key: cord-0861908-4py4snrw
authors: Lewis, Greyson R.; Marshall, Wallace F.; Jones, Barbara A.
title: Modeling the dynamics of within-host viral infection and evolution predicts quasispecies distributions and phase boundaries separating distinct classes of infections
date: 2021-12-20
journal: bioRxiv
DOI: 10.1101/2021.12.16.473030
sha: a30d5cc65949807cb77c3e4820b16a59f4971414
doc_id: 861908
cord_uid: 4py4snrw

We use computational modeling to study within-host viral infection and evolution. In our model, viruses exhibit variable binding to cells, with better infection and replication countered by a stronger immune response and a high rate of mutation. By varying host conditions (permissivity to viral entry T and immune clearance intensity A) for large numbers of cells and viruses, we study the dynamics of how viral populations evolve from initial infection to steady state and obtain a phase diagram of the range of cell and viral responses. We find three distinct replicative strategies corresponding to three physiological classes of viral infections: acute, chronic, and opportunistic. We show similarities between our findings and the behavior of real viral infections such as common flu, hepatitis, and SARS-CoV-2019. The phases associated with the three strategies are separated by a phase transition of primarily first order, in addition to a crossover region. Our simulations also reveal a wide range of physical phenomena, including metastable states, periodicity, and glassy dynamics. Lastly, our results suggest that the resolution of acute viral disease in patients whose immunity cannot be boosted can only be achieved by significant inhibition of viral infection and replication. Author summary Virus, in particular RNA viruses, often produce offspring with slightly altered genetic composition. This process occurs both across host populations and within a single host over time. Here, we study the interactions of viruses with cells inside a host over time. In our model, the viruses encounter host cell defenses characterized by two parameters: permissivity to viral entry T and immune response A). The viruses then mutate upon reproduction, eventually resulting in a distribution of related viral types termed a quasi-species distribution. Across varying host conditions (T, A), three distinct viral quasi-species types emerge over time, corresponding to three classes of viral infections: acute, chronic and opportunistic. We interpret these results in terms of real viral types such as common flu, hepatitis, and also SARS-CoV-2019. Analysis of viral of viral mutant populations over a wide range of permissivity and immunity, for large numbers of cells and viruses, reveals phase transitions that separate the three classes of viruses, both in the infection-cycle dynamics and at steady state. We believe that such a multiscale approach for the study of within-host viral infections, spanning individual proteins to collections of cells, can provide insight into developing more effective therapies for viral disease.

Introduction 1 Viral infections are ubiquitous across the tree of life. Though great progress has been 2 made in the prevention and treatment of diseases caused by viruses, a deep 3 understanding of how they infect hosts and evolve within those hosts remains elusive. 4 Of the many ways in which viruses cause disease and evade a host's immune response, 5 one common theme is a high rate of mutation during replication, leading to the 6 formation of a cloud of genetically similar viral progeny known as a quasispecies [1] [2] [3] [4] [5] . 7 Over the past four decades, a growing body of work has demonstrated experimentally 8 the existence of viral quasispecies in infections for diseases including polio, hepatitis C, 9 SARS-CoV-2, and others [6] [7] [8] [9] [10] [11] [12] [13] . Additionally, it was found that different viruses within 10 a quasispecies can exhibit a wide range of infectiousness, virulence, and replicative 11 fitness [6, 14, 15] . Complementary to these developments, the theoretical foundations of 12 quasispecies, proposed originally by Eigen, have been the subject of extensive study in 13 mathematical biology and physics, leading to exact solution methods and applications 14 ranging from B-cell receptor diversity to intra-tumor population dynamics [16] [17] [18] [19] [20] . 15 Approaches for understanding within-host viral infections based in theory and 16 computation have grown substantially in both number and complexity [21] . 17 Computational approaches range from agent-based models (e.g. [22] ), to simulation of 18 partial differential equations (e.g. [23] ), to multi-compartment hybrid models (e.g. [24] ), 19 and beyond. Although less common, approaches based on statistical mechanics have 20 previously proven effective and provided insight into both biology and physics [25] . A 21 recent study describes a model linking fitness distributions to the probability of causing 22 a pandemic [26] . Our work is built off of that by Jones et al. [27] , in which statistical 23 mechanics and thermodynamics are used to better understand viral quasispecies 24 infections within hosts at steady state. 25 In this work, we model and study the dynamics of viral quasispecies inside an 26 individual host via a 3-step process (Fig. 1) . Using a novel, exact calculation method, 27 we determine the distribution of viral subclasses ("match numbers" m) across a 28 finite-sized and self-replenishing pool of infectible host cells that have a limited viral 29 capacity. After infection, the host mounts an immune response, clearing viruses in 30 proportion to their match number. Viruses that survive immune clearance undergo 31 replication with a probability that increases with match number, inducing a 32 population-level tug-of-war via the opposing pressures of replication and the immune 33 response. Viruses that do replicate are subject to the additional pressures of mutation, 34 while those that do not will remain inside cells into the next round of infection, 35 effectively shrinking the available pool of cells that newly produced viruses can infect. 36 We allow only one mutation per virus per iteration, and our mutation matrix has a 37 natural fixed point around m = 14, a moderate value. 38 To describe a broad variety of host environments, we model host cells as exhibiting a 39 specified level of viral permissivity T : low values of T allow only viruses with the In this figure, we show the cycle composing a single iteration of our model. This cycle consists of three steps: infection, immune response, and reproduction with mutation. Viruses that reproduce and mutate from the last step form the cohort infecting cells at the beginning of the next iteration. In our model, one iteration corresponds to a single time step. space of possible host-virus interactions. The combined pressures of permissivity, 47 immunity, error-prone replication, mutational drift, and the pool of infectible cells' 48 limited size lead to a phase space with a rich variety of viral behaviors and physical 49 phenomena. We find that the T-A phase space is separated into three main non-extinct 50 regions; and that those regions correspond well to the physiological classes of acute, 51 chronic, and opportunistic infections. We also find that the boundaries in phase space 52 separating the three infection regimes range from first to infinite order, and that 53 infections near the first-order transition exhibit behavior characteristic of "glassy" 54 dynamics. Lastly, we make two predictions: that chronic viral infections are unlikely to 55 be cured through increasing immunity alone, and that the transition from acute to 56 chronic infection behavior is discontinuous across different viral subtypes. 57 Model, methods, and materials 58 Our system consists of a finite pool of cells and viruses that may infect those cells. 59 Viruses can reside either inside cells or in the environment/extracellular space. The In order to capture essential information about variation within and across viral 70 quasispecies, we describe each virus in our model with a "match number" m. The 71 match number characterizes a cell-virus interaction and can be interpreted as a measure 72 of the binding strength of an idealized viral surface protein to a simple host-cell receptor 73 protein. In our model, all host cells express a single relevant receptor type of sequence 74 length 50 with unchanging identity, while viruses exhibit a single surface protein with 75 varying composition and length 100. The optimal alignment (which may contain gaps) 76 between these two sequences sets the value of m, which ranges from 0 (minimal 77 alignment) to 50 (maximal alignment). The match number enables us to group 78 equivalent interactions among a statistically large number (in total, 26 100 ) based on an 79 extended amino acid set [27] ) of possible viral surface protein compositions. decreased infective and reproductive ability. The third pressure is mutation, as random 85 sequence changes to viral offspring can lead even the most infectious virus to have less 86 effective progeny. The fourth, an entropic pressure, is the mutationally-based limitation 87 on variety of offspring that a virus can produce -there are far more potential progeny 88 for some match numbers than others, given a finite alphabet for constructing proteins. 89 The final pressure is induced by the finite pool of cells, as new viruses must compete 90 with both each other and latent intracellular viruses for the limited number of cells to 91 infect. Throughout this work, we see the effects of the interplay between these pressures. 92 Our model differs from standard quasispecies modeling approaches in four key ways. 93 Firstly, we include a finite and self-replenishing pool of cells in which viruses may 94 remain latent across multiple infection cycles. Secondly, we use a generalized Hamming 95 class (the "match number") to group viruses with identical numbers of mutations from 96 the optimal sequence, which may not be present at the initial infection. Thirdly, we 97 include mutational backflow, as bidirectional mutation across match number classes is 98 prominent across a large range of m. Finally, we perform simulations with large 99 numbers of cells and viruses over long enough timescales to reveal multiple 100 time-dependent features and to ultimately reach steady state. 101 We now go into more detail about the individual components of the viral life cycle. 102

