A rate equation approach to model the denaturation or replicationbehavior of the SARS coronavirus

J. Liu

Abstract As a newly emerging virus, little is known aboutthe SARS coronavirus, whose outbreak has brought awayseveral hundred people’s lives over the world in the year of2003 and is seriously imperiling the human health.Revealing the denaturation and replication mechanisms ofSARS coronavirus has great importance for successfullyfighting SARS. However, experiments related to SARScoronavirus are extremely dangerous and thereforerestricted only to certain specific labs with high safetystandard. Clearly, predicting the behaviors of SARS coro-navirus in a wide variety of environmental conditions,which are not easily accessible, are thus critically neces-sary. In this study, we proposed to quantify the survivaltime of SARS coronavirus either in vitro or in vivo, throughintroducing thermal rate process models established fromthe well-known Arrhenius law. The complex physical andchemical behaviors of the SARS coronavirus can then beattributed to its activation energy, frequency factor,damage function as well as the surrounding environmentalconditions. Based on the first data on stability and resis-tance of SARS coronavirus measured by members of WHOlaboratory network, the rate coefficients involved in theabove equations were estimated for the first time. Predic-tions on the survival time of SARS coronavirus in differenttemperature scale were then performed. It was foundtheoretically that, such survival time falls in an extremelywide range, say from several seconds in high temperatureto an almost infinitely long time in a low temperatureenvironment, which has already or is being supported bythe currently available tests data. Applications of thepresent theory to interpret several existing phenomenawere presented and their implementations in developingnew technical ways for SARS prevention and clinicaltherapy were discussed. Uncertainties involved in thetheoretical models were also analyzed and predicted.Parametric studies were performed to test the effects of therate coefficients to the survival time of SARS coronavirus.Some important factors, which can significantly vary the

denaturation or replication process of SARS coronaviruswere pointed out. Through regulating the parametersinvolved in the equation, certain potential therapies eitherthrough drug delivery or engineering approach to treat theSARS disease can possibly be established. Extension of thepresent model for further studies was also suggested. Thisstudy opens a new theoretical way for probing into thecomplex behaviors of SARS coronavirus.

Modellierung der Denaturierung oder Replizierungvon SARS-Korona-Viren

Zusammenfassung Der Kenntnisstand uber die Eigen-schaften des in 2003 neu aufgetretenen SARS KoronaVirus, der einige Hundert Menschenleben gekostet hat, istrelativ gering. Die Ermittlung des Denaturierungs- undReplizierungsmechanismuses des SARS Virus ist fur seineBekampfung von hoher Bedeutung. ExperimentelleUntersuchungen an diesem extrem gefahrlichen Virusdurfen nur durch Laboratorien mit einem hohen Sicher-heitsstandard erfolgen. Die Vorhersage des Verhaltens desSARS Virus in unterschiedlichen Umgebungsbedingungenist dabei erforderlich. In der vorliegenden Studie wird dieUberlebensdauer des Virus unter Labor- und realenBedingungen durch Anwendung der bekannten Arrhe-nius-Beziehung fur temperaturabhangige Vorgangeermittelt. Das physikalische und chemische Verhalten desSARS Virus wird anhand der zugrundeliegenden Modell-Parameter beschrieben. Basierend auf den ersten Mes-sungen von Mitgliedern des WHO-laboratory-networkuber die Stabilitat und Widerstandsfahigkeit des Viruswurden erstmalig die Geschwindigkeitskoeffizienten desBerechnungsmodells bestimmt. Vorhersagen der Uberle-bensdauer des SARS-Virus unter unterschiedlichen Tem-peraturbedingungen wurden ausgefuhrt. Das sich hierausergebende, sehr unterschiedliche Ausmaß der Uberle-bensfahigkeit in Abhangigkeit der Umgebungstemperaturist durch den Vergleich mit verfugbaren experimentellenErgebnissen bestatigt worden. Die Anwendung der vor-gestellten Modellierung zur Interpretation realer Phano-mene und zur Entwicklung technischer Maßnahmen zurVorbeugung und klinischen Therapierung von SARS wirddiskutiert. Der Einfluß von Unsicherheiten des Modellswird analysiert und abgeschatzt. Parametrische Studiensind durchgefuhrt worden, um den Einfluß der Ge-schwindigkeitskoeffizienten auf die Uberlebensdauer desSARS Virus darzustellen. Einige wichtige Einflußgroßenauf die Denaturierung und Replikationsfahigkeit des SARSVirus werden aufgezeigt. Durch eine Variation der Mo-dellparameter kann die potentielle Wirksamkeit medika-

Forschung im Ingenieurwesen 68 (2004) 227–238 � Springer-Verlag 2004DOI 10.1007/s10010-004-0130-2

227

Received: 24 March 2004

J. LiuCryogenics Lab, P. O. Box 2711,Technical Institute of Physics and Chemistry,Chinese Academy of Sciences, 100080 Beijing,People’s Republic of Chinae-mail: jliu@cl.cryo.ac.cn

This work is supported by the National Natural Science Foun-dation of China under Grants 50325622 and 50176053.

mentoser oder physikalischer Therapien abgeschatzt wer-den. Erweiterungsmoglichkeiten des vorgestellten Modellswerden vorgeschlagen. Die vorliegende Studie ermoglichtneue, theoretische Vorgehensweisen zur Untersuchung deskomplexen Verhaltensmusters des SARS Virus.

