Combination anti-coronavirus therapies based on nonlinear mathematical models
J. A. Gonzalez, Z. Akhtar, D. Andrews, S. Jimenez, L. Maldonado, T. Oceguera, I. Rondon, O. Sotolongo-Costa
CCombination anti-coronavirus therapies based on nonlinear mathematical models
J. A. Gonz´alez, a) Z. Akhtar, D. Andrews, S. Jimenez, L. Maldonado, T.Oceguera, I. Rond´on, b) and O. Sotolongo-Costa Department of Physics, Florida International University, Miami, Florida 33199,USA Department of Biology, College of Arts and Sciences, University of Miami,Coral Gables, Florida 33146, USA Medical Campus, Miami Dade College, 950 NW 20th Street, Miami,Florida 33127, USA Departamento de Matem´atica Aplicada a las TT.II, E.T.S.I Telecomunicaci´on,Universidad Politecnica de Madrid, 28040-Madrid, Spain Department of Biological Sciences, Florida International University, Miami,Florida 33199, USA Department of Physics, University of Guadalajara, Guadalajara, Jalisco,M´exico C. P. 44430 School of Computational Sciences, Korea Institute for Advanced Study, Seoul 0245,Republic of Korea, Universidad Aut´onoma del Estado de Morelos, Cuernavaca, M´exico,C.P. 62209 (Dated: 1 March 2021)
Using nonlinear mathematical models and experimental data from laboratory andclinical studies, we have designed new combination therapies against COVID-19. a) Electronic mail: [email protected] b) Electronic mail: [email protected] a r X i v : . [ phy s i c s . m e d - ph ] F e b urrently there are no approved treatments for the SARS-CoV-2 infection.Moreover, scientists do not know any treatment that would consistently cureCOVID-19 patients. This paper is an argument for combination therapiesagainst COVID-19. We investigate a nonlinear dynamical system that describesthe SARS-CoV-2 dynamics under the influence of immunological activity andtherapy. Using the nonlinear mathematical model and experimental data fromlaboratory and clinical studies, we have designed new combination therapiesagainst COVID-19. The therapies are based on antivirals in combination withother therapeutic approaches. The general therapeutic plan is the following:Gene therapy and/or Antivirals plus Immunotherapy and Anti-inflammatorydrugs and/or drugs that control Cytokine Storms plus Cytotoxic therapies. Webelieve that these new therapies can improve patient outcomes.I. INTRODUCTION Emerging viral diseases have caused significant global devastating pandemics, epidemics,and outbreaks (Smallpox, HIV, Polio, 1918 influenza, SARS-CoV, MERS-CoV, Ebola, andSARS-CoV-2).Currently there are no approved treatments for any human coronavirus infection. Moreover,scientists do not know any treatment that would consistently cure COVID-19 patients. Theworld is facing a general catastrophe as people see the reality of alarming rises in infections,a building economic crisis, a shortage of ventilators, the lack of coronavirus testing, andmany other disasters. The governments are desperate to find a solution. In some cases, theyare even promoting unproven “remedies”. The novel coronavirus presents an unprecedentedchallenge for everybody, including the scientist: the speed at which the virus spreads meansthey must accelerate their research. We need a treatment that is 95% effective in order tosafely open the countries and save the world from an economic catastrophe.There is a wealth of literature dedicated to mathematical modeling of the virus-immune-system interaction. See, for example . There are many treatments in development. However,most of them have drawbacks .This paper is an argument for combination therapies against COVID-19. We have shown2efore that combination therapies can be better than monotherapies. For instance, forsome cancer tumors, the immunotherapies do not work at all . We have proposed to use acombination of therapies that could eradicate the cancer completely . In the present paperwe will design new therapies based on antiviral agents in combination with other therapeuticapproaches. These new therapies should improve patient outcomes. II. GROWTH MODELS
There are several famous equations that have been used to describe cell population growth:exponential, Gompertz, logistic, and power-law equations .In reference , a biophysical justification for the Gompertz’s equation was presented.The deduction is based on the concept of entropy. The entropy definition used in Ref is the well-known Boltzmann-Gibbs extensive entropy. Gonzalez et al. have used the newnon-extensive entropy in the derivation of a new very general growth model .The exponential, logistic, Gompertz, and power laws are particular cases of the new equa-tion. The new model has the potential to describe all the known and future experimentaldata .The non-extensive parameter q plays an important role in the new model. Suppose weare studying virus population dynamics.Different types of viral infections can possess different values of the non-extensive para-meter q .Boltzmann-Gibbs statistics satisfactorily describes nature if the microscopic interactionsare short-range and the effective microscopic memory is short-ranged, and the boundaryconditions are nonfractal.There is a large series of recently found natural systems that present anomalies that violatethe standard Boltzmann-Gibbs method.A non-extensive thermostatistics, which contains the Boltzmann-Gibbs as a particle case,was proposed in a series of papers . 3owadays, scientists have produced a large amount of successful applications of the newtheory . These are mostly phenomena in complex systems.The mentioned thermodynamic theory contains the non-extensive entropy: S q = k − (cid:80) wi =1 p qi q − , (1)where k is a positive constant, w is the total number of possibilities of the system, w (cid:88) i =1 p i = 1 , q ∈ R , (2)This expression recovers the Boltzmann-Gibbs entropy, S = − k (cid:80)(cid:80)(cid:80) wi =1 p i ln p i , in the limit q →
1. Parameter q characterizes the degree of non-extensivity of the system.This can be seen in the following rule: S q ( A + B ) /k = [ S q ( A ) /k ] + [ S q ( B ) /k ] + (1 − q ) [ S q ( A ) /k ] [ S q ( B ) /k ] , (3)where A and B are two independent systems in the sense that P i,j ( A + B ) = P i ( A ) P j ( B ) .We could say that parameter (1 – q ) characterizes the complexity of the system.The case − q ≥ implies that the system is resilient.For example, this condition indicates that the virus infection will lead to a drug-resistantdisease. In particular, this disease can become resistant to the attack of the immune systemand conventional therapy.The new generalized equation for population growth is the following dXdt = kX ∞ q − (cid:20) − (cid:18) XX ∞ (cid:19) q − (cid:18) − XX ∞ (cid:19) q (cid:21) , (4)where X ( t ) is the growing population, k is certain free parameter, and X ∞ is the asymptoticvalue of X ( t ) when t → ∞ .We have already remarked that this is a very general model that contains most knowngrowth models .Now, we will show that this model is universal in the sense