Infection begins with viruses emerging from the environment and hopping from one cell 104 to the next, trying to infect at each one. At each cell, the virus engages in a wide range 105 of binding poses, testing the interaction of its binding protein at each alignment. If the 106 virus infects, it hops no further, inhibiting other viruses from entering that same cell: in 107 our model we allow a maximum of one virus per cell. However, our model is statistical, 108 and therefore the distribution of viruses in cells after infection consists of two

where m is the viral match number and T is the permissivity. We plot e m in Fig. 2C . behavior are shown in Eqs. 2a-2c, and plotted for a single virus attempting to infect 120 cells of varying prior occupancy level in Fig. 2A (derivation in S1 Appendix, section 1). 121 We note that these equations are an exact representation of the process described above, 122 whereas previous work used approximate expressions to derive analytic results [27] .

The result is calculated recursively, so that the infection probability laid down by the 124 first virus, plus the viruses retained after the previous infection cycle, serve as the 125 infected cells that the next virus encounters. For example, the third virus contends with 126 the effects of the first and second viruses, plus those remaining in cells from the 127 previous infection cycle, and so forth, until all N viruses at that stage of cycle have had 128 a chance to infect. In order to calculate infection probabilities in a statistical manner 129 consistent with our model, we average over both the order in which viruses encounter 130 cells and the possible sequences in which viruses emerge. We then arrive at the 131 following recursive formulas:

Thus, for a system with N viruses, the probability of cells being occupied by viruses after infection is

Here, k m represents the net probability that a single cell will be infected by a virus 133 of match number m after the k-th virus has tried to infect. As defined above, P m is the 134 probability distribution over match numbers of viruses in the environment before the 135 infection cycle begins, c is the number of cells, and k ranges from 0 to N (the number of 136 viruses in the environment before the infection cycle begins). 137 We note that the number of viruses in the environment after replication and mutation is represented by a real number. However, Eq. 2d requires an integer number of viruses. Therefore, to determine the value of N for real N , we linearly interpolate the value of N between the two whole numbers of viruses nearest to N :

where bN c is the greatest integer less than or equal to N , and dN e is the least integer 138 greater than or equal to N .

Immune response

Our model of the immune response depends only on the match number and the system immune clearance intensity, A. The function does not change within a simulation: its static nature assumes that prior infections have created a fixed memory. Those viruses with a higher match number are those that are most likely to have infected in the past and are therefore most likely to be recognized during a new infection. We take the immunity function to have a sigmoidal shape with an inflection point ⌫ fixed at m = 6 amino acids [27] that sets the 50% immune intensity response:

We note that immune clearance is greatest for m corresponding to maximal 141 infectious and reproductive ability, setting up a dynamic evolutionary pressure between 142 T and A. The maximal strength of the immunity we set to A, with A ranging from zero 143 to one (Fig. 2B ). The probability of destroying a virus only approaches 1 with A = 1 and 144 moderate match number. Taken together, the viruses that remain after immunity are

Replication with mutation 146 If a virus is able to both infect a cell and evade the immune response, it engages in the 147 third and final stage in its life cycle: replication and mutation. We take the probability 148 to replicate to be e m , the same as the infection probability for a single empty cell, as 149 both replicative and infective ability are essential to a virus's function.

As depicted in Fig. 2C , viruses with m = 50 replicate with probability 1, but for all 151 other match numbers, the probability is less, although at no match number is it exactly 152 zero. However, for any non-infinite permissivity, the probability of replication with very 153 low match numbers is exponentially small. As e m takes on values from 0 to 1 (with 154 predictable behavior at both extremes), we can derive a natural quantitative scale for The distribution of viruses that remain latent or reproduce and escape from cells are, respectively,

Here, R is the probability that cells remain infected after viruses have a chance to reproduce, and F is the probability that a cell will have had its viruses reproduce. For viruses that do replicate, we set the fecundity (', number of new viruses produced from each progenitor virus) to 20, giving the total number of viruses present in the environment at the beginning of the next infection cycle:

Each viral offspring is mutated at exactly one position in the viral protein 159 sequence [28] . The one amino acid out of 100 on the virus that mutates is chosen at 160 random. This single change can either keep the number of matches to the cellular 161 receptor protein the same, increase it by one, or decrease it by one. Depending on the 162 match number, the probabilities for these three possibilities can shift -large match 163 numbers will have a lower chance for still higher match, and low match numbers a lower 164 chance for still lower match. These transition probabilities were estimated using 165 high-performance computing (HPC) calculations to generate the mutation matrix M 166 ( Fig. 2D ; described in S1 Appendix, section 2; originally in [27] ). The probability 167 distribution of viruses in the environment at the beginning of the next infection process 168 is, then,

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All newly replicated viruses escape from the cell into the extracellular environment 170 for the next cycle. The fixed pool of cells is replenished with uninfected cells to keep the 171 number constant at c, and the infection process starts again.

Computational implementation 173 Considered across repeated cycles of infection, immune response, and replication, the equations above can be seen as difference equations, which we evaluate through iterative calculation. We run simulations for 100,000 iterations, as this almost always guaranteed convergence in our simulations. We define convergence at the latest iteration i such that, for all following iterations, simulation. In addition, given that viral population sizes are real-valued scalars, we set 177 a extinction cutoff of 10 8 such that a simulation is terminated when the environmental 178 viral load sinks beneath that cutoff.