List of symbolsA surface area of dropletC specific heat of dropletC(0) virus concentrations at initial stateCðtÞ virus concentrations at time tEa activation energy (J/mol)Eac activation energy of virus½E0� total concentration of enzyme½Ea� concentration of active enzyme½Ei� concentration of denatured enzymeFd injury fractionh heat convection coefficientK chemical reaction rateK2 rate constant for immune reactionKc reaction equilibrium constantKd equilibrium constantP collision number or frequency factor (s�1)Pc frequency factor of virus (s�1)R constant of perfect gas (8.31 J/mol/K)½S� virus concentrationsS�½ � concentration of all protein substancesS0½ � protein concentration

t time (s)t0 survival time (s)T absolute temperature (K)V volume of dropletV1 reaction rateV2 reaction rate

Greek symbolsa ratio for volume of all protein substances to

that of SARS coronavirusq density of dropletX injury functionD uncertainty of variableDH variation of enthalpyDS variation of entropy

Subscriptscrit critical valuei initial stateopt optimum1 surrounding air

1IntroductionNamed by the world health organization (WHO), severeacute respiratory syndrome–SARS is a newly emergingworld-wide infectious disease [1–3], which is also com-monly called atypical pneumonia in China [1–4]. Since itsoutbreak, SARS has brought about serious threatening onthe human health. Up to now, investigating the patho-genesis and treatment therapy on SARS keeps receiving

great attentions over the world. Some encouraging pro-gresses have been achieved in the SARS research such asidentification of cause of disease [5–7], gene sequencing[8–10], virus detection [11], epidemiology analysis [12,13], drug and bacterin development [14], pathology [15],and clinical therapy [16, 17] etc. It now becomes clear that,the pathogeny to induce SARS is a brand new type ofcoronavirus, whose behavior turns out rather unique andhad never been found before in nature. However, sincelittle is known about the properties of SARS and noeffective drugs are available, successful treatments on thisdisease at the present stage are extremely difficult, if notimpossible. Clearly, revealing the physical and chemicalbehaviors of the SARS coronavirus either in or outside thehuman body will be very beneficial for fighting the SARS.

One of the latest advancements made by the WHOmulticentre collaborative network (MCN) on SARS is thesuccessful test on the survival time of SARS coronavirusunder several different environmental conditions [18].Owing to this contribution, we now get to know that thesurvival time for the SARS coronavirus in the air spansfrom hours to days or even longer time. Such a huge dif-ference in time scale is rather perplexing, whose inter-pretation is waiting for great efforts. In fact, the survivalbehavior of SARS coronavirus is a complex issue, whichdepends both on the surroundings and the virus itself [19].To reveal the intrinsic mechanisms, a systematic theoryconcise in concept would be very desirable. In this study,we proposed to quantify the denaturation or replicationprocess of the SARS coronavirus through a rate equationtheory and new models capable of comprehensivelycharacterizing the physical and chemical behaviors ofSARS coronavirus will be established.

2Rate process model for survival time of SARS coronavirusIt is well known that, virus is a non-cellular organism, whorealizes its surviving by attacking a hosting cell. The viruscould not divide itself automatically [20]. As an organismconsisted by certain special nucleic acid and protein, thedenaturation of SARS coronavirus can be regarded as atemperature dependent rate problem, just as any naturalsubstances do. In fact, it has long been a basic way toaddress the thermal injury of biological tissues by usingthe rate equation. Henriques and Moritz [21] were the firstto characterize the thermal injury as a first order chemicalreaction process. The concise equation given by them latebecame the start point to analyze the thermal injury pro-blem of biological tissues. Xu and Qian [22] made newprogress in better understanding the thermal injurymechanisms by introducing the denaturation of enzymeprotein. Recently, Bhowmick and Bischof [23] performedan experimental test on the injury problem of the cellmembrane. Typical biophysical and biochemical eventsoccurring in thermal injury at macro or cellular levelgenerally include [24]: thermal denaturation of proteins,alterations in metabolic process, thermally induced alter-ations in physical or chemical characteristics of cells likehyperpermeability of membrane, intracellular ionic con-centration and nuclear degradation, loss of birefringenceproperties in muscle and collagen, loss of hemoglobin

228

Forsch Ingenieurwes 68 (2004)

from red blood cells. Similarly, the denaturation process ofthe SARS coronavirus in the air or its hosting cell shouldalso follow the same mechanism as above, although mayvary in the rate coefficients. The rate process model isoriginally established from the chemical reaction dynamics[25–28]. Referring to Fig. 1, one can see that for a typicalreaction, the thermally activated reactant should overcomean energy barrier Ea in order to generate products. Cor-respondingly, possibilities for the virus to replicate or dieout also depends on the magnitude of its stored energy,and the premise for this to happen is that the storedenergy should be higher than that of its critical reactionvalue, i.e. the activation energy. If treating the denatur-ation process of the virus as a first order process anddefining the injury function as X, one can write itschanging rate as,

_X ¼ dX=dt: ð1ÞIntegration of the above equation over the whole period

of thermal denaturation is X, i.e.

X ¼Z t

0

dX=dtð Þdt: ð2Þ

Then, variation of the virus appears as a temperaturedependent rate process whose chemical reaction rate K(i.e. injury rate) can be described by the well-knownArrhenius equation [25]:

dXdt¼ K ¼ P exp � Ea

RT

� �ð3Þ

where, R is constant of the perfect gas (8.31 J/mol/K), Ea

the activation energy (J/mol),P the collision number orcommonly called as frequency factor (s�1), T the absolutetemperature (K), and t the time; X stands for injury ratio,when X = 1, it is generally regarded as that the object hasalmost entirely denatured (or injured) [27, 28]. Generally,for different biological organisms or environmental con-ditions, the magnitudes for P and Ea could vary in a verylarge range.

