Thermal inactivation is an important mechanism for decontamination of viral pathogens, including the novel coronavirus SARS-CoV-2. Relevant contexts include decontamination of surfaces, proposed thermal treatments such as heated ventilation, the possible influence of summer weather, and fever as a physiological response. Indeed, moderate fever, defined as body temperatures between 38.3ºC and 39.4ºC, is lately considered a favorable response of the body against SARS-CoV-2 (1). Fever has been associated with improved survival rates during typical complications (2) , and may be significant at early stages of infection as well.
Coronaviruses are particularly prone to heat damage. Both the membrane proteins (3) (4) (5) and nucleocapsid protein (6) are easily affected by heat.
Experimental studies of thermal inactivation have tabulated results for a number of coronavirus types. One aim has been to clarify safe protocols for decontamination of shared instruments or personal protective equipment (7) . Another is to extrapolate an inactivation time from measurements at high temperature to more modest temperatures. A number of heuristic models have been suggested as a basis for fitting the data (8) . The Arrhenius equation for a thermally activated process is seemingly attractive as a physical model, and was used by two groups to extrapolate from isolated measurements to a continuous temperature variable (9, 10) . The treatments remain phenomenological, however, so the extracted parameters cannot properly be assigned a physical interpretation. Most importantly, the activation barrier is a free energy with an entropic contribution rather than a fixed energy.
We propose that the scaling in time of a rate-limited process of irreversible virus inactivation should be described instead as a second-order, entropy-driven phase transition, consistent with the coil-globule transition in protein dynamics (11) (12) (13) . On this basis we derive a simpler formula relating the inactivation rate to temperature with two fitting parameters that can be determined from data available in the literature. Together with new data (5, 14) available on SARS-CoV-2 hosted in cell cultures, we can predict the period in which moderate fever is effective against the virus. During the course of illness the patient may be exposed to repeated infections from viral residues, hence it is important to know the significant duration of one fever episode.
J o u r n a l P r e -p r o o f Thermal Inactivation Scaling Model While we do not know the mechanism of the virus inactivation in detail, we can approach the problem of thermal damage from a biophysical perspective. RNA is stable at moderate temperatures (5), so one expects loss of protein function. In general, protein inactivation may arise from various mechanisms including hydrolysis, oxidation, aggregation, kinetically trapped conformations, and denaturation by unfolding (15) . Moderate heating is known to cause irreversible inactivation of membrane proteins on the SARS viruses (3, 4) , as well as the nucleocapsid protein N (6). The incipient process of inactivation at temperatures so close to physiological is expected to involve a conformation rather than chemical modification. In particular, we consider the exposure of hydrophobic regions to the solvent as a plausible root of inactivation (16, 17) . The question considered is how temperature affects the rate of inactivation.
As a concrete example we consider the globule-coil transition in proteins as a critical path in the inactivation process. Corona viruses are characterized by elaborate spike proteins protruding from the membrane (18) , denaturation of which would prevent interactions with cellular receptors that are essential for infection. Loss of protein structure is often considered in two steps (11, 13) ; first the melting of order in the globular domains, and then expansion to a random coil. The melting stage has been identified as a first order phase transition (11) , which has been attributed to vibrational states (19) . The second stage involves a second order phase transition related to conformational entropy of the polymer (12, 19) , balanced by intramolecular interactions (20) .
The kinetics of a multi-stage process depend primarily on the rate-limiting step, the origin of which may be either an activation energy barrier or a geometric obstruction, i.e., any spatial constraint affecting entropy. A melting transition is favorable above a certain temperature; the melting rate is normally constrained by thermal diffusion necessary to supply the latent heat. Latent heat is not a relevant activation energy barrier since the free energy does not change along the process. Therefore, the second stage in a globule-coil transition is the rate limiting process, once allowed. On the other hand, most often the mechanism of inactivation involves denaturation due to structural changes, possibly a transition from a "correctly" folded state located at a minimum in free energy to a random coil state at another local minimum in free energy (22, 23) . The bottleneck in such a transition depends on a free energy barrier ∆ = ∆( − ) between the rest states, where S denotes entropy. Thus, thermal fluctuations provide a certain chance to overcome an internal energy barrier, but equally could allow a biological system to transit a temporary decrease in entropy, to resolve entanglement for example. This description conforms to the stability of the molten globule, which suggests that the entropic drive of the polymer to coil is balanced by a certain energy that increases with conformational disorder. We identify the order parameter as # = √3, according to N bending points added to the polymer, which becomes more flexible with respect to the native fold. As a perturbation to a particular geometry at equilibrium, i.e., the native fold, the N bending points will raise elastic energy proportionally to N and will raise the entropy proportionally to N according to the number of random walk segments. Similarly, the Landau description can build on Flory theory (20) if we define # = 4−∆ / , where ∆ is a negative change in the average size of the polymer with respect to , the size at rest in athermal solvent. Progression toward the random coil should decrease the polymer size and increase energy due to shortening distance between repulsive residue groups.