For this work, we run simulations across the combination of parameters shown in 180 Table 1 . We initialize simulations with either a uniform or "natural" match-number 182 distribution (Figs. 3A, B) . The uniform distribution is defined in the standard manner, 183 with each value of m assigned equal probability. The "natural" initial distribution is 184 defined as the steady-state distribution for A = 0 and T ! 1, which is also the 185 stationary eigenvector of the mutation matrix. Using these two distributions enables us 186 to compare effects of the mean and variance of the initial match-number distributions. 187 We also note that in three limits (very low T , near infinite T , and near-complete 188 cellular occupation after the immune response), we are able to solve for the steady state 189 of the system analytically, suggesting that it is unlikely that there are additional, 

We have implemented simulations in Python 3 using Numpy accelerated with Numba 197 package [29, 30] . Where referenced in the code corpus, snippets from Stack Overflow there are a small subset of cases in which we see two separated peaks arise. 208 We begin by considering steady-state results. In Fig. 4 , we plot the end results of simulating the modeling equations above (Eqs. 2a-8) 212 for 100,000 iterations. We see at steady-state that the match-number distributions 213 coalesce into quasispecies distributions, in some places bimodal. We also find that the 214 model goes to the same steady state determined by T and A, regardless of initial 215 conditions of viral load and match-number distribution (as long as viruses have not gone 216 extinct); this result is not a given for our model, as the infection process is non-linear. 217 In particular, the Natural initial condition leads to a substantially larger extinction 218 region than does the uniform initial condition, suggesting that quasispecies require more 219 viruses at higher match numbers for survival at low permissivity. We note that 220 extinction only seems to occur at low permissivity; at high enough T , no value of 221 immunity is able to completely suppress the viral population. Match number distributions at the ends of simulations for a subset of sampled permissivity and immune intensity are shown for uniform (black) and natural (dashed-red) initial distributions. For all simulations, we sample permissivity on a logarithmic scale because e m is an exponential function of permissivity (Eq. 1). Each distribution is normalized to sum to 1, so broader distribution necessarily exhibit lower peaks. For each distribution, the x-axis ranges in match number m from 0-50, and the y-axis ranges ranges in probability from 0-1. The bimodal distributions are enclosed by dark-gray dashed lines and extinction is indicated by the lack of a plotted distribution.

Inspection by eye suggests three general regions of self-consistency. Low permissivity 223 leads to distributions centered around high match numbers, regardless of immunity.

High permissivity and moderate-to-high immunity lead to distributions centered around 225 low match numbers. High permissivity and low immunity lead to distributions centered 226 around the natural distribution (3B).

At low permissivity, two effects dominate: the need for viruses to reach very high 228 match numbers in order to both infect and replicate, as well as the effectively 229 m-independent suppression of high match-number viruses by the immunity. 230 Accordingly, we observe two trends reflecting these pressures. Firstly, we see that the 231 peaks of the distributions increase toward higher m as immunity increases, with the 232 distribution peaks centered at lower m for higher permissivity. Secondly, we find that 233 the widths of the distributions decrease as immunity increases and permissivity 234 decreases. These trends are linked and reflect a strategy of the quasispecies to more 235 efficiently increase its population size to overcome increasing immunity. In particular, 236 the distribution peaks shift towards higher m and become more narrow in order to take 237 advantage of the sharp increase in replicative ability as match number increases in this 238 permissivity regime (Fig. 2C) . Eventually, this strategy fails and viruses are forced into 239 extinction at the highest levels of immunity.

In contrast, in the high-permissivity and higher immunity regime, immunity 241 dominates other pressures in the system. Here, match number distributions are centered 242 at low m, tending toward lower m as immunity increases. We remind readers that the 243 dynamic range of immunity spans from m = 0 to m ⇡ 15 (Eq. 4 and Fig. 2B ). Thus, as 244 immunity increases, viruses are able to evade immune clearance by shifting towards 245 lower and lower m. We also see that, as permissivity decreases, the peaks of match 246 number distributions shift to lower m, showing the stronger effects of immune pressure 247 over permissivity in this regime. In this regime, permissivity is high enough that viruses 248 with low m are still able to infect and reproduce, enabling viruses to pursue both sets of 249 strategies as environmental conditions change. However, as the permissivity further 250 decreases, this strategy becomes untenable as viruses at low m become unable to 251 replicate, leading to the emergence of higher m viruses at lower permissivity.

Lastly, we address the high-permissivity, low-immunity region. Here, the pressures of 253 permissivity and immunity are minimal, enabling entropic mutational pressures to 254 dominate. Accordingly, we see that match number distributions in this region closely 255 resemble the Natural initial distribution, which is defined as the stationary distribution 256 of the mutation matrix. 257 We highlight that, broadly across phase space, match number distributions tend to 258 increase in width as immunity decreases. This observation is consistent with previous 259 clinical findings that immuno-compromised or -suppressed patients exhibit greater 260 diversity of viral variants than those with stronger immune systems [13, 31, 32] . 261 We draw attention to the unexpected emergence of bimodal distributions (boxed by 262 gray dashed lines in Fig. 4 ) at T = 13.5 and A = 0.57, 0.71. Of note, the peaks of these 263 distributions appear to be centered at the locations of the unimodal peaks at higher and 264 lower T . 265 We also see two major permissivity-dependent trends that span multiple regions.

The first, at low immunity, is that distribution peaks shift smoothly to higher match 267 numbers and become more narrow as permissivity decreases. This reflects the steadily 268 growing pressure of permissivity, which increasingly allows only higher match-number 269 viruses to infect and reproduce as T decreases. The second, at moderate to high 270 immunity, is that the distribution peaks shift initially towards low m and then suddenly 271 jump to increasingly high m as permissivity decreases. We can now interpret the Given the emergence of three distinct regions, a sudden jump in behavior, and the 276 appearance of bimodal distributions, we are prompted to explore the possibility that T 277 and A define a phase space exhibiting critical behavior. Following prior work on a 278 similar model [27] , we define an order parameter:

where P m is the environmental match-number probability distribution at the end of a 280 simulation. As before, we normalize the order parameter to fall between 0 and 1. We Purple regions with white X's denote viral extinction before the end of simulations. The white line is a spline fit to locations of maximal difference in order parameter at fixed permissivity or immunity. Red squares denote points where P m is bimodal. A reminder that the natural scale for permissivity is between T ⇡ 1 and T ⇡ 70, where most changes in behavior are expected to occur.