Combining (2) and (3), one can obtain the Henriquesequation to quantify the thermal injury of the biologicalmaterial [21], i.e.

X ¼Z t

0

P � exp �Ea=RTð Þdt: ð4Þ

In this equation, only two coefficients Ea and P areneeded to perform a comprehensive analysis on the de-naturation problem of the biological samples. Such inte-gration equation is very easy to quantitatively characterizethe denaturation of virus.

Here, the physical meaning for X can be expressed asthe logarithm of the ratio of the concentration of the initialvirus particles and that of the surviving virus particles, i.e.

XðtÞ ¼ lnCð0ÞCðtÞ

� �ð5Þ

where, C ((0) and C (t) are virus concentrations at initialstate and that at time t, respectively.

Its meaning can further be interpreted as follows.Assuming the thermal process as a first order activity, onehas [26]

dCðtÞdt¼ �KCðtÞ: ð6Þ

For time dependent rate parameter K (due to variationof temperature), the virus concentration should read as

CðtÞ ¼ Cð0Þe�R t

0KðtÞdt

: ð7ÞTherefore, it can be seen that the thermal injury in factaccords with (4), i.e.

XðtÞ ¼ lnCð0ÞCðtÞ

� �¼Z t

0

Kdt ¼Z t

0

P � exp � Ea

RT

� �dt:

ð8ÞThen the injury fraction can be described as [26]

Fd ¼Cð0Þ � CðtÞ

Cð0Þ ¼ 1� e�X: ð9Þ

The above dynamics model indicates that, if the tem-perature is lower than certain critical value, the rate of theinjury accumulation could be omitted. On the contrary,the injury rate will significantly increase. This criticaltemperature is generally defined as that when dX=dtbecomes equal to 1.0, i.e. [25]

dXdt¼ P expð�Ea=RTcritÞ ¼ 1 i:e: Tcrit ¼

Ea

R lnðPÞ :

ð10ÞIn other words, if the temperature is higher than this

value, the thermal environment can easily kill the SARScoronavirus.

3Equation for in vitro survival time of SARS coronavirusThe above equation is very useful in characterizing thesurvival time of SARS coronavirus. As will be shown later,it can provide predictions close to the reality and help get aclear and concise view on the complex behaviors of SARScoronavirus. With an extremely small size (in micro/nanoscale), the SARS coronavirus would quickly reach a ther-mal equalibrium with the environment, which can beillustrated as follows. Assuming that a liquid droplet (suchas sputum) containing virus was dropped to the sur-rounding air with temperature of T1, its temperatureresponse can be described by a lumped system model, i.e.

�hAðT � T1Þ ¼ qVCdT

dtð11Þ

Fig. 1. Energy state diagram of a biochemical process

229

J. Liu: A rate equation approach to model the denaturation or replication behavior

where, q; C are respectively the density and specific heat ofthe droplet, A;V the surface area and volume of the dro-plet, h the heat convection coefficient between the dropletand the surroundings.

Solution to (11) can easily be obtained as:

T ¼ T1 þ ½Ti � T1� exp � hA=qVCð Þt½ � ð12Þwhere, Ti is initial temperature of the droplet.

For an extremely small liquid droplet, the second term atthe right hand side of (12) will become zero within seconds.Compared with the long survival time of the SARS corona-virus, which is generally in minutes or even days, thetemperature of the SARS coronavirus can be treated as aconstant without causing evident error. Therefore integra-tion of (4) will lead to the following theoretical equation forthe survival time t0 of SARS coronavirus in the air as,

t0 ¼ X � expðEa=RTÞ=P: ð13ÞIt indicates that, the SARS coronavirus is very sensitive to

the temperature such that the survival time t0 exponentiallydepends on the temperature T, which accords well withthe currently available observation. Eq. (13) still reveals that,the higher activation energy or the lower temperature orfrequency factor for the virus, the longer time for it to live.

Should be pointed out that, application of (13) oftenimplies approximations due to uncertainty of the ratecoefficients. The reason mainly comes from the experi-ments, which may have a low testing accuracy or justrecord insufficient information, especially because defini-tion of some important values such as critical survival rate,and details on the substrate are only approximately true.Besides, experimental tests up to several days could notguarantee a constant environmental temperature. These allcontribute to the statistical errors and may lead toinaccurate estimation on the rate coefficients. To test theeffects from each of the parameters to the survival time in(13), t0 was expressed as the following function

t0 ¼ f ðEa; P;T;XÞ: ð14ÞAssuming Dt is uncertainty caused by the approximate

parameters, then one has

Dt

¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiot0

oEaDEa

� �2

þ ot0

oPDP

� �2

þ ot0

oTDT

� �2

þ ot0

oXDX

� �2s

ð15Þwhere, D stands for uncertainty for each of the variables.

Eq. (15) can further be expressed as [29]:

Substituting (13) into (16) leads to

Dt

t0¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiEa

RT

� �2 DEa

Ea

� �2

þ DT

T

� �2" #

þ DP

P

� �2

þ DXX

� �2

vuut :

ð17Þ

Therefore, if uncertainties for the four parameters Ea; P;Tand X are given, their effects on the survival time can beevaluated and its uncertainty limit will thus be predicted.