discussed in Ref. and includesmany others classes of models as particulars cases. For early stages of the infection, (4) can be written in the following form dXdt = α q (cid:18) XX ∞ ( q ) (cid:19) q (cid:34) − (cid:18) XX ∞ ( q ) (cid:19) − q (cid:35) , (4a)4 I P II Y X FIG. 1. Phase space representation of the dynamics of the dynamical system (5) , (6). In this case,the fixed point P I is a stable node, the point P II is a saddle, and the point where X ≈ X ∞ is astable node (not shown). The blue line is the stable manifold of the saddle point P II . And the redline is the unstable manifold of the mentioned saddle point. The blue line is a global separatrixof the dynamics. All initial conditions that are on the “left” of the blue line will lead to a phasetrajectory that tends to a point where X = . All initial conditions that are on the “right” of theseparatrix will lead to a phase trajectory that tends to the point where X ≈ X ∞ . This pictureoccurs when q > , bV > af . where α q = kqX ∞ ( q )1 − q , X ∞ ( q ) = q q − X ∞ .An analytical solution to equation (4a) can be expressed as (cid:18) XX ∞ ( q ) (cid:19) − q = 1 − (cid:34) − (cid:18) X X ∞ ( q ) (cid:19) − q (cid:35) e − qkt (4b)where X is the initial condition so that X ( t = 0) = X .We can re-write solution (4b) as r ( τ ) = 1 − e − qτ (4c)where r = (cid:16) XX ∞ ( q ) (cid:17) − q and τ = kt − ln (1 − r ) /q .This calculation shows that our model represents a universal growth law .So even this general class of growth laws are a particular case of equation (4).5 II. THE MODEL
In the present paper, we will investigate the following dynamical system dXdt = kX ∞ q − (cid:20) − (cid:18) XX ∞ (cid:19) q − (cid:18) − XX ∞ (cid:19) q (cid:21) − bXY − c ( t ) X, (5) dYdt = d ( X − eX ) Y − f Y + V − c ( t ) Y, (6)where X denotes the virus population and Y denotes the population of lymphocytes. Equa-tion (5) describes the reproduction of the virus. The virus is killed when it meets agents ofthe immune system (term − bXY ).The reproduction of the agents of the immune system is described by the term d ( X − eX ) ,where initially the presence of the virus stimulates the reproduction of Y ( t ) . When virusload is very large, the person is so sick that the reproduction of Y ( t ) is inhibited. The term − f Y corresponds to the natural death of lymphocytes. The term V represents an externalflow of lymphocytes.The term − c ( t ) X stands for virus-killing process due to different therapies. The term − c ( t ) Y shows that therapies can also affect other normal cells (including the immunesystem).The system (5) and (6) is inspired by models of the immune system developed in references and . However, instead of the exponential growth assumed in , we are using our growthmodel given by Eq. (4). IV. INVESTIGATION OF THE MODEL
First, we will consider the case where q > , X ∞ e >> , c ( t ) = 0 , c ( t ) = 0 .Let us define a = qkq − . The dynamical system (5)-(6) can have, in principle, four fixedpoints P I = ( X , Y ) = (0 , Vf ) (7)6 I P II P IV Y X FIG. 2. This situation is topologically equivalent to that shown in Fig. (1) However, now we cansee the fixed point where X ≈ X ∞ . (Far right). P II = ( X , Y ) , (8)where e < X < X ∞ , Y = ab , P III = ( X , Y ) , (9)where < X < e , Y = ab , P IV = ( X , Y ) , (10)where X = X ∞ ,The conditions for the existence of points P II and P III are the following inequalities14 e − h > , (11) h > , (12)where h = (cid:0) f ab − V (cid:1) bead The eigenvalues of the Jacobian matrix corresponding to the fixed point P I are λ ( I )1 = a − bVf , (13) λ ( I )2 = − f. (14)7f af < V b , the fixed point P I is a stable node and the fixed point P II is a saddle (Seefigures (1) -(2)) . If af > V b , and h − e < , then the four fixed points exist and arenon-negative. Both fixed points P I and P II are now saddles. Between these two points,there is the point P III , which is stable (See Fig. (3) and Fig.(4))).If af > V b , and h − e > , then there are only two fixed points: point P I whichis now unstable and point P IV , which is stable. As a result, most trajectories tend topoint P IV (with maximum virus population) . This is not a very favorable situation for thepatient. (See Fig. (5))In the neighborhood of point P II , the separatrix of the saddle can be approximated bythe straight line Y = − (cid:32) λ ( II )2 bX (cid:33) X + a + λ II b (15)Any point corresponding to initial conditions of the Cauchy problem on the right of theseparatrix leads to a dynamics where the trajectory approaches the point of maximum virusload (point P IV ).On the other hand, if the initial conditions correspond to a point located on the left of theseparatrix, the system will evolve to a stable fixed point.Using (15), we can calculate the threshold or critical virus population that would lead to adynamics approaching point P IV : X crit = (cid:32) aλ ( II )2 (cid:33) X . (16)When X ∞ is small, the outcome can be very favorable.For instance, when X ∞ < e + 2 f d , (17)all the phase trajectories tend to the fixed point P I ( X = 0) .We can also apply the isocline method in order to further investigate the system. A carefulanalysis of the behavior of the phase trajectories allows us to conclude that the condition d > ef, (18)8 y x FIG. 3. Phase space representation of the dynamics of system (5) ,(6) . In this case, the fixed point P I is unstable, the fixed point P II is still a saddle. Now there is a new fixed point P III that is astable node (for which X > ). Now the separatrix is represented with a red line. All the phasetrajectories that are on the “left” of the separatrix are approaching the point P III (where X > ).All the phase trajectories that are on the “right” of the separatrix are approaching the fixed pointwhere X ≈ X ∞ . Note that there are trajectories for which X ( t ) is monotonically increasing from asmall value until it reaches its maximum (Point P III ). There are other trajectories for which X ( t )reaches a maximum, and, later, it decreases until it enters the point P III . Conditions bV < af and h − < are satisfied. is favorable for the patient. This is a sufficient condition to avoid an uncontrollable rise ofthe virus population leading to the point P IV .In many cases, it is convenient to re-write the system (5) -(6) as one equation where theonly unknown is X ( t ) , d Xdt + (cid:2) f − d ( X − eX ) (cid:3) dXdt − X (cid:18) dXdt (cid:19) = − dU ( X ) dX (19)In general, it is useful to discuss the dynamics of virus population as a general equation ofthe following type d Xdt + F dis (cid:18) X, dXdt (cid:19) = − dUdX (20)See Refs for a simple explanation. Equation (20) is equivalent to a Newton’s equation fora “fictitious” particle moving in the potential U ( X ) under the action of nonlinear damping.9he potential U ( X ) can have minima and maxima. So we can conceive the situationwhere the “fictitious” particle is trapped inside a potential well. The particle needs to jumpover a barrier