A first-order phase transition can be defined as finite discontinuity in the order 295 parameter (e.g., the change in density upon the melting of ice into liquid water), while a 296 December 17, 2021 12/28 continuous (higher-order) phase transition occurs where the order parameter has a 297 discontinuity in a higher derivative (e.g., a ferromagnetic transition) [33] . A crossover 298 region, also known as an infinite-order phase transition, is a region in which the order 299 parameter of a system undergoes a continuous change between two qualitatively distinct 300 states [34] (e.g., the BEC-BCS transition of ultracold Fermi gases [35] ). We note that 301 our model is in the grand canonical ensemble, with an effective bath of "particles"

(possible viral sequences) drawn from during replication and returned during infection 303 and the immune response. This bath is therefore of size large enough (26 100 ⇡ 10 140 ) to 304 consider the system in the thermodynamic limit, allowing the possibility of phase 305 transitions.

First-order phase transitions in nature often exhibit the coexistence of both states 307 near their transition boundary [33] . We also see this in our results: as shown in Fig. 4 , 308 the match number distributions of viruses at the first-order phase boundary are bimodal 309 (enclosed by a dashed-gray box in Fig. 4 and red squares in Fig. 5 ), with each of the two 310 peaks located at the same position as the unimodal steady-state peaks on either side of 311 the boundary. We interpret this as the coexistence of two different quasispecies.

In addition to the first-order phase transition, there is a second transition that spans 313 moderate-to-high permissivity between low and moderate immunity, which we refer to 314 as the "vertical" phase boundary. This boundary differs in that it changes from second-315 to higher-order as permissivity increases, smoothly shifting to an infinite-order crossover 316 region as T approaches infinity. Even in that regime, the order parameter changes by 317 almost 40% across a region spanning only 0.2 in A. Hence, the vertical boundary 318 divides phase space at moderate-to-high T into two distinct and separate regions. As 319 described earlier, it marks the boundary between viral match-number distributions that 320 have order parameters more strongly determined by mutational pressure and those 321 determined by immune pressure.

In particular, we note that we do not have a critical point where the phase boundary 323 makes a bend from horizontal to vertical. We do not see a supercritical phase around 324 the cusp, as the crossover region that extends diagonally from that point to A = 0,

T ⇡ 5 appears to be continuous for all derivatives. However, we do see some unusual 326 oscillatory behavior in that region for very small A around T = 5, which we discuss 327 later.

The phase portraits for the two initial conditions exhibit one major difference: in the 329 low-permissivity region (lower half of the phase portrait), we see a dramatically smaller 330 zone of extinction (white X's) for the uniform initial distribution than for the "natural" 331 initial distribution, which lacks viruses at high m.

As noted before, the two phase portraits exhibit identical steady-state results for the 333 order parameters (and indeed, as we shall see below, viral population size) in 334 non-extinct regions. These results, when considered with similar observations for other 335 measurements shown throughout this work, give credence to the existence of a unique 336 non-zero steady-state for each permissivity-immunity pair, a seemingly emergent result 337 that we did not expect given the non-linearity of the model. Based on the results of this section, we conclude that the three phases correspond to 343 three real disease classes: "acute," "chronic," and "opportunistic."

In Fig. 6 we show the steady-state values of these measures for uniform and natural 345 initial distributions. Here, "viral load" is defined as the ratio of the number of viruses at 346 a point in phase space to the maximum allowable viral population (c' = 20, 000 in the 347 environment and c = 1, 000 in cells); the total viral load is defined as the sum of these 348 ratios and is the total viral occupation after infection and the immune response. We 349 find that both the intracellular and extracellular viral load landscapes are generally Comparing the viral loads in the environment (Fig. 6A ) and inside cells (Fig. 6C ) 354 shows that, unexpectedly, the total viral load (Fig. 6D) does not depend on either shift in viral load between cells and the environment takes place roughly around the 360 crossover region (from T ⇡ 10 to T ⇡ 20, see Fig. 6B ). As permissivity decreases, 361 making infection and replication more difficult, viruses that enter cells are more likely 362 to remain in cells rather than replication at these low levels of immunity. This has two 363 effects: fewer new viruses enter the environment at the end of each iteration, and a 364 dwindling number of cells are available to infect at the beginning of each cycle. 365 Together, these lead to a shift in viral occupation from the environment into cells.

In contrast, we see in phase I that the total viral load depends heavily on immunity. 367 In fact, there is a linear dependence of total steady-state viral load on immunity, as we 368 show analytically in the supplement (S1 Appendix, section 3). Unexpectedly, at very 369 low permissivity (T < 0.25) for the uniform initial distribution, the environmental viral 370 load (Fig. 6A) jumps from low values at A = 0 to high values at A ⇡ 0.02, then 371 decreases as immunity increases. To understand this further, at the very lowest value of 372 T = 0.1, we are able to use perturbation theory techniques to analytically predict the 373 viral loads: we predict a maximum in viral load that depends on permissivity and find 374 that the results from perturbation theory at T = 0.1 match the simulations very well 375 (error < 1%). We include a derivation and further details in the appendix (S1 Appendix, 376 section 4; Supplementary Figs. S1 Fig, S2 Fig) .

From these calculations we learn that the change in viral load at low T as immunity 378 increases is the combined result of four pressures: the finite number of infectible cells, 379 low permissivity, mutation, and immunity. 380 We explain these results semi-quantitatively here. At A = 0 and very low T , the In phase III we also see a broad region with high viral load, but in this case it is for 399 the viral load inside cells, and as a function of permissivity rather than immunity. We 400 refer specifically to the region above the horizontal phase boundary, roughly bounded by 401 T = 11 and T = 20 and A > 0.2. We explain this in the following way. Starting from 402 high permissivity, as T is lowered at fixed A, low-m viruses gradually lose the ability to 403 replicate but maintain their ability to infect (Figs. 2A, C) . This leads to large Finally, we focus on the high permissivity regime (Phase II, T > 130), where at zero 410 immunity the environmental viral load reaches its maximum, as all viruses that infect 411 are able to replicate. As immunity increases, the environmental viral load initially 412 decreases linearly, but the high level of permissivity quickly enables viruses with low 413 match number to survive and replicate regardless of immunity. The viral load in cells is 414 very low in this region because all viruses are likely to replicate, regardless of m.

In summary, our analysis of the steady-state results has led to the following findings. 416 First, there are three distinct phases of viral types, separated by phase transitions of 417 varying order and crossover regions. Secondly, the phase transitions and crossover 418 regions seen in the order parameter emerge in the population size landscapes as well.

Thirdly, the size and location of extinction regions varies with the initial conditions.

Lastly, viral extinction is more likely to occur at lower permissivity and higher 421 immunity.

These results lead us to postulate that each of the three phases corresponds to a 423 disease type affecting humans in real life. Firstly, we suggest that the class of viruses 424 below the horizontal phase boundary (Phase I), due to their persistent reproductive 425 ability at nearly all immunities, corresponds to "acute" viral infections brought on by 426 viruses that are highly adaptable (via mutations) and generally infectious, such as the 427 influenza viruses, rhinoviruses, and others, including SARS-CoV-19 [36, 37] . Also, in 428 this region, the total viral load exhibits a strong inverse dependence on immunity, 429 concluding with regions of total viral extinction for high immunity. Notably, the region 430 of extinction is larger in the case of the more lifelike natural distribution, which does 431 not contain contributions from the extremes of the match number spectrum.