4Equation for in vivo survival behavior of SARS coronavirusThe discussion given above is only useful when theSARS coronavirus lives in the air. For the process that thevirus attacks its hosting cell, then replicates or dies, morefactors should be taken into concern. Analysis performedbelow will be very beneficial to better understand the ef-fects of the concentrations of the virus and immune cell etcto the infection mechanisms of SARS disease. Generally,when virus enters the lung, it would try to replicate itselfby attacking the hosting cell. Without adopting effectivemeasure to prevent this from happening, the virus con-centration would become higher and higher until causeinjury to the lung tissues [12]. Therefore, quantitativeevaluation on the early damage of the lung and then pre-dict the damage time will be very important for a suc-cessful treatment planning on SARS. Clearly, anappropriate theoretical model would be desirable for suchpurpose. In the following, a theoretical model will beestablished to characterize the replication or deathbehavior of the SARS coronavirus in human body.

As is known, replication of virus must recur to theenzyme of its hosting cell. Assuming this enzyme-cata-lyzed process follows a first order equation, then itsreaction rate V1 is proportional to the virus concentrations[S] and that of the active enzyme ½Ea�, i.e. [22]

V1 ¼ K½S�½Ea�: ð18ÞWithin the scale of the physiological temperature, the

active enzyme ½Ea� and the denatured enzyme ½Ei� obeysthe following equilibrium relation:

½Ea�ÐKd

½Ei� ð19Þwhere the equilibrium constant Kd depends on temperatureand can be determined via the van’t Hoff equation [22],

Kd ¼ ½Ei�=½Ea� ¼ exp DS=R� DH=RTð Þ ð20Þwhere DH and DS are respectively the variations of the enth-alpy and entropy during the enzyme denaturation process.Since the total amount of the active enzyme and the non-active enzyme is a constant, i.e. ½Ea� þ ½Ei� ¼ ½E0�, one has

½Ea� ¼ ½E0�=½1þ Kd�: ð21ÞThen, the chemical reaction rate of virus replication

under enzyme catalysis was obtained as

V1 ¼ K½E0�½S�=½1þ expðDS=R� DH=RTÞ�: ð22ÞIt indicates that, there exists an optimum temperature

Topt where the reaction rate is the highest, and its value can

be determined by solving equation dV

dT

���T¼Topt

¼ 0. Generally,

this value is close to the normal physiological temperature

Dt

t0¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffio ln t0

o ln Ea

DEa

Ea

� �2

þ o ln t0

o ln P

DP

P

� �2

þ @ ln t0

@ ln T

DT

T

� �2

þ @ ln t0

@ ln XDXX

� �2s

: ð16Þ

230

Forsch Ingenieurwes 68 (2004)

37 C. If substituting Topt into (22), one can further obtainthe optimum reaction rate Vopt. However, expressions forboth Topt and Vopt will not be given here for brief.

The enzyme catalysis would help to realize a virusreplication process. In contrast to this, phagocytosis by theimmune reaction due to body response will restrain thegrowth of the virus, although the detail is not clear pre-sently. For a generalized purpose, assuming that the cellconcentration to induce a first order immune reaction is[C], the reaction rate will then also be proportional to theconcentrations of the virus and the immune cells, i.e.

V2 ¼ K2½S�½C� ð23Þhere, K2 is the rate constant for the immune reaction, [C]can be regarded as a transient value and follows the rela-tion as:

½C� ����!Kc ½Ci� ð24Þwhere, the reaction equilibrium constant Kc can beapproximately described by the Arrhenius equation,

Kc ¼ Pc exp �Eac=RTð Þ ð25Þwhere, Eac, Pc are respectively the activation energy andfrequency factor of virus subject to immune reaction.

Concentration of the immune cells can be expressed as

d½C�dt¼ �Kc½C� ð26Þ

i..e

ln½C�=½C0� ¼ �Z t

0

Kcdt ¼ �t � Pc � expð�Eac=RTÞ:

ð27ÞSince mechanism for enzyme-enabled reaction

differs from that of the phagocytosis reaction, theoptimum temperatures for the two processes are alsodifferent. If certain external heating can be applied toenhance phagocytosis while restrain enzyme enabledreaction, then a useful way to kill the SARS virus canseems be established.

With the above relations, one can predict the survivaltime of the SARS coronavirus in the human body. Assumingthe initial concentration of virus invading the human bodyis ½S0�, and the rate for the enzyme enabled reaction is V1, therate for the phagocytosis reaction induced by immune cell isV2, then the transient virus concentration [S] can bedetermined by the following equation,

d½S�=dt ¼ ðV1 � V2Þ

¼ ½S� K½E0�1þ expðDS=R� DH=RTÞ � K2½C�� �

ð28Þ

where the transient value of [C] is determined from (27).Clearly, if temperature and the concentration of

immune cell [C] are both constant, the virus concentrationcan be obtained as

ln½S�=½S0� ¼ t0K½E0�

1þ expðDS=R� DH=RTÞ � K2½C�� �

:

ð29Þ

If most of the virus denatured, i.e. the current concen-tration [S ] significantly deviate from its initial value [S o],then the damage can be regarded to happen. From (5), onecan find that the damage function for the virus reads asX ¼ ln½S0�=½S�. Then the survival time for the virus toinvade the human body is obtained as

t0 ¼ X K2½C� �K½E0�

1þ expðDS=R� DH=RTÞ

� ��1

: ð30Þ

If parameters involved in the above equation can beestimated through experiment, one can predict thebehavior of the SARS coronavirus and thus the SARSpathology development in the human body. However, dueto difficulty to carry out an in vivo test, there is currently astrong lack of experimental data for the survival time ofthe SARS coronavirus in human body. This study stillcould not provide a calculation on such situation.