for the virus population to continue increasing. Studying the relative heightsof the barriers, we get the condition9 e ( V b − af ) + 2 ad > , (21)when this condition is satisfied, the “right” barrier of the potential well is higher than the“left” barrier. This case is more favorable for the patient.A careful analysis shows that the condition2 d > ef, (22)is very favorable for the patient.The general meaning of conditions (18), (21) and (22) is that the comparison between thevalues of d and the product ef can decide the outcome.Let us analyze now the case q ≤ . When q ≤ , (23a)the point P I will be always unstable. This means that it is almost impossible to reduce thevirus population to zero. This finding will play a very important part in the design of newtherapies.Let us discuss time-dependent therapy against COVID-19.Let us consider the dynamical system (5)-(6) with time-dependent therapy c ( t ) and c ( t ) = (cid:15) c ( t ) where (cid:15) < .Using ideas from , we can obtain the following result. If q < , it is very difficult to curethe virus disease. If we have a target decay for the virus population X ( t ) , then c ( t ) mustbehave as c ( t ) = (cid:20) kX − q ∞ − q (cid:21) (cid:18) X ( t ) − q (cid:19) (23b)For instance, if we require the virus population to be reduced following a power law, say X ( t ) ≈ α/ t γ , then the therapy must behave as c ( t ) = t γ (1 − q ) . The exponent gamma10 y x FIG. 4. This dynamics is similar to that shown in Fig. (3). However, here the fixed point P III isa stable focus. represents the rate of decay of the virus population.We have designed therapies using the following late-intensification schedules: c ( t ) = c ln( e + δt ) and (23c) c ( t ) = c [1 + δt ] γ (1 − q ) (23d)where c ( t ) = c is a well-known constant-dose treatment taken by a patient for several days.(See Ref. ). Our logarithmic late-intensification schedule has been very successful (See andreferences quoted there in). The traditional therapy is changed only slightly.However, the results are spectacular. V. MONOTHERAPIES
When q > , the parameters of the system and the initial conditions play an importantrole in the outcome. The virus-host interaction is decisive. There are situations where theimmune system by itself can reduce the virus population to zero. Under other circumstances,the virus population will increase to numbers that can threaten the patients survival. If weapply conventional antiviral therapies with c ( t ) = c in the system (5) and (6) (where c is a constant), for q > , the cure can be accelerated .If q ≤ , then for any value of c , the virus population is never reduced to zero. The fixed11 I P IV Y X FIG. 5. In this case, the fixed points P II and P III disappear. Point P I is unstable. All the phasetrajectories tend to the point P IV , where X ≈ X ∞ . Conditions af > Vb and h − > aresatisfied. point P I is always unstable.The medical significance of this result can be expressed employing this statement: when q ≤ the disease is resistant to the immune response and the action of conventional therapy. VI. COMBINATION THERAPIES
Our analysis shows that for q ≤ , the virus can develop resistance both against theattack of the immune system and all conventional monotherapies with constant doses ofthe medication. All this investigation leads to combination therapies.First, we have to use therapies that change the parameters in such a way that fixed point P I becomes asymptotically stable (a stable node). Then we need to apply therapies that willhelp the phase trajectory to go to the point P I . The condition q > should be completedwith the stability of fixed point P I : bV > qkfq − , (23e)and complemented with condition (18).This means that immuno-therapy is also very important for the development of antiviral12herapies.This work can guide physicians to rationally design new drugs or a combination of alreadyexisting drugs for the development of antiviral therapies.Condition (23e) shows that the killing ability of the immune system and the externalflow of lymphocytes should be stronger than the virus replication and the natural death ofimmune system agents.Additionally, condition (18) says that the reproduction of the lymphocytes should bestronger than the inhibition of the immune system due to the general health weaknessescreated by the disease.The perfect strategy is to use a therapy that can change q so that the fixed P I canbe, in principle, stable. Of course, this does not guarantee that the point P I is stable.The condition q > is a necessary condition for the stability of point P I . However,it is not a sufficient condition.Later we need another therapy that will change the other parameters (see section 3) sothat the fixed point P I is actually asymptotically stable. This step is probably satisfiedwith an immuno-therapy.Finally, we need a treatment c ( t ) that definitely kills the virus, leading the phase tra-jectory to the fixed point. The ideal candidate for the first task could be a gene-targetedtherapy. On the other hand, we believe there are antivirals that can be utilized in order toaccomplish this goal. VII. SOME BIOPHYSICAL ANALYSIS
Drug repurposing for SARS-CoV-2 is very important for our world. It can represent aneffective drug discovery strategy from existing drugs. It could shorten the time and reducethe cost compared to de novo drug discovery .13hylogenetic analysis of 15 HCoV whole genomes reveal that SARS-CoV-2 shares thehighest nucleotide sequence identity with SARS-CoV .A molecular docking study has been published by Abdo Elfiky .The results show the effectiveness of Ribavirin, Remdesivir, Sofobuvir, Galidesivir, andTenofovir as potent drugs against SARS-CoV-2 since they tightly bind to its RdRp. Addi-tional findings suggest guanosine derivative (IDX-184), Sefosbuvir, and YAK as top seedsfor antiviral treatments with high potential to fight SARS-CoV-2 strain specifically. VIII. REAL EXPERIMENTS AND CLINICAL STUDIES
We have reviewed the medical literature on COVID-19 treatments. There is experimentalevidence supporting combination therapies
However, medical practice has been concen-trated mostly on monotherapies.
Even when combination therapies have been used, in many cases, the combinations havenot been optimized. We believe we can improve the treatment outcomes using our results.Combination of antivirals is the most common therapeutic set.
In many cases, the usedantivirals were previously developed for other viruses (e.g. SARS, MERS, Ebola, Flu, andHIV).
We present a summary of the studies about COVID-19 treatments.Therapy 1: Lopinavir + Ritonavir (Antivirals against HIV) + Oseltamivir (Flu) Lopinavir/Vitanovirare approved protease inhibitors for HIV.Results: There is some scientific evidence that this combination can work
Inconsistent results in some completed clinical trials.Therapy 2: Antiflu Arbidol + Anti-HIV antiviral Darunavir.Results: There is some scientific evidence that this combinations can help.