Secondly, we see in Phase III, without the pressure from low permissivity, viruses 433 with low steady-state match number are able to survive while consistently evading the 434 immune response. Even if immunity is increased in this phase, we find that there is no 435 clear resolution of viral infection, consistent with the physiology of long-term, chronic 436 diseases. We thus associate this phase with "chronic" viral infections, as this behavior 437 matches well onto the set of viruses that stay primed in the body for many years, such 438 as varicella-zoster (chicken pox/shingles), hepatitis B and C, and HSV [38] [39] [40] [41] . 439 Lastly, in Phase II, we see that low immunity and high permissivity result in a high 440 environmental and total viral load. These viruses also do not go extinct, regardless of 441 initial conditions. However, even a small increase in immunity leads to a relatively large 442 decrease in viral load. As such, we associate this phase with "opportunistic" viral 443 infections with high viral loads that only generally present in immune-suppressed and 444 -compromised individuals, such as cytomegalovirus and JC virus [8, 42] . 445 Moving forward, we therefore refer to the three phases as "acute" (Phase I), 446 "opportunistic" (Phase II), and "chronic" (Phase III). We now go on to explore the Using infection dynamics data from studies of a range of viral diseases, including flu, 457 and SARS-CoV-19, we can approximate the length of a full infection and replication 458 cycle (i.e. cellular generation interval) as 0.5 days [43, 44] . Based on this approximation, 459 we find that "acute" infections resolving in total viral extinction generally correspond to 460 an illness duration of less than one month, similar to what is observed in the leads to more rapid viral clearance, as is to be expected for acute viral infections. In 463 contrast, for "chronic" infections in the upper-right region of the phase diagram, the 464 time to steady state ranges from hundreds to thousands of iterations. Using the 465 approximate cellular generation interval for HIV-1 (2.6 days) for a single iteration in 466 this region, we find that the system exhibits the months-to years-long timescale typical 467 of chronic and persistent viral infections [47, 48] . For the "opportunistic" phase, we find 468 times to steady state that are close to but shorter than those of the "chronic" region.

These results give confidence that the model reflects biologically relevant dynamics and 470 reinforces the association between different phases and different disease types. 471 We move on to other observations regarding Figs. 7A, B. In line with earlier results, 472 we see that the order parameter-derived phase boundary (white spline) overlays neatly 473 on the landscape of iterations to steady state, reinforcing the finding that nearly every 474 property of this system reflects the phase transitions.

The most marked feature of this data occurs along the horizontal phase boundary, 476 where we see a series of narrow horizontal regions exhibiting an unusually large number 477 of iterations to convergence (yellow to yellow-green stripes). These long-lived 478 simulations all exhibit bimodal match-number distributions, both at steady state 479 (Fig. 4) and during the dynamics (Figs. 7C, D) . We note that some points in these 480 regions exhibit meta-stability (see S3 Fig) lasting many iterations (tens of thousands, in 481 some cases) before finally and suddenly converging to steady state.

The metastable states explored around these values of T and A exhibit probability 483 distributions and viral loads very close to those at steady state. We can therefore view 484 the progression of these simulations as natural experiments in which the metastable 485 states serve as initial conditions that are small perturbations from steady state. Seen in 486 this way, the very long time to steady state serves as evidence of "critical slowing down," 487 a phenomenon typically seen near phase transitions in a wide variety of dynamical 488 systems [49] . 489 As mentioned, we also see, after large numbers of iterations, that metastable states 490 abruptly shift to steady state. This phenomenon is known in the context of glassy 491 dynamics as "quakes" [50] . Glassy dynamics typically involve a competition between Thus, we can see that viral extinction in this model is a dynamic process that is 507 strongly dependent on both the immunity and the initial conditions. 508 Finally, we consider the time to steady state of viruses that survive in the "acute" 509 region. At A = 0 and T ⇡ 3 5, we unexpectedly find an entire region of phase space in 510 which the steady state of the system is oscillatory (bright yellow, S4 Fig) . There, the 511 system reaches a dynamic steady state with effectively infinite convergence time: no 512 single steady-state quasispecies distribution is associated with those values of T and A. 513 Instead, the system oscillates in viral load by up to 200 viruses and in mean match 514 number by m = 1 (more details in S4 Fig) . Considered in the broader context of the 515 "acute" region, we can interpret the periodic behavior as the dramatic culmination of a 516 trend that appears throughout the entire region: the time to steady state is very low at 517 high A and low T , and progressively increases as one diagonally approaches the 518 subregion with oscillatory steady states. There, the pressures of permissivity and 519 mutation (energy and entropy) are effectively equal, resulting in a finely balanced 520 two-state system, moving slowly over time between one state and the other.

Scaled distribution dynamics 522 We now consider the dynamics of the quasispecies distributions, shown in Figs. 7C, and 523 D. The dynamics of the distributions give additional insight into two patterns in 524 steady-state results seen earlier: 1) In the low T "acute" region, the peaks of 525 steady-state distributions increase to higher m as A increases at fixed T ; 2) In the 526 higher T "chronic" region, the peaks of steady-state distributions shift toward lower m 527 as permissivity decreases at fixed A.

At very low T , in the "acute" phase, we see considerable mutation, as is 529 characteristic for many viruses associated with acute infections. The pressure of 530 permissivity dominates, preventing all but the highest match-number viruses from 531 replicating. At lower A (A < 0.5), the viral quasispecies begin at high m but are able to 532 shift over time to lower m by mutation. As immunity increases, however, viruses have 533 less and less time to shift to lower m before being eliminated by the immunity, leading 534 to steady-state distributions stuck at higher m. We note a similar effect as permissivity 535 increases at low A, where the entropic pressure of mutation is increasingly able to push 536 the virus towards m = 14, the point of highest viral sequence degeneracy (the number of 537 different ways to form a virus of match number m).