Similar as above, due to uncertainty of the interiorenvironment of human body such as variations in pHvalue, temperature, drug concentration, body response,substance variation even experimental measurements,uncertainty should also be considered when applying theabove equation to predict the behavior of SARS corona-virus to attack the hosting cells. For such in vivo situation,the survival time can be expressed as [29],

t0 ¼ f Ea; Eac; P; Pc;T;X;DS;DH;K2;C; :::ð Þ ð31ÞCombination of (30) and (31) will also lead to a uncer-tainty equation similar to (17). But it will not be presentedhere for brief.

Clearly, uncertain factors for the survival time of theSARS coronavirus in vivo appear much complex than thatof in the air. This makes it hard to accurately predict thedisease development. Such situation should be paid withspecial attentions in the near future.

5Estimation of rate process coefficientsof SARS coronavirusBefore one can perform a prediction on the survival time ofthe SARS coronavirus, the rate coefficients P and Ea must beobtained. Up to now, only very limited parameters for thebiological samples are reported [25, 27]. Especially, there is astrong lack of such data for the bio-samples in cell level.Even for ordinary virus, no rate coefficients are available, letalone the corresponding information for the brand newSARS coronavirus. Generally, estimation of Ea and P can beperformed by systematically measuring the survival time ofvirus subject to different temperatures. Then based on alarge quantity of the data of t0 versus T, Ea and P can becalculated through a least square fitting on the (13). Theprocedure can be as follows: placed the virus samples in aspecific environment, and applied it with surface heating orcooling and kept the virus for a specific time, then completethe viability test. The same experiments were repeated manytimes under different constant temperatures. The criticaldamage degree in each case was chosen from a series ofdamaged virus samples, i.e X ¼ 1, or Cðt0Þ ¼ 36:8%Cð0Þ.Then through rewriting (13) to its logarithm form, one has

231

J. Liu: A rate equation approach to model the denaturation or replication behavior

lnðt0Þ ¼Ea

R

1

T� ln P ð32Þ

From the intercept between lnðt0Þ and the critical valueof 1/T, one can get P; while from the slope of the curve, Ea

can be obtained.In practice, due to high infection of SARS coronavirus,

experiments are restricted only to certain labs with highsafety standard. Fortunately, the WHO MCN for SARS hasexperimentally obtained the first data on stability andresistance of SARS coronavirus, which makes the presentstudy possible. Here, we will make an analysis on theirdata as modified in Table 1, where the results were par-ticularly chosen from those in neutral environment.Although these data may contain approximation, it doesprovide great valuable knowledge on the survival behav-iors of the SARS coronavirus in various environments.

Based on the data in Table 1, the curve for theln t0 � 1=T can be drawn as Fig. 2, where different symbolsrepresent data from different labs. It shows that, these dataappear as some what a little disperse, which indicates thatthe experimental test may contain uncertainty. Theoreti-cally (see Eq. (32)), relation between ln to and 1/T follows alinear equation. However, due to approximation in themeasured data, this relation does not strictly hold.Therefore, the presently estimated rate coefficients are infact only mean values. Through averaging the data fromdifferent labs, we obtain the straight line in Fig. 2, which isjust the rate equation for the survival time of the SARScoronavirus in the atmospheric exvironment, i,e,

ln t0 ¼ 11199:5=T � 26:4476 ð33ÞComparing between (32) and (33), one can obtain for

the first time the activation energy and the frequencyfactor for the SARS coronavirus respectively as follows,

Ea ¼ 11199:5� 8:31 ¼ 0:9307� 105ðJ=molÞ;P ¼ 3:429� 1011ðs�1Þ: ð34ÞThe above values appear much smaller than that of the

ordinary bio-samples. The fact for the small activationenergy implied that it is easy for the SARS coronavirus toreact to its contacting objects, which also explain thephenomena that this virus can easily be destroyed by anordinary disinfector [18]. Meanwhile, it also indicates thatthe SARS coronavirus has a high infection tendency. But ifits replication path can be successfully blocked off, theSARS coronavirus could also not survive for a long time.Effects of the frequency factor appear opposite to theabove.

6Results and discussion

6.1Prediction of survival time of SARS coronavirusHaving obtained the above rate coefficients, one can thenperform a theoretical prediction on the survival time of theSARS coronavirus in a wide variety of temperature ranges.Depicted in Fig. 3 are the predicted survival times of the

Table 1. Experimental conditions and survival time of SARS coronavirus (from [18])

Lab� Substrate Condition Survival time

GVU Virus spiked in baby stool Ph 6 – 7 (regarded hereat room temperature)

3 h

QMH Stool Room temperature At least 2 daysUrine Room temperature At least 24 hVirus culture

medium+1% bovine serumOn plastic surface

in Room temperatureAt least 2 days

Virus culturemedium+1% bovine serum

30 – 37�C At least 1 h

Virus culturemedium+1% fetal calf serum

56�C Degration of titre over time (1000infections virus units in 15 min)