Therapy 3: Lopinavir/Ritonavir + Ribavirin.This is an anti-HIV therapy used in SARS.14esults: There is some scientific evidence that this combination could work .Therapy 4: Remdesivir + Lopinavir/Ritonavir + Interferon beta.Remdesivir interferes with virus RNA polymerases to inhibit virus replication, and was usedfor Ebola virus outbreak.Results: The combination is being tested for MERS. There is some scientific evidence thatthis combination could work for other coronaviruses Therapy 5: Cloroquine, Hydroxycloroquine.This is an antimalarial drug.Results: Inconsistent results in completed clinical trials , and .Therapy 6: Hydrocloroquine + antibiotic azithromycin.Results: Dedier Raoolt and coworkers published results of a completed clinical trial that“proved” efficacy . However, nowadays, this therapy is considered controversial.Therapy 7: Convalescent plasma.Convalescent plasma from cured patients provides protective antibodies againstSARS-CoV-2.Results: Proven efficacy Therapy 8: “Natural killer” cell therapy.Natural killer cell therapy can elicit rapid and robust effects against viral infections throughdirect cytotoxicity and immunomodulatory capability. Results: Being tested in clinical trials , and .Therapy 9: EIDD-2801.This is an antiviral.EIDD-2801 is incorporated during RNA synthesis and then drives mutagenesis, thus inhibi-ting viral replication .Results: Being tested in clinical trials .15herapy 10: Remdesivir.This is an antiviral that interferes with virus RNA polymerases to inhibit virus replication.Results: Approved by FDA Inconsistent and conflicting results in completed clinicaltrials. . This is a promising drug.Therapy 11: Ivermectin.Results: Leon Caly et al have observed that the FDA-approved drug Ivermectin inhibitsthe replication of SARS-CoV-2 in vitro. The authors have shown that this dug actually“kills” the virus within 48 hours. This drug is being used extensively and massively in somecountries (e.g. South America), in some cases, as a national policy. However there are nocompleted clinical trials.Therapy 12: Human monoclonal antibodies.Results: Chuyan Wang et al have found a human monoclonal antibody (47 D
11) that neu-tralizes SARS-CoV-2 Being tested in clinical trials.Therapy 13: Mesenchymal Stem Cells. This is a cell therapy. MSCs have regenerativeand immunomodulatory properties and protect lungs against ARDS.MSC therapy can inhibit the over activation of the immune system and promote repairimproving the microenvironment. They regulate inflammatory response and promote tissuerepair and regeneration . Results: Proven efficacy in completed clinical trials .Therapy 14: Lopinavir/Ritonavir + Ribavirin + Interferon beta-1b1.Results: Fan-Ngai Hung et al have published positive and promising results of a clinicaltrial . IX. NEW COMBINATION THERAPIES
We have estimated the parameters of the model (equations (5) and (6)) using publisheddata from the dynamics of different kinds of biological populations
These data include sets virus population and cell populations.In all the cases, the growth of the studied populations represents the most relevant behavior16f the disease.Some examples are the virus population and the neoplastic cell population. Normally, theseare the only published data.The value of the virus population density is provided by the viral load = number-of-copies/mL.Actually, what we usually know is log [number-of-copies/ mL , or log [number-of-copies/ cells].Sometimes the parameters are estimated using best-fit solution functions.In other cases, approximate values of the parameters are obtained from some characteristicsof the dynamics like the slope of the tangent line to the graphed function, the fixed points,the stability conditions, and the eigenvalues of the Jacobian matrix.Often, data about the immune behavior is not explicitly available. So, we use a ver-sion of the model that consists of one nonlinear differential equation only for X ( t ) .However, that equation contains the parameters that characterize the immune system.Thus, these parameters can be estimated, too.We have observed several patterns in the virus dynamics First pattern: the viral load increases rapidly and reaches a peak. Then the viral loaddeclines due to the action of a strong immune system. The final viral load cannot be de-tected. We assume it is zero. (See Fig. (1)).Second pattern: the viral load increases rapidly and reaches the peak, followed by a plateau.The plateau can be short or long. After the plateau, the viral load declines to zero. (Inthis case, the dynamics reaches a fixed point. Then, the parameters of the immune systemchange (e.g. b , V )). Then the fixed point P I is stable again.Third pattern: the viral load increases rapidly and reaches a peak, followed by a plateauwith a large value of the virus load. The plateau never ends. The patient dies. The viralload never declines. Fig. (5).Fourth pattern: the viral load increases rapidly and reaches a peak. Then the viral load17eclines. The decline is followed by a long plateau. The value of the virus load is muchsmaller than the peak. However it is far from zero. (See Fig. (3))These behaviors can also occur under the action of therapy Let us introduce the units of the variables and parametersDefine the variable X ( t ) = log [ number of copies / mL ].[ X ( t )] = uv . Thus X = 1 uv if (the number of copies) /mL = 10. Here uv stands for unitof viral load where X given in units of log [number-of-copies/mL]. Y ( t ) is the number of lymphocytes/ mL , [ Y ( t )] = 1 nc , where 1 nc = 1 lymphocyte /mL ,[ a ] = 1 /day , [ f ] = 1 /day , [ b ] = 1 / ( nc ) day , [ V ] = nc/day , [ d ] = 1 / ( uv ) day , [ a ] = 1 /uv , q isdimensionless.Let us discuss some particular examples of real virus population growth.Example 1 (Patient 14 in Ref. ) X ∞ = 8 uv, k = 0 .
04 (1/day), q = 2, ( a − V b/f ) = − .
71 (1/day), ( d − ef ) = 0 . ) X ∞ = 8 . q = 2, k = 0 .
05 (1/day), e = 0 . b = 0 .
028 (1/(nc) day), f = 0 .
251 (1/day), V = . ((nc)/day), a = 0 . d = 0 .
119 (1/(uv) day).In this case, the viral load will reach the maximum. Then the virus load will decline. But itwill not approach zero. The value of X ( t ) will be approximately constant for a long time.In the dynamical system this is a stable fixed point. The real data shows a long plateauwhere uv < X < uv . The known data does not show an end to this plateau.Example 3 (patient 1 in Ref. ) X ∞ = 7 . X is given in units of log [number-of-copies / q = 0 . k = 0 . f − V b/a = 0 .
25 (1/day), d − ef = 0 . / e ) − (cid:112) (1 / e ) − h = 4 . ∗ . Initially, the viral load increases and reaches the maxi-mum, followed by a plateau. Both the model and the real data agree with this.18ater this patient was treated with Remdesivir. Now we re-estimate the parameters X ∞ = 7 . ∗ , here X is given in units of log [number-of-copies / q = 2, k = 0 .
01 (1/day), a − V b/f = − . d − ef = 0 . q and ( a − V b/f ).The cases of patients 2 , are very similar to Example 1.The immune system is able to eradicate the virus without external therapy.Example 4 (patient 3 from Ref. ) X ∞ = 7 . ∗ , q = 0 . k = 0 .
04 (1/day), f − V b/a = 0 .
46 (1/day), d − ef = − .
