strategies.
261
262
The relationship between virulence and transmission depends on control conditions in individual
263
hosts.
If
immune control is ideal for the host, repl
ication is quickly blocked by interferon and
264
neither serious symptoms nor substantial transmission occur. With interferon suppression,
265
transmission peaks at low virulence. With interferon suppression and host variation, however,
266
transmission is higher and
peaks at higher virulence. This effect is amplified when high
-
α
267
individuals interact, leading to both high presymptomatic shedding and high susceptibility.
268
.
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