NIID Virus culture +2%fetal calf serum

4�C At least 4 days

Virus culture +2%fetal calf serum

37�C Less than 4 days

Virus culture +2%fetal calf serum

56�C Less than 30 min

UnivM Virus culture 4�C At least 21 daysCUHK �virus in phosphate

buffered saline (PBS)�virus in sterilized stool

Room temperature on PBS StoolPlastered wall 24 h 36 hPlastic surface 36 h 72 hFormica surface 36 h 36 hStainless steel 36 h 72 hWood 12 h 24 hCotton cloth 12 h 24 hPig skin � 24 h � 24 hGlass slide 72 h 96 h

*CUHK: Chinese University Hong KongGVU: Government Virus Unit, Dept. of Health, Hong Kong, SAR ChinaQMH: Queen Mary Hospital, The University of Hong Kong, Hong Kong, SAR ChinaNIID: National Institute of Infectious Diseases, Tokyo, JapanUnivM: University Marburg, Germany

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SARS coronavirus in the temperature of 0� C–80�C. Itindicates that, the SARS coronavirus can survive in anextremely wide range of time, say, from several minutes tohundreds of hours. Particularly, at a low temperature,SARS coronavirus can even live much longer. For anenvironment at temperature close to 0�C (such as in arefrigerator), the survival time of the SARS coronaviruscan be as long as 540 h. While at a lower temperature suchas –80�C, this time will sharply increase to about5.375 � 108 days. In other words, the SARS coronavirusmay live forever. Therefore, as a newly emerging virus, it isnot realistic to expect that the SARS coronavirus wouldsoon completely disappear, since the virus could easilyremain alive through the winter and then resuscitate in awarmer weather via various ways such that drying theSARS coronavirus in summer may help it realize a safedormancy and then preserve itself from being thermallykilled. Since the survived SARS coronavirus may induce anew cycle of SARS epidemic, the situations to confront theSARS disease are still very serious unless bacterin will befound soon. Therefore, establishing certain effective way toprevent SARS in advance in the season of its high infectionis very necessary. From Fig. 3, one can observe that, theSARS coronavirus is very sensitive to the temperature. Forexample, at the room temperature around 25�C which is

comfortable for human body, the SARS coronavirus canlive up to 16 h. Such time scale has already been longenough for the virus to spread to a large area of regions.Clearly, temperature over 20�C has produced certain effectto decrease the viability of the virus. To show its detail, weredraw the survival time curve at temperatures above 20�Cin Fig. 3 as Fig. 4 and define the temperature above 20�Cas thermally important temperature. In the following, wewill make a few discussions on the survival behaviors ofSARS coronavirus at several special temperatures. If theenvironmental temperature is at 37�C, Fig. 4 shows thatthe survival time for the SARS coronavirus is t0 ¼ 3:97 h.Therefore, at a hot season such as summer, it is generallynot easy for the SARS coronavirus to survive, and possi-bility for inducing the disease epidemic will be signifi-cantly decreased. On the other hand, if intentionallyheating the virus by using the tumor hyperthermia tem-perature 42�C, which is bearable by human body, thepredicted survival time for the SARS coronavirus ist0 ¼ 2:24 h (see Fig. 4). Consequently, treatment of SARSthrough high temperature heating is possible, as long asthe heating can be kept to a long enough time. Becausehyperthermia therapy is a relatively safe and non-noxiousway, it is worthy of trying to cure SARS, especially since nomiracle drug and definite treatment strategy are availableat the present time. Besides, if the temperature was furtherincreased to a higher however still bearable level, thetreatment time can be evidently shortened. For example, ifchoosing heating temperature 56�C to do calculation, thepredicted survival time for the SARS coronavirus is onlyt0 ¼ 0:49 h. In addition, if certain assistant drug wasdesigned to decrease the activation energy or increase thefrequency factor of the SARS coronavirus, the virus couldpossibly be killed by an ordinary heating under a relativelylow temperature. In this sense, combining drug deliveryand hyperthermia to cure SARS is worth of investigation.We have proposed an engineering approach to restrain thereplication of SARS coronavirus through directly venti-lating hot oxygen to the human alveolus [30]. Should bepointed out that, the above analysis is performed based onthe data of SARS coronavirus in air. Factors in humanbody are rather complex such as that there exist bothfactors favorable for immune or drug to react and that forthe virus to replicate. Therefore, when developing a

Fig. 2. Tests data on absolute temperature and survival time ofSARS coronavirus (from WHO measurements [18])

Fig. 3. Temperature dependent survival time of SARS coronavi-rus

Fig. 4. Thermally important temperature scale to kill SARS cor-onavirus

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treatment therapy in clinics, such comprehensive factorsshould be fully considered. In certain extreme situationsuch as in a high temperature environment, the SARScoronavirus can only survive for a short period. FromFig. 4, when the temperature approaches 80�C, the pre-dicted result shows that the SARS coronavirus will dena-ture within 3 min. This conclusion has great importancefor practice. We have proposed to heat a polluted room tocompletely kill the virus inside [31]. It seems now clearthat, 80�C is an appropriate temperature to efficiently killthe virus. Killing the SARS coronavirus to clean the roomis a new way of disinfection and produces no pollution tothe environment. Such method is worth of investigation.Overall, the above results predicted from (33) accord wellwith the currently available measurement and observedphenomena [18]. Although the theoretical equation isfounded from limited data, its efficiency in prediction issatisfactory. Theoretically, the time for the SARS corona-virus to live falls in an extremely wide range, but not justthe currently reported value. This time depends on themagnitudes of the P, Ea and T and the physical andchemical factors capable of changing these parameters.However, we should point out that, the experimental testsare still far from being enough, since different environ-mental factors and substrates will have effects on thesurvival time of the virus. Therefore, to completelyunderstand the survival behavior of the SARS coronavirus,strictly defining the test conditions and perform corre-sponding work are still very necessary.