46 (1/(uv) day).The viral load reaches a maximum, followed by a plateau. This is an 80 years old man witha very depressed immune system (he had had thyroid cancer).This patient was sick with COVID-19 for 24 days. He was medicated with Remdesivirstarting on day 16.The viral load decreased slightly. However, the immune system was too weak.The viral load never reduced to zero. The patient died on day 24. This patient probablyneeded a combination therapy containing antivirals, immunotherapy and a virus-killingmedication.Example 5 (patient DF from Ref. ) X ∞ = 8 . q = 0 . k = 0 .
02 (1/day), b = 0 .
024 (1/(nc)day), d = 0 . f = 0 . e = 0 .
02 (1/(uv)), V = 0 . ) X ∞ = 9 . q = 1 . k = 0 .
01 (1/day), a − V b/f = − .
35 (1/day), d − ef = 0 . q .The immune system was working well. The viral load is eradicated. This is seen inthe dynamics of the model and in the real clinical data.We have investigated all the data published in Refs .For instance, in Ref. , the authors studied 52 patients. The cases are very similar to theexamples and patterns that we have described here. They found mild, severe, critical, anddeadly cases.In general, considering all the literature here are interesting points that must be re-marked.Some older patients with rapid evolution towards critical disease with multiple organ failurepresented a long sustained persistence of SARS-CoV-2.This persistent high viral load is explained by the ability of the SARS-CoV-2 to evade theimmune response .SARS-CoV-2 might be able to inhibit immune system signaling pathways, resulting ina malfunctioning of the immune system.In most critical patients, the blood viral load was never eliminated .This can be explained with the stable fixed points of our model.The results of our investigation of the model, the virus kinetics research, and the data fromlab experiments and clinical studies lead us to the following strategy to cure COVID-19:A combination of antivirals can change the virus reproduction capabilities (parameter k ) and drug resistance (parameter q ). This can make the fixed point P I stable.A combination of immunotherapies can boost the immune system (parameters ( b, d, V ).The agents of the immune system can reduce the virus load. (See Figs. (1) -(4)).20 virus-killing therapy.Even if the point P I is stable and the immune system is working, it is possible that the virusdynamics is not riding a phase trajectory that is approaching the fixed point P I , where X = 0. For instance, if the initial condition is on the “right” of the separatrix of the saddlepoint P II , then X ( t ) is not approaching the point X = 0.A virus-killing therapy can change the position of the initial point ( X , Y ) , in such a waythat this point will be on the “left” of the separatrix (See Fig. (1) ).Now there is always a phase trajectory that will drive the viral load, X ( t ) , to the pointwhere X ( t ) = 0. The particular medications that will be used in every combination areselected from the set of drugs already tested in clinical trials.The ideas discussed in the first 6 sections of the paper lead to the conclusion that weneed a combination therapy that contains at least some the following features:(A) A combination of drugs that impair somehow the biophysics of the virus replication,infection and/or treatment resistance.(B) A combination of drugs that enhance the immune system ability to provide enoughagents and their capability to fight the virus + anti-inflammatory drugs.(C) A cell-killing therapy.Our paper is not only about mathematical models. We have critically reviewed all thepublished data about possible medical treatments against COVID-19.We have used a method that we have developed called Complex Systems Investigationto analyze the data.Complex Systems Investigation contains ideas from Nonlinear Dynamical Systems, In-verse Problems, and Experimental Design Mathematics. Our results show that a successfultreatment should be a combination of therapies as that shown in Fig. (6)This is just a useful therapeutic plan. We will see later that the role of a cytotoxictherapy sometimes can be played by an immunotherapy or an antiviral. Fig. (6) shows avery general plan. 21 IG. 6. General Therapeutic Plan
Now we will present several concrete combination therapies. There are certain observations that support the existence of synergism between Remdesivir and monoclonal antibodies.Considering the fact that our investigation leads to a combination of antivirals, immunother-apy, and virus – killing medications, the mentioned synergism help us build the treatmentshown in Fig. (12).The simplest of our designed therapies is shown in Fig. (7) and Figures (8)-(15) showdifferent alternative treatments. FIG. 7. One of the simplest therapies: Remdesivir plus Immunotherapy plus Monoclonal antibodies
Baricitinib is an important anti-inflammatory drug. It has also anti-viral effects . FIG. 8. Anti-HIV cocktail plus Immunotherapy plus Ivermectin IG. 9a. Antiviral combination plus Interferon beta plus Natural Killer Cell TherapyFIG. 9b. Antiviral combination plus Interferon beta plus Natural Killer Cell Therapy plus Anti-inflammatory drug
A commonly used steroid, Dexamethasone, can control the cytokine storms and can reducethe risk of death.Consider all the calculations presented together with the previous examples. We have donea similar research work with all the experimental data available in the references
Sometimes, the data is very fragmented. In some cases, we only know the input, the medi-cations, and the output.For instance, consider a patient with the following estimated parameters before therapy: X ∞ = 9 . uv , q = 0 . k = 0 .
04 (1 /day ), b = 0 .
01 (1 / ( nc ) day ), d = 0 . / ( uv ) day ), e = 0 .
03 (1 / ( uv )), V = 0 . nc ) /day , X = 4 uv .Evidently, the patient has a bad prognosis. There is no way that this viral load will decreaseunder natural circumstances. We will apply the therapy shown in Fig. (10).Our result is the following:The first round (antiviral combination: Remdesivir + EIDD – 2801) will produce the pa-rameters: q = 1 . k = 0 .
01 (1 /day ). These are the only parameters that can be changedwith the given antivirals.After the immunotherapy (Convalescent plasma + Interferon beta), we get( a − Vb / f ) = − .
82 (1 /day ) and d − = 0 .
71 (1 / ( uv ) day ).Additionally, the virus-killing medication (Natural Killer Cell Therapy) will reduce the ”ini-tial” viral load to the value X < . < X crit . Now there is a phase trajectory that can23 IG. 10. Powerful combination therapy that should kill the virus and save lives. The treatmentincludes a cocktail of antivirals: Remdesivir plus EIDD-2801, Immunotherapy, Natural Killer Celltherapy, Mesenchymal Stem Cells, a corticosteroid, and an anti-coagulant.FIG. 11. This is a combination of antivirals, an immune system booster, monoclonal antibodies,azithromycin, and the controversial hydroxychloroquine carry the viral kinetics to the stable fixed point P I with the value X = 0. (See Fig. (1)).We can cure this patient with fulminant COVID – 19 infection!. We believe these resultscan explain the clinical outcomes observed in references .24 IG. 12. This therapy contains a combination of antivirals, monoclonal antibodies, an immune sys-tem booster, convalescent plasma, Natural Killer Cell Therapy, and an immune system modulator.This is a powerful combination.FIG. 13. This is a next-door therapy. Any hospital should be able to provide this treatment, whichcould save patients’ lives.
