key: cord-0708815-uphmbcgi
authors: Sarma, Anish A.; Sarma, Aartik; Csete, Marie; Lee, Peter P.; Doyle, John C.
title: Control-theoretic immune tradeoffs explain SARS-CoV-2 virulence and transmission variation
date: 2021-04-26
journal: bioRxiv
DOI: 10.1101/2021.04.25.441372
sha: 886a47c21de71817a6e2713bde21d0ca1ed28cdc
doc_id: 708815
cord_uid: uphmbcgi

Dramatic variation in SARS-CoV-2 virulence and transmission between hosts has driven the COVID-19 pandemic. The complexity and dynamics of the immune response present a challenge to understanding variation in SARS-CoV-2 infections. To address this challenge, we apply control theory, a framework used to study complex feedback systems, to establish rigorous mathematical bounds on immune responses. Two mechanisms of SARS-CoV-2 biology are sufficient to create extreme variation between hosts: (1) a sparsely expressed host receptor and (2) potent, but not unique, suppression of interferon. The resulting model unifies disparate and unexplained features of the SARS-CoV-2 pandemic, predicts features of future viruses that threaten to cause pandemics, and identifies potential interventions.

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 SARS-CoV-2 infection (3, (8) (9) (10) . The complexity of the immune 28 response has impeded a unified 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) (12) (13) (14) , to immune 32 biology to analyze SARS-CoV-2 virulence and transmission. 33 34

We use control 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 minimizes 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 transmission 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

v is a vector of viral loads, u is the immune action, and new virus enters the system as δ. 48 AΔ and BΔ are sets of time-varying matrices describing uncertain linearized dynamics. We 49 leverage theorems guaranteeing that the best-case K always 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). We next consider the effects of immune control with innate extracellular effectors. 73

Higher α requires stronger immune responses to achieve a comparable effect on viral load 74 ( Figure 2D ). The immune response creates symptoms, which enable behavioral measures to 75 avoid infection (23). We use a highly simplified model of avoidance and isolation, emphasizing 76 the consequences of biological variation. We define transmission RCL, where w(t) is a warning 77 signal and γ(t) = exp (-pw(t) ). Initially, we take w(t) to be a scaled norm of the immune response, 78 so that symptoms promote avoidance and isolation.

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

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 

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hosts. If immune control is ideal for the host, replication 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