Wetting by the solvent entails another order parameter 5 , which varies between 0 (native hydrophobic configuration) and 1 (fully wet or solvated). Wetting involves rearrangement of both the solvent molecules and the distances between chain segments, which carries an entropy cost |Δ 5678 |. In addition, wetting by water supresses both the hydrophobic interactions (e.g. clustering of hydrophobic amino acid residues) and electrostatic interactions (28) . Hence we assume that -# . fully vanishes when the polymer is wet, since its long range repulsive interactions are suppressed. Therefore, the general free energy of the system is found to be Landau's parameters and T is the native reaction rate. As noted in ref (15) , results of membrane protein inactivation fitted to Arrhenius model often yield E in a range between 2.0 ~ 3.0 ⋅10 5 Jmol −1 , while in ref (9) E is between 0.8⋅10 5 Jmol −1 and 2.2⋅10 5 Jmol −1 . We can fit the same data referenced in that paper to our model. In Kelvin units, using R=8.314 J⋅K −1 mol −1 , T m =310 ⁰K, we find for the range of general inactivation that U lies between 1.28 and 1.46, while the value of U among corona viruses is between 1.10 and 1.31. Based on the Arrhenius fitting parameters A and E for corona viruses found in ref (10) and T m =310 ⁰K we find that the range of T (present model) is between 0.005 and 0.050 min -1 , in contrast to the unphysically large range of the attempt frequencies A (see table 1 ). In practice, the discrepancy between predictions of the energy barrier and free energy barrier models is minor compared to experimental uncertainty of the data at hand. Other heuristic fitting formulas also provide reasonable agreement with the limited data (8) (9) (10) . However, the underlying physical models are very different, and with further development could provide a key, for example, to evaluate the risk of possible mutations in the virus. Relevance to SARS-CoV-2 inactivation by fever
We focus our attention on temperatures near physiological, which are relevant to fighting the virus inside the body.
Measurements on thermal inactivation of SARs-CoV-2 are found in ref (5), performed by a group of FDA researchers on sputum samples spiked with the virus. Table 2 shows the reduction in viral infectivity after exposure to different temperatures. The uncertainty in measurements of the log 10 reduction is expected to be around 1 log unit, so the expressed uncertainty in inactivation rate is about 30% or 0.15 in log scale. The data points are shown in Fig.2 , fitted with Arrhenius model and our power law model. The latter is fitted with 2 points compared to 3 in the Arrhenius model, which make the slight difference in the curves. Ref (5) provides evidence of temporary increase in the number of copies in the samples, suggesting unaccounted replication at some stage of the test. Ref (14) reports only a monotonic decrease in population; however, the rate of decrease is smaller in cell culture than in serum. These aspects of the data remain unclarified, but basically introduce an uncertainty in the measurement. One explanation for the good coping of young children with Covid-19 is their rapid fever response to infection (35) . The possibility to increase core body temperature is still being J o u r n a l P r e -p r o o f Thermal Inactivation Scaling investigated (1), yet the recommendation to avoid antipyretic agents in cases of moderate fever seems justified. Another option is to provide hot air breathers to patients over longer times and at lower temperatures.
Supporting material Supporting calculation tables can be found online.
Author Contributions S.S and M.E. performed the research and wrote the article, S.S handled the calculations. [19] and calculations in the text. The transition from point p1 to p2 is a first order transition in which the free energy remains unchanged. The change in interaction energy of the molten globule between p2 and p3 is approximately balanced with entropy and is calculated according to Landau 