Regeneron pharmaceuticals has developed monoclonal antibodies to treat MERS. This com-pany is already working on similar antibodies that might work against SARS-CoV-2.Lopinavir/ritonavir + arbidol improved pulmonary computed tomography images .Interferons + Natural killer cells are promising. Interferons can enhance natural killercells cytotoxicity. Mesenchymal stem cells will act against inflammatory factors (cytokinestorms).MTHFV1 is a gene indispensable for viral replication in bat and human cells.Carolacton is a MTHFV1 inhibitor. It is a natural bacteria-derived product .This is a good candidate for the first round in the combination therapy (see Fig. (14)).A candidate for natural killer cell therapy is CYNK-001 .25 IG. 14. This could be a perfect realization of the General Therapeutic Plan (See Fig. (6)): Genetherapy → Immunoterapy + an Anti-inflammatory drug → Cytotoxic therapyFIG. 15. The combination Remdesivir plus LAM-002 should cripple the virus, the immunotherapiesshould kill the virus, and dexamethasone should relieve the inflammation and avoid the cytokinestorms. The whole combination should control the immune system disorders, allergic reactions,and the breathing problems. This therapy should save lives and help patients recover faster.
The most powerful therapy is shown in Fig. (10) . Probably this therapy should beused in the most severe critical fulminant cases.On the other hand, Fig. (13) shows the next-door therapy. In principle, all elementsshould be available right now in every American city.
X. DISCUSSION
Remdesivir is considered the most promising drug for COVID-19 and MERS.However, the clinical trials have produced conflicting results. Sometimes the results are26ncouraging, sometimes there are no significant benefits at all. Sometimes the people arestill dying even taking remdesivir.Our response to this paradox is that remdesivir will work as part of a combination therapy.Our result is that the idea of using remdesivir and some immunotherapies in combinationwould have profoundly excellent prospects. (See figures (6)-(15)).We have tried to construct the combinations using drugs that have shown proven effi-cacy in completed clinical trials and/or laboratory experiments .Parameter q can be changed using drugs that change the nature of the virus.Parameter q is related to the nature and structure of the virus.For instance, the drug EIDD-2801 interferes with a key mechanism that allows the SARS-CoV-2 virus to reproduce in high numbers and cause infections.EIDD-2801 is incorporated during RNA synthesis and then drives mutagenesis, thus in-hibiting viral replication. So, this antiviral changes the nature of the virus.Parameters q is related to the explosive reproduction of the virus and it is related tothe difficulty to eradicate the virus.The action of EIDD-2801 and Remdesivir is different. Remdesivir shuts down viral replica-tion by inhibiting a key enzyme, the RNA polymerase.Both Remdesivir and EIDD-2801 can change parameter q . Antivirals keep the virus fromfunctioning and/or reproducing. If we combine them, we can increase the probability thatthey will do the job of changing the biophysics of the virus. Then we can add immunother-apies to eradicate the virus.These two antivirals are much more potent if given early. In general, this is the case27or most antivirals.Some physicians can have concerns because, for them, it is not clear whether severalcombinations of medications and the high doses of the drugs in question could cause sideeffects.Our research leads to the following solution to these problems: the addition of new drugsto the therapy and the total increase of doses can be administered using late-intensificationschedules (e.g. logarithmic or power-law therapies .Our stable fixed point with a small but finite virus population explains the followingmystery: why a lot of patients who recovered from Coronavirus have retested positive .The existence of a finite minimum of the virus load in order to start an infection (Eq.(16)) explains that there is a threshold value for a person exposure to sick people so thatthe person becomes infected. Our findings can also inform vaccine development. A vaccineworks by training the immune system to recognize and combat viruses.Some precedents.Therapy of HIV is complicated by the fact the HIV genome is incor-porated into the host cell genome and can remain there in a dormant state for prolongedperiods until it is reactivated. Some scientists believe that it is not possible to actuallyeradicate the virus completely.Our research shows that this is a very striking example where q ≤ . Following ourideas, it is possible that HIV can be completely eradicated. AZT was the first antiviralagent used for the treatment of HIV and was introduced in 1987. However, it became clearthat mono therapy with AZT did not provide durable efficiency and hardly made any dentin the mortality rate.Later, different studies showed that combination therapy with two nucleotide analogueswere better than monotherapy with only one.28fter several experimental breakthoughs, a combination therapy known as HAART (highlyactive antiretroviral therapy) using two or three agents became available. By combiningdrugs that are synergistic, non-cross-resistant and no overlapping toxicity, it may be possibleto reduce toxicity, improve efficacy and prevent resistance from arising.All the antiviral drugs and therapeutic methods now known were discovered by randomsearch in the laboratory.We believe that using mathematical biophysics it is possible to create a rational approachfor the discovery of new antiviral compounds and the design of the optimal combinationtherapy. XI. REMARKS • We have developed a mathematical model to describe the SARS-CoV-2 viral dynamics.The model is a nonlinear dynamical system. • We have investigated the dynamical system theoretically and numerically. • We have found conditions for the stability of the fixed point that corresponds to thecomplete eradication of the virus. • We identified the separatrix that separates the initial conditions that lead to themaximum value of the viral load from the initial conditions that lead to a limitedgrowth of the virus population. • We have studied the global dynamics of the dynamical system. We can predict theevolution of any initial condition. • The fixed point X = is stable when q > , (24) V b >af (25)29
If the following conditions are satisfied af > V b, (26) h − e > , (27)then the separatrix does not exist and there are no restrictions to the growth of theviral load. This is a terrible situation. • Furthermore, condition q ≤ means that the virus cannot be eradicated by the im-mune response or using any conventional monotherapy. • Let us discuss the biological meaning of the following conditions
V b > af, (28) d > ef, (29) q > . (30)In the real-life scenario, conditions (28)-(30) mean that the immune system is workingwell and the virus infection is not drug resistant. The combination therapy must beable to generate conditions (28)-(30). • Our study provides explanations to several phenomena that have been observed duringthe experimental studies of SARS-CoV-2 virus. • We have critically reviewed the experimental and clinical literature about COVID-19. • Using the results from the investigation of the model and experimental data fromlaboratory and clinical studies, we have designed new combination therapies againstCOVID-19.
XII. CONCLUSIONS
J. H. Bergel et al have published a paper in The New England Journal of Medicine withthe information about the NIAID-supported study titled “Remdesivir for the treatment ofCOVID-19”. 30IAID director had said that remdesivir will become the standard care of COVID-19.The drug shortened the course of illness from an average of 15 days to about 11 days.However, it is clear that the drug is not enough to help patients.The medication is not a cure and it does not act quickly. There is high mortality de-spite the use of remdesivir. So, remdesivir is not sufficient to cure patients.It seems that remdesivir does not cause an excess of side-effects.Our take is that remdesivir alone is not enough. Many other treatments, given as monother-apies, have failed to provide the promised results.Our conclusion is that we need new scientifically designed combination therapies.Using mathematical models and experimental data from laboratory and clinical studies,we have been able to design new therapies, which, we expect, will cure the patients. (Seefigures (6)-(15)).The new therapies also should be validated in double-blind, placebo-controlled trials with alarge number of patients. XIII. DATA AVAILABILITY
The data that supports the findings of this study are available within the article.