6.2Effects of the rate coefficientsIn the following, we will perform an analysis on theuncertainty of the survival time predicted from (33). Forthis purpose, uncertainties for the activation energy,temperature, frequency factor and degree of denature areall prescribed as 10%. Then the uncertainty of the pre-dicted survival time at room temperature 25�C isDt=t0 ¼ 0:88: Presented in Fig. 5 are estimated survivaltimes for more situations. From that, one can find that thetemperature and activation energy affect most to accu-rately predict the survival time. From (17), we can draw aconclusion that at a high temperature, the accuracy forpredicting the survival time is also high, while at lowtemperature, the prediction accuracy will become worse oreven have several times of errors. What appear morereliable from the theoretical prediction is the averagesurvival time which overall reflects the reality. However,the real value usually fluctuates around this averagemagnitude. This conclusion would be very useful in cor-rectly interpreting the current or future measured results.

When in the air or in human body, if certain disinfector,acid and alkali substance, drug, bacterin, or some chemicaland physical factors may induce variations of the activa-tion energy and frequency of the SARS coronavirus, it ispossible a simple heating process would kill the virus. Onthe contrary, the survival and replication capability of theSARS coronavirus may be significantly enhanced, itsability of anti-drug, heat resistance and infection willincrease. Theoretically speaking, all these situations arepossible. Parametric studies given in Fig. 6 show that, for

different rate coefficients, survival time for the SARScoronavirus can vary in orders even under the sameenvironmental temperature. Therefore, if one wishes tocomprehensively analyze or test the survival behavior ofthe SARS coronavirus, the environmental conditionsshould be strictly defined.

6.3Effects of enzyme denaturation and immune reactionFrom Eq. (30), one can find that, one of the most importantfactors to enable virus replication and hyperplasia is theenzyme. When a small part of the enzymen becomesirreversibly denatured, it may make it impossible for theSARS coronavirus to replicate. Generally, denaturation ofenzyme is related to the temperature, pH value and drugapplication etc. If certain approaches were taken tointervene these factors, new therapy on the SARS can pos-sibly be established. On the other hand, the phagocytosis onthe virus due to immune regulation from the human bodywill also help to kill the virus. In fact, this is one of the mostimportant factors for many SARS patients to survive.Presently, due to lack of specific medicine, the clinicaltreatment on SARS is still above all a kind of supportingtherapy, whose essence is to regulate the immune reaction ofthe human body and thus finally destroy the SARS coro-navirus. However, if inappropriate regulation was made, itmay even enhance the replication of the virus and then makesituation more serious. Therefore, study on the mechanismsof the SARS occurrence and development should be paidwith much more attentions.

6.4Denaturation of SARS coronavirus and its affinitywith human bodyIt becomes gradually clear now that it is very easy for theSARS coronavirus to deposit in the areas of the respiratorytract, alimentary canal and neural elements, which is

Fig. 5. Uncertainty of predicted survival time (Dt=t0) under dif-ferent temperature level and subject to uncertain rate coefficients

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perhaps one of the reasons for such virus to have a stronginfection. As frequently observed, the SARS coronavirustends to adhere to these tissues. Therefore, revealing thecharacteristics of the physiological solutions in these areaswill help to better understand the attacking behaviors ofthe SARS coronavirus. Similarly, we can also explain theother SARS phenomena. For example, the reason leads tovirus aberrance is perhaps because its rate process hasbeen changed forever. For example, some studies showthat, the infection ability for the filial generation becomesweak, which may reflect that the process has changed therate coefficients such as P and Ea. It looks the filial gene-ration does have a decreased activation energy or a pos-sibly increased frequency factor. However, the SARScoronavirus does can change to a converse aberrance. This

is because an extremely small change in its structure willpossibly vary its activation energy. When the aboveparameters appear to induce an increased viability of theSARS coronavirus, its survival time will be lengthened andthe infectivity will be improved. This should be born inmind with caution and effective prevention measureshould be made.

6.5Effects of acid-alkali environment and sterile liquidThe rate equation established above has only consideredthe effect of a single temperature. And the estimated ratecoefficients are obtained by using the data measured inneutral environment. More complex environmental situ-ations were not taken into account. In the coming future, it

Fig. 6. Effects of rate coefficients on the sur-vival time of SARS coronavirus

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is worthwhile to establish a similar rate equation, whichhowever includes more factors such as pH value etc. Thenthrough a limited tests on the survival time of the SARScoronavirus subject to different thermal as well as pHconditions, the corresponding rate coefficients can beestimated which would be beneficial for predicting morewide behaviors of the SARS coronavirus. Such effort wouldbe of significance for the clinical treatment as well asprevention of SARS. For example, if one could change theacid and alkali conditions at the human tissues easilyattacked by the SARS coronavirus, it will help to enhancethe killing performance of the delivered drugs and therapy.For the prevention, it would be beneficial to select certaindisinfector, which is more environmentally friendly andcauses less pollution.

6.6Cause of prolonged survival time of SARS coronavirusin an agglomerate with protein substancesThe ability for the SARS coronavirus to survive in the airsignificantly depends on the surroundings. However, ifvirus was contained in a protein environment such assaliva or respiratory secretion, its survival time willbecome much longer [32]. As is well known, the phlegm orsaliva consists of a lot of protein and nutrimental sub-stance. One of the reasons for the SARS coronavirus to livelonger in such situations is perhaps because these com-pounds tend to maintain large activation energy Ea or alow frequency factor P. As predicted in Fig. 6, if only thesetwo parameters change a little, they will be significantenough to induce a variation in orders of the survival timeof the SARS coronavirus.