In the higher-T "chronic" region, the pressure from immunity is greater than that of 539 permissivity, enabling only low-m viruses to survive. The peaks of steady-state 540 distributions in this region shift toward lower m as T is decreased at moderate A 541 because, even with reduced infection and replicative capabilities, viruses are best able to 542 survive by evading the immune response. The dynamic behavior in this region varies 543 according to initial conditions because, while the "uniform" distribution already has 544 low-m viruses, the "natural" initial distribution does not and thus needs to evolve 545 viruses with that property. Notably, the "uniform" distribution simulations show very 546 low-m viruses initially infecting, which then mutate and shift over time to increase m 547 and reach their steady states. We see in particular near the horizontal phase boundary 548 that, although higher-m viruses infect early on, they are cleared by the immune 549 response, enabling the more evasive low-m viruses to remain in cells and replicate, 550 albeit to a limited extent. This is an analogous effect to what we see in the "acute" 551 region, where viruses with the other extreme value of m initially dominate and then 552 slowly evolve into a less extreme steady state. 553 We also note that bimodality in the distribution dynamics is much more common The overarching theme for the following results is that the phase boundary divides 569 parameter space into regions of distinct relationships between the viral loads inside and 570 outside cells for both initial conditions. We find two trends: the viral loads in the 571 environment and within cells are either correlated or anti-correlated. In the 572 "opportunistic" and "acute" regions, we observe anti-correlated behavior for both initial 573 conditions. In these regions, the total viral load reaches its steady-state value during the 574 fast timescale. Therefore, during the majority of the dynamics shown in Fig. 8, a   575 change in the environmental viral load must lead to an equal and opposite change in the 576 intracellular viral load. By contrast, we observe correlated scaling of the viral loads in the "chronic" phase. 578 In this phase, heightened immune clearance and lower match numbers slow the growth 579 of the total viral load, which can only converge at the rate of its slowest constituent 580 component. We note the remarkable scaling at high permissivity and high immunity, 581 where the viral loads within cells and in the environment grow in lockstep, in particular 582 for the natural initial condition (Fig. 8B) . Importantly, the viral load dynamics in this 583 phase show how a high total viral load can be reached in non-immunocompromised 584 humans.

Finally, we note that it is not universally true that the within-cell and environmental 586 viral loads grow monotonically throughout disease progression. Rather, we see a In this work, we have shown how a collection of viruses that infect cells, face an immune 596 response, replicate, and mutate, can lead to a system exhibiting multiple behaviors 597 corresponding to three distinct phases of a phase diagram separated by both first-order 598 and continuous phase transitions. These phases emerge in a space we construct from the 599 two main parameters of our model, permissivity (T ) and immunity (A), and exhibit 600 self-consistent properties unique to each phase.

Collectively, the pressures of permissivity, immunity, mutation, entropy, and the 602 available pool of infectable cells compete, leading to three distinct classes of viral 603 populations that correspond to the three different phases. In the high-permissivity and 604 low immunity phase (Phase II), viruses are primarily directed to steady state by entropy 605 in the form of mutational burden. In the high permissivity and moderate-to-high 606 immunity phase (Phase III), viruses are driven to steady state by an attempt to evade 607 the immune response (through decreased m), enabled by the uniformly high infection 608 rate at high permissivity. In the low permissivity phase (Phase I), infection and 609 replication are only possible at higher match number, leading to an 610 immunity-dependent push toward steady-state quasispecies distributions with 611 decreasing width and increasing match number. Taken to the extreme, the restrictions 612 of low permissivity and high immunity lead to the complete extinction of viruses, even 613 for the uniform initial distribution. 614 We link the three general classes of viruses from our model to infection types 615 affecting humans in real life: acute, chronic, and opportunistic. Firstly, we suggest that 616 the class of viruses below the horizontal phase boundary (Phase I), due to their 617 persistent replicative ability at nearly all levels of immunity, corresponds to "acute" 618 viral infections brought on by viruses that are highly adaptable and generally infectious, 619 such as the influenza viruses, rhinoviruses, and many coronaviruses [36, 37, 52, 53] . 620 Secondly, viruses at a broad range of immunity above the horizontal phase boundary 621 (Phase III) can be identified with "chronic" infections: low match numbers enable them 622 to persistently evade the immune response. Such behavior matches well onto the set of 623 viruses that stay primed in the body for months to years, such as varicella-zoster 624 (chicken pox/shingles), hepatitis B and C, and HSV [38] [39] [40] [41] . Lastly, at high permissivity 625 and low immunity (Phase II), viruses exhibit moderately low match numbers, -compromised individuals, including cytomegalovirus and JC virus [8, 42] .

Our results reinforce this interpretation in several ways. Firstly, in the "chronic" 630 phase, we find that there is no clear resolution of viral infection even if immunity is 631 increased, consistent with the physiology of long-term, chronic diseases. We contrast 632 that with the "opportunistic" phase, where even a small increase in immunity leads to a 633 relatively large decrease in viral load, as is to be expected for this type of disease.

Finally, in the "acute" phase, the level of immunity has a large effect on the rate of 635 recovery, as evidenced by the corresponding decrease in time to steady state and 636 increase in extinction probability. This reflects the typical resolution of acute viral 637 infection after the development of an immune response. 638 We can also ask what the model would suggest to drive viruses causing these 639 diseases to extinction. Firstly, we judge it more likely that an individual receives an 640 initial bolus of viruses that more closely resembles the "natural" distribution than the 641 "uniform" distribution. For the "natural" distribution, there is a large region of viral 642 extinction in our phase space. With this in mind, we can now analyze infection types. 643 For an "acute" infection, the only guaranteed paths in our model to viral extinction are 644 to decrease permissivity and/or increase immunity. However, in those patients for whom 645 an increase in immunity is not possible, our model suggests that only a drastic decrease 646 in T , likely via inhibiting general aspects of viral replication, will significantly decrease 647 viral load. For the other two types of infections, only by decreasing T to meet the 648 horizontal phase boundary can full extinction be attained. Supporting that idea, we 649 note that for patients with "chronic" diseases such as hepatitis C, increasing the level of 650 immunity is often insufficient on its own to cure the underlying disease, a result also 651 found in our model [54, 55] .

For a broader initial distribution of viruses, our model leads to a non-zero viral load 653 in steady state for all but the highest levels of immunity. Based on our model, we posit 654 that wellness may be attained without the total clearance of viruses. While high levels 655 of viral load more often correspond to severe disease, low viral loads may not lead to 656 any symptoms; indeed, prior work has shown that a non-negligible proportion of the 657 population tested positive for flu antigen but did not exhibit any symptoms [56, 57] .

In considering future work to improve the model, we suggest four high impact 659 modifications corresponding to increasing levels of complexity. Firstly, rather than 660 keeping secondary parameters constant, such as the mutation rate and fecundity, we 661 believe that it would be useful to see the effects of varying individual parameters across 662 different phases and within individual simulations, reflecting the variation in mutational 663 and replicative rates across different viruses [58, 59] . 664 Secondly, we propose implementing an immune module that is capable of actively 665 updating and expanding its memory. In its present form, our model's immune response 666 exhibits a static memory with a solely match number-dependent level of response 667 representing a fixed past history. In humans, however, the adaptive immune system 668 actively recognizes new threats, generates a response, and stores the capability to 669 respond in memory [60] . In its simplest form, adaptive immunity could be modeled by 670 dynamically changing A during a single simulation. A truly adaptive immune system 671 would likely substantially increase the size of extinction regions, as static quasispecies 672 would be hard pressed to exist for extended periods of time. Finally, we discuss modifications that can be made to inform the development of In this section we derive equations for the infection rate in our model. To start, we will 2 calculate the probability that a single virus will infect any one single cell of the c total 3 cells encountered within the host. We do this because, in this framework, each virus will 4 either infect a single cell or not infect at all. It may be the case that some proportion of 5 cells are already occupied, which will reduce the probability that a virus infects a cell. 6 The probability of a single virus infecting is 12 We can now calculate the probability that the virus infects any one cell in the pool. 13 This value is equal the sum of the probabilities that: the virus lands and infects at the 14 first cell (cell 1); that it does not infect at cell 1, but does at cell 2; that it does not 

Here we can see that the probability for a virus to infect at any one cell is equal to 18 one minus the probability that it does not infect at any cell. We note that the spatial 19 distribution of the cells has disappeared from these expressions, consistent with model. 20 Substituting in the original variables gives 

How do we account for the distribution of viruses? The probability that a virus 22 emerging from the collection of viruses in the environment of being match number m is 23 equal to P m . Once the virus has begun attempting to infect cells, the distribution has 24 no further bearing on that particular virus's ability to infect, as it already has a match 25 number m. Therefore, the probability that the first virus infects with match number m 26 is

To convert this probability of the first virus infecting a particular cell into one of 28 infecting any of the cells in the infectible pool, we divide by c to obtain the total 29 additional infection probability added by the first virus, ⇤ 1 m :

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The total probability per-cell, including the previously occupied cells, is now updated to be:

In our system, the initial cellular occupation is given by the viruses remaining in 31 cells, R :

Once the first virus has had an opportunity to infect, the second virus emerges and 33 attempts to infect. The existing level of cellular occupation that it encounters must now 34 take into account both the pre-infection probability of occupation and the probability 35 that the first virus successfully infected:

This process continues until the N -th virus has had an attempt to infect:

Thus, for a system with N viruses, the distribution of viruses that have infected cells is

We note that the number of viruses in the environment after reproduction and mutation is a real number. However, Eq. 6b requires an integer number of viruses. Therefore, to determine the value of ⇤ for real N , we linearly interpolate the value of ⇤ between the two whole numbers of viruses nearest to N : (N bN c) ) · ⇤ bN c .

2 Mutation Matrix where m is the match number and w is the degeneracy of the chosen host cell receptor 39 sequence (with value 0.7867 from previous work). 40 3 Derivation of linear dependence of total viral load 41 on immunity in Phase I 42 For nearly all simulations in this regime, the total infection probability at steady state is approximately 1 ( P m I m ⇡ 1). Also, the match-number distributions at steady-state are centered and concentrated at relatively high m (m > 30), for which the immune response is effectively constant and equal to A. Therefore,

demonstrating permittivity-independent linear scaling of total viral load with immunity. 43 Within the same phase, the viral load inside cells contrasts with that in the 44 environment by showing a super-linear and monotonic decrease from high to low value 45 as immunity increases. This can be explained as follows. As immunity increases, the 46 match number distributions shift towards higher m. At such low permittivity, the 47 probability that viruses remain inside cells (1e m ) exhibits exponential decay as a 48 function of increasing m. Therefore, as immunity increases, the probability that viruses 49 remain inside cells decreases super-linearly. 50 This result provides another explanation for the scaling behavior of the viral load in 51 the environment: subtracting a super-linearly decaying function from a linearly 52 decaying function gives a maximum in the difference. 53 4 Low-permissivity perturbation theory derivation 54 and results 55 In order to understand the limiting behavior of the model, we applied perturbation 56 theory appropriate for the low-permissivity regime to derive steady-state solutions for 57 the within-cell and environmental population sizes at steady state. 58 At sufficiently low permissivity, only viruses with m = 49 and m = 50 should make 59 non-negligible contributions to the system, so we constrain our consideration to include 60 only those types of viruses. Also, given such low permissivity, we expect the replicative 61 ability of m = 49 viruses to be negligible. In order to stay in the perturbative regime, we 62 must remain at sufficiently low permissivity such that m = 49 viruses do not reproduce, 63 as their reproduction would lead to an expansion of the variety of viruses present in this 64 system (thereby contradicting our initial assumption requirement for only two types of 65 viruses). Additionally, any viruses that remain inside cells after reproduction must have 66 match number equal to m = 49, as m = 50 viruses always reproduce with probability 1 67 and escape into the environment: thus, the entire within-cells viral population is exactly 68 the population of m = 49 viruses inside of cells. Therefore, whenever viruses replicate 69 into the environment, they must come from m = 50 progenitors in a proportion 70 determined by the mutation matrix: P (50 ! 49) ⇡ 0.39 and P (50 ! 50) ⇡ 0.61.

To derive steady state conditions at any immunity, we assume that the system is 72 near steady state and has some population inside cells (after reproduction) equal to µ.

During the next iteration, just after infection by viruses in the environment, the cells 74 will be fully occupied if there were sufficiently many infecting viruses with m = 50 75 (m = 50 viruses have a near-certain probability of infecting unoccupied cells).

To calculate the updated infection probability of the m = 49 and m = 50 viruses, we 77 need to know the ratio of their infective abilities and also have some knowledge of their 78 infective abilities as a function of the prior level of infection, µ. 79 We first note that, when µ = 1, there is no additional infection. With this in mind, Note, however, that these equations would imply that the infectibility of m = 50 and 84 m = 49 have a constant ratio for any initial level of infection µ. We know this is not 85 correct, because for µ close to 1, the ratio of µ 49 /µ 50 = b 0 /a 0 must be much less than 86 its value when µ = 0. Given that new infection by m = 49 viruses is more sensitive to 87 the prior level of infection than that by m = 50 viruses ( Fig. 2A of main text), we now 88 try the next-simplest approximation and say that:

We now impose the sum rule, which says that the new total infection by m = 49 and 90 m = 50 viruses is equal to 1: To test whether this is a reasonable approximation for infection at very low 93 permissivity, we need to choose a permissivity satisfying the aforementioned constraints 94 but is also sufficiently large that simulations at that permissivity will reach steady-state 95 in a reasonable amount of time. Accordingly, we choose a sample permissivity of 96 T = 0.1, as e 49 ⇡ 4.5 ⇤ 10 5 .

To test this, we calculate the cell occupancy just after infection for m = 49 and 98 m = 50 viruses. In particular, we assume 1000 cells and 10,000 viruses (however, the 99 calculation is insensitive so long as the number of viruses exceeds 3,000 for c = 1000).

The output of the infection process gives 

Given this result, we can now determine the self-consistent, steady-state solutions for 107 the viral populations. We take b 1 to be the value calculated for µ = 0 so as to be 108 consistent with the perturbative expansion. 109 We begin with some prior infection level µ. As determined above, the cell occupancy 110 just after infection is:

where b 1,µ=0 = 0.01423388 for T = 0.1.

Immune activity reduces the viral population as expected: As only m = 49 viruses will remain inside cells, the final within-cell viral population is:

To determine the steady-state values, we now set the final within-cell population to 114 µ, giving us a final, self-consistent equation for the within-cell population at steady 115 state in the perturbative regime:

This is a standard quadratic equation in µ with solutions:

Inspection of the roots shows that the '+' root exceeds 1, while the ' ' root falls 118 between 0 and 1, so we use the ' ' root moving forwards.

Given that the predicted within-cell population at steady-state is equal to µ ⇤ (A), the corresponding predicted environmental population at steady-state (scaled to 0-1) is:

With these results in hand, we compare our predictions to the data from our 120 simulations, as shown in Supplementary Fig. 2 . We see that there is excellent agreement 121 between the predictions and the data, except for where the simulations go extinct.

Relative percentage errors do not exceed 1% for either computed quantity. 123 The figures demonstrate that our approximation, assuming that cells are fully occupied after infection, deviates from the data at both A = 0 and A ⇡ 1. At A = 0, further analysis of the simulation dynamics shows that the system continues to evolve very slowly towards a state of full cell occupancy, at which point no further viruses from the environment can infect and reproduce, causing the environmental viral load to go to 0. At A ⇡ 1, study of the infection equations show that complete filling after infection breaks down when N ⇡ 2c. Translating this into the self-consistent steady-state equations gives a critical A ⇤ at which this happens: 

Selforganization of Matter and the Evolution of Biological Macromolecules

Molecular Quasi-Species

A Principle of Natural Self-Organization

RNA Virus Quasispecies Populations Can Suppress Vastly Superior Mutant Progeny

RNA Virus Evolution via a Fitness-Space Model

Quasispecies Diversity Determines Pathogenesis through Cooperative Interactions within a Viral Population

Hepatitis C Virus (HCV) Circulates as a Population of Different but Closely Related Genomes: Quasispecies Nature of HCV Genome Distribution

Deep-Sequence Identification and Role in Virus Replication of a JC Virus Quasispecies in Patients with Progressive Multifocal Leukoencephalopathy

Naturally Occurring SARS-CoV-2 Gene Deletions Close to the Spike S1/S2 Cleavage Site in the Viral Quasispecies of COVID19 Patients

Influenza A Virus Subpopulations and Their Implication in Pathogenesis and Vaccine Development

Evolution of Viral Quasispecies during SARS-CoV-2 Infection

SARS-CoV-2 Exhibits Intra-Host Genomic Plasticity and Low-Frequency Polymorphic Quasispecies

Quasispecies Dynamics and Molecular Evolution of Human Norovirus Capsid P Region during Chronic Infection

Poliovirus Intrahost Evolution Is Required to Overcome Tissue-Specific Innate Immune Responses

Intra-Host Diversity of SARS-Cov-2 Should Not Be Neglected: Case of the State of Victoria

Mathematical Models of Quasi-Species Theory and Exact Results for the Dynamics

Rigorous Mathematical Analysis of the Quasispecies Model: From Manfred Eigen to the Recent Developments

Coevolution of Quasispecies: B-Cell Mutation Rates Maximize Viral Error Catastrophes

Theoretical Models of Generalized Quasispecies

Genetic Instability and the Quasispecies Model

Trends in Mathematical Modeling of Host-Pathogen Interactions

Agent-Based Modeling of Host-Pathogen Systems: The Successes and Challenges

Host Modeling. Infectious Disease Modelling

Linked Within-Host and between-Host Models and Data for Infectious Diseases: A Systematic Review

The Quantitative Theory of Within-Host Viral Evolution

A Simple Model for How the Risk of Pandemics from Different Virus Families Depends on Viral and Human Traits

Statistical Mechanics and Thermodynamics of Viral Evolution

Closing the Gap: The Challenges in Converging Theoretical, Computational, Experimental and Real-Life Studies in Virus Evolution. Current Opinion in Virology

Array Programming with NumPy

A LLVM-based Python JIT Compiler

Antigenic Drift of Influenza A (H3N2) Virus in a Persistently Infected Immunocompromised Host Is Similar to That Occurring in the Community

Within-Host and Between-Host Evolution in SARS-CoV-2-New Variant's Source

Theory of First-Order Phase Transitions

Unified Approach to Crossover Phenomena

Crossover from BCS-superconductivity to Bose-condensation

Dynamics of Influenza Virus Infection and Pathology

Pathogenesis of Rhinovirus Infection. Current Opinion in Virology

Chronic Hepatitis B

Viral Hepatitis C. The Lancet

Herpes Simplex Virus Infections. The Lancet

Limits and Patterns of Cytomegalovirus Genomic Diversity in Humans

Kinetics of Influenza A Virus Infection in Humans

SARS-CoV-2 (COVID-19) by the Numbers. eLife

Table: Infectious Diseases -HealthEd

Infection Exposure Questions

Universal/Standard Precautions

HIV-1 Dynamics in Vivo: Virion Clearance Rate, Infected Cell Life-Span, and Viral Generation Time

Early-Warning Signals for Critical Transitions

Universal Features of Annealing and Aging in Compaction of Granular Piles

Theoretical Perspective on the Glass Transition and Amorphous Materials

The Role of Viruses in the Etiology and Pathogenesis of Common Cold

Encyclopedia of Virology

Treatment of Chronic Non-A,Non-B Hepatitis with Recombinant Human Alpha Interferon

Introduction to Hepatitis C Virus Infection: Overview and History of Hepatitis C Virus Therapies

Estimates of the US Health Impact of Influenza

Comparative Community Burden and Severity of Seasonal and Pandemic Influenza: Results of the Flu Watch Cohort Study. The Lancet Respiratory Medicine

Viral Mutation Rates

Cell Biology by the Numbers

Adaptive Immunity

Acknowledgments 719 We acknowledge support from the Center for Cellular Construction (NSF DBI-1548297), 720 and from NIH grant R35 GM130327 (WFM). WFM is a Chan Zuckerberg Biohub 721 investigator. This work was performed at the Aspen Center for Physics, which is 

Additional tests for stability (simulation out to one million iterations) has show that the 129 system stays at the apparent final state to accessible limits. 

Additionally, we find that the observed orbit is stable. In order to test the stability, 142 we induce large perturbations towards the end of simulations to try to send the If an analogous first-order phase transition is present in the experimental system, it 160 should appear in the form of a sharp extinction of one viral quasispecies, followed by the 161 emergence of a region favorable to a drastically different viral type on the other side of 162 the boundary, with the possibility of a limited amount of simultaneous presence in the 163 transition region. A complementary experiment would involve rapidly shifting a 164 steady-state viral quasispecies from a high permissivity host strain to a low permissivity 165 strain, which we predict will lead to rapid viral extinction. To avoid the complexities of 166 mammalian viral-host interactions, the aforementioned experiments may be able to be 167 performed in a suitable bacteriophage-bacteria system. If that is the case, it would be 168 interesting to see if the transition in quasispecies type corresponds to a transition