REFERENCES A. K. Abbas, K. M. Murphy and A. Sher, “Functional diversity of helper T lymphocytes”,Nature 383, 787 (1996). 31. Fenton and S. E. Perkins, ”Applying predator-prey theory to modelling immune-mediated, within-host interspecific parasite interactions”, Parasitology 137, 1027 (2010).S. Alizon and M. van Baalen, ”Acute or chronic? Within-host models with immune dy-namics, and infection outcome”, The American Naturalist 172, E244 (2008).J. D. Murray, ”Mathematical Biology”, Springer-Verlag, Berlin, (1993).R. Antia, J. C. Koella and V. Perrot, ”Models of the within-host dynamics of persistentmycobacterial infections”, Proceedings of the Royal Society of London B 263 (1996).S. Bassetti, W. E. Bischoff and R. J. Sherertz, ”Are SARS superspreaders cloud adults?”,Emerging Infections Diseases 11, 637 (2005).R. Callard and A. J. Yates, ”Immunology and mathematics: Crossing the divide”, Im-munology 115, 21 (2005).M. A. Nowak and R. M. May, ”Virus dynamics: Mathematical principles of immunologyand virology”, Oxford University Press, Oxford (2000).A. S. Perelson, ”Modelling viral and immune system dynamics”, Nature Reviews Immunol-ogy 2, 28 (2002).Z. Shen, et al. , ”Superspreading SARS events”, Emerging Infections Diseases 10, 256(2004).C. L. Ball, M. A. Gilchrist and D. Coombs, ”Modelling within-host evolution of HIV:mutation, competition, and strain replacement”, Bulletin of Mathematical Biology 69,2361 (2007).L. N. Cooper, ”Theory of an immune system retrovirus”,Proceedings of the NationalAcademy of Sciences of the USA 83, 9159 (1986).D. Wodarz, et al. ”A new theory of cytotoxic T-lymphocyte memory: implications for HIVtreatment” Philosophical Transactions of the Royal Society B: Biological Sciences 355, 329(2000).F. Dubois, H. V. J. Le Meur and C. Reiss, ”Mathematical modeling of antigenecity forHIV dynamics”, Maths In Action, 3, 1 (2011). K. Kupferschmidt and J. Cohen, “Race to find COVID-19 treatments accelerates”. Science367, 1412,(2020). H. P. de Vladar and J. A. Gonz´alez, “Dynamic response of cancer under the influence ofimmunological activity and therapy”. Journal of Theoretical Biology 227, 335 (2004).32
J. Gonz´alez, H. P. de Vladar and M. Rebolledo, “New late-intensification schedules forcancer treatments”. Acta Cient. Venez. 54, 263 (2003). J. A. Gonz´alez, et al. “New combination therapies for cancer using modern statisticalmechanics”. arXiv:1902.00728 (2019). J. A. Gonz´alez and I. Rond´on, “Cancer and nonextensive statistics”. Physica A 369, 645(2006). C. P. Calder´on and T. A. Kwembe, “Modeling tumor growth”. Mathematical Biosciences103, 97 (1991). E. M. F. Curado and C. Tsallis, “Generalized statistical mechanics: connection with ther-modynamics”. Journal of Physics A: Mathematical and General 24, L69 (1991). C. Tsallis, “Introduction to Nonextensive Statistical Mechanics: approaching a complexworld”, Springer Science and Business Media, (2009). C. Tsallis, “Possible generalization of Boltzmann-Gibbs statistics”. Journal of StatisticalPhysics 52, 479 (1988) R. Botet, M. Ploszajczak and J. A. Gonz´alez, J. A. “Phase Transitions in NonextensiveSpin Systems”. Phys. Rev E 65, 015103 (2001) G. B. West, J. H. Brown, B. J. Enquist, “A general model for ontogenetic growth”, Nature413, 628 (2001). C. Guiot, P. G. Degiorgis, P. P. Delsanto, P. Gabriele and T. S. Deisboeck, “Does tumorgrowth follow a universal law?, J. Theor. Biol. 225, 147 (2003). I. Rond´on, O. Sotolongo-Costa, J. A. Gonz´alez, and J. Lee, ”A generalized q growth modelbased on nonadditive entropy”, Int. J. Modern Phys. B. 34, 29, 2050281 (2020). N. V. Stepanova, “Course of the immune reaction during the development of a malignanttumor”. Biophysics 24, 917 (1979). Y. M. Romanousky, N. V. Stepanova, N. V. and D. S. Chernavsky, “Mathematical Bio-physics”, Nauka, Moscow (1984). J. A. Gonz´alez and J. A. Holyst, ”Solitary waves in one-dimensional damped systems”,Phys. Rev. B 35, 3643 (1987). J. Cohen, “Can an anti-HIV combination or other existing drugs outwit the new coron-avirus?”, Science, Jan 27, (2020). C. Chu, et al. , “Role of lopinavir/ritonavir in the treatment of SARS: initial virologicaland clinical findings”, Thorax 59, 252 (2004).33 T. P. Sheahan, et. al. , “Comparative therapeutic efficacy of remdesivir and combinationlopinavir, ritonavir, and interferon beta against MERS-CoV”. Nature communications 11,222 (2020). P. Gautret, et al. , “Hydroxychloroquine and azithromycin as a treatment of COVID.19:results of an open-label non-randomized clinical trial”, International Journal of Antimi-crobial Agents, 56, 1 (2020). V. Richardson, “Hydroxychloroquine rated most effective therapy by doctors for coron-avirus: Global survey”. The Washington times (Thuesday, April 12, 2020). M., Klein, New York Post (April 4, 2020). K., Thomas, New York Times (April 2, 2020). M. Wang, et al. “Remdesivir and chloroquine effectively inhibit the recently emerged novelcoronavirus (2019-nCoV) in vitro”. Cell Research 30, 269 (2020). K. Seley-Radke, “Could chloroquine treat coronavirus?”, Scientific American (March 27,2020). K. Duan, et al. “The feasibility of convalescent plasma therapy in severe COVID-19 pa-tients: a pilot study”. medRxiv (2020). C. Shen, at al. , “Treatment of 5 critically ill patients with COVID-19 with convalescentplasma”. JAMA, 323, 1582(2020). A. Maxmen, “How blood from coronavirus survivors might save lives”, Nature 580, 16(2020). P. Keith, at al. “A novel treatment approach to the novel coronavirus: an argument forthe use of therapeutic plasma exchange for fulminant COVID.19”. Critical Care 24, 128(2020). T. P. Sheahan, et al. , “An orally bioavailable