On the other hand, the reason for the SARS coronavirusto survive longer can be regarded as a kind of concen-tration damage. If treating the SARS coronavirus depositedin the phlegm or saliva as also a part of this protein likeagglomerate, then only when all the protein substances inthe agglomerate were denatured can the packed SARScoronavirus be regarded as completely destroyed. In thisway, one can predict the prolonged survival time of theSARS coronavirus deposited in phlegm or saliva. Substi-tuting (5) into (13), one obtains,

t0 ¼ ln S�½ �= S½ �ð Þ � exp Ea=RTð Þ=P ð35ÞHere, S�½ � is concentration for all the protein sub-

stances in the phlegm agglomerate. Compared with pro-tein concentration S0½ � of the SARS coronavirus existingalone, there satisfies such a relation, i.e. S�½ � ¼ a S0½ �,where a is ratio for the volume of all protein substancesto that of the SARS coronavirus. Since the virus wascontained in the phlegm or saliva, if only 90% of theprotein substance become denatured, can it be said thevirus was killed. Assuming the SARS coronavirus onlytakes up 10% volume of the total phlegm or saliva, onehas a ¼ 10: Then the increased survival time for the viruswill be ln 10 times of its original value. For example, anoriginal survival time of 5 h will now increase to5ln10 = 11.5 h. Therefore, cleaning away the environ-mentally existing phlegm or saliva is very necessary forthe successful control of the SARS infection.

6.7Implementation of rate equation theory to therapyand detection of SARSThe above study demonstrates for the first time that, therate equation has flexible adaptability to characterize thesurvival time of the SARS coronavirus. Once the activationenergy and frequency factor in a specific environment areavailable, one can perform predictions on the survivalbehavior of the SARS coronavirus in a wide variety ofsurrounding situations. The present theory appears usefulnot only for better understanding the spreading and sur-vival mechanisms of the SARS coronavirus, it also suggestsan important clue to develop efficient drugs, bacterin andclinical therapy. For example, if certain strategy capable ofvarying the activation energy, frequency factor and tem-perature of the SARS coronavirus can be found, it is verypossible to set up a useful way for fighting the SARS. Infact, such effort has been reflected in the current SARSresearch activities. Compared to the ways of developingbacterin and drug, physical therapy appears much directin its output results. For example, by testing the killingeffect of a group of various physical factors to the in vitroand in vivo SARS coronavirus, it is possible to quicklyscreen out useful engineering approaches for treating theSARS. The currently available approaches worthy of tryingare as follows [31]: Based on the temperature sensitivity ofthe SARS coronavirus, one can develop various heatingmethods for the hyperthermia and disinfection purposes;While from the non-thermal effects produced by theelectromagnetic wave (static or dynamic electric stimula-tion, radio frequency), the optical wave (laser or violet),acid and alkali environment (by varying the pH value), themass diffusion (high concentration oxygen or humantolerable disinfector), or delivered target substance (drugor bacterin) etc., one can also find useful ways to intervenethe rate process of the SARS coronavirus and then suc-cessfully prevent or cure it. Similarly, if certain specificsubstance was found to react with the samples containingSARS coronavirus and produce an easily detectablephysical chemical mass, one can establish a quick diag-nostic method on the SARS in clinics.

7ConclusionsIt is a good way to quantitatively characterize thedenaturation or replication behavior of SARS coronavi-rus through the rate process theory. Based on the limitedexperimental data on survival time, the activation energyand frequency factor for the SARS coronavirus in aspecific environment can be estimated and then appliedto predict more information inaccessible through expe-riment. The currently available experimental data for thesurvival times of SARS coronavirus is very limited, andthe measurements only reflect certain specific cases. It isvery necessary to perform systematic tests on the sur-vival times of SARS coronavirus subject to electric,magnetic, acoustic, optical, thermal or acid-alkali effectswith different strengths. Only in this way, can the databank for the activation and factor be established. Thecurrent data for the survival times of the SARS coro-

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navirus provided by different labs appears a little dis-perse from each other. Their accuracy is in need ofre-evaluation. One possible way can be that, the same SARScoronavirus from the only source should be distributed tomultiple labs over the world for independent tests on thesurvival time of the sample under the same experimentalconditions strictly defined. In this way, deviation from eachtest can be tested and thus allows for a comprehensiveunderstanding on the SARS coronavirus. Such strategy willalso be helpful for testing the variability of SARS corona-virus. The causes to kill the SARS coronavirus can be at-tributed to the effects of changing the activation energy andfrequency factor. Any drug or engineered way capable ofinducing a death of SARS coronavirus will be beneficial forthe treatment of SARS. Therefore, it is critical to performexperimental study on the survival or replication of SARScoronavirus either in vivo or vitro. Through a compre-hensive analysis on the effects of various physical andchemical factors, one can establish the possible drug orengineering way. Although the present study is mainly forthe SARS coronavirus, behaviors for any general virus toattack its host cell are approximately similar to each otherand therefore can be described by the rate process theory.Analysis given in this paper will be beneficial in quantify-ing the physical or chemical behavior of SARS coronavirus.Up to now, little is known in this area. The present studyprovides a new theoretical foundation.

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