broad-spectrum antiviral inhibits SARS-CoV-2 in human airway epithelial cell cultures and multiple coronaviruses in mice”. ScienceTranslational Medicine, 12, 541, (2020). National Health Commission of the People’s Republic of China, “Diagnosis and treatmentplan of Corona Virus Diseases 2019”. Global Health Journal. A. Philippidis, “Catching up to coronavirus: Top 60 treatments in developments”, GENGenetic Engineering and Biotechnolgy News, March 18 (2020). A. Philippidis, A. “Vanquishing the virus: 160+ COVID-19 drug and vaccine candidatesin development”. GEN Genetic Engineering and Biotechnolgy News, April 13 (2020).34 M. Wang, et al. , “Remdesivir and chloroquine effectively inhibit the recently emergednovel coronavirus (2019-nCoV) in vitro”. Cell Research 30, 269 (2020). M. L. Holshue, et al. , “First case of 2019 Novel coronavirus in the United States”. NewEngland Journal of Medicine 382, 929 (2020). Z. Leng, et al. “Transplantation of ACE2- Mesenchymal Stem Cells Improves the Outcomeof Patients with COVID-19 Pneumonia”. Aging and Disease 11, 216 (2020). G. Yu, “How a 100-year-old vaccine for tuberculosis could help fight the novel coronavirus”.CNN.com, April 19 (2020). A. Patri and G. Fabbrocini, “Hydroxychloroquine and ivermectin: A synergistic combi-nation for COVID-19 chemoprophylaxis and treatment? J. Am. Acad. Dermatol., June(2020). Y. Zhou, at al. , “Network-based drug repurposing for novel coronavirus 2019-nCoV/SARS-Cov-2”. Cell Discovery 6, 14 (2020). A. A. Elfiky, “Ribavirin, Remdesivir, Sofosbuvir, Galidesivir, and Tenofovir against SARS-CoV-2 RNA dependent RNA polymerase (RdRp): A molecular docking study. Life sci-ences, 253, 117592 (2020). J. Grein, et al. , “Compassionate Use of Remdesivir for Patients with Severe Covid-19”.The New England Journal of Medicine, 382, 2327 (2020). A. Feverstein and M. Harper, “Early peek at data on Gilead coronavirus drug suggestspatients are responding to treatment”. STAT, April 16 (2020). E. Silverman, A. Feverstein and M. Herper, “New data on Gilead´s remdesivir, releasedby accident, show no benefit for coronavirus patients company still sees reason for hope”.STAT, April 23 (2020). L. Caly, et al. “The FDA-approved drug ivermectin inhibits the replication of SARS-CoV-2in vitro”. Antiviral Research 178, 104787 (2020). C. Wang, et al. “A human monoclonal antibody blocking SARS-CoV-2 infection”. NatureCommunications 11, 2251 (2020). Z. Leng, R. et al. , “Transplantation of ACE2- Mesenchymal Stem Cells Improves theoutcome of patients with COVID-19 Pneumonia”Aging and Disease 11, 216 (2020). I. Fan-Ngai Hung, et al. , “Triple combination of interferon beta-1b, lopinavir-ritonavir,and ribavirin in the treatment of patients admitted to hospital with COVID-19: an open-label, randomised, phase 2 trial”. The Lancet, 395, 1695 (2020).35 H. Kuchler and D. P. Mancini, “Fauci praises remdesivir after data show it speeds recov-ery”. Financial Times, April 30 (2020). Y. Wang, et al. , “Remdesivir in adults with severe COVID-19: a randomised, double-blind, placebo-controlled, multicentre trial”. The Lancet, 395, 1569 (2020). F. Cantini, et al. , “Baricitinib therapy in COVID19: A pilot study on safety and clinicalimpact”. Journal of Infection, 81, 318 (2020). L. Deng, et al. , “Arbidol combined with LPV/r versus LPV/r alone against corona virusdisease 2019“, Journal of Infection 81, e1 (2020). J. Zhang, B. Xie, and K. Hashimoto, “Current status of potential therapeutic candidatesfor the COVID-19 crisis” Brain, Behavior, and Immunity. P. Hancocks, Y. Seo and J. Houingsworth, “Recovered coronavirus patients are testingpositive again. Can you get reinfected? CNN.com, April 18 (2020). J. H. Beigel et al. “Remdesivir for the treatment of COVID-19 – Preliminary Report”.The New England Journal of Medicine, May 22 (2020). R. W¨olfel et al. , “Virological assessment of hospitalized patients with COVID-19”, Nature581, 465 (2020). Y. Pan et al. , “Viral load of SARS-CoV-2 clinical samples”, Lancet Infect. Dis. 20, 411(2020). V. J. Munster et al. , “Respiratory disease in rhesus macaques inoculated with SARS-CoV-2”, Nature (accepted) (2020). M. M. B¨omer et al. , “Investigation of a COVID-19 outbreak in Germany resulting froma single travel-associated primary case: a case”, Lancet Infect. Dis. (accepted) (2020). Jin Yong Kim et al. , “Viral load kinetics of SARS-CoV-2 infection in first two patients inKorea”, Journal of Korean Medical Science 35, e86 (2020). Kelvin Kai-Wang To et al. , “Temporal profiles of viral load in posterior oropharyngealsaliva samples and serum antibody responses during infection by SARS-CoV-2: an obser-vational cohort study”, The Lancet Infectious Diseases 20, 565 (2020). Shufa Zheng, et al, “Viral load dynamics and disease severity in patients infected withSARS-CoV-2 in Zhejiang province, China, January-March, 2020: a retrospective cohortstudy”, BMJ 369, m1443 (2020). Nelson Lee, et al. , “Viral loads and duration of viral shedding in adult patients hospitalizedwith influenza”, The Journal of infectious Diseases 200,492 (2009).36 Fei Zhou et al. , “Clinical course and risk factors of adult in patients with COVID-19 inWuhan, China: a retrospective cohort study”, Lancet 395, 1054 (2020). Francois-Xavier Lescure et al. , “Clinical and virological data of the first cases of COVID-9in Europe: a case series”, Lancet Infect. Dis. 20. 697 (2020). L. Zou et al. , “SARS-CoV-2 viral load in upper respiratory specimens of infected patients”,N. Engl. J. Med (accepted) (2020). Victor M. Corman et al, “Viral shedding and antibody response in 37 patients with Eastrespiratory syndrome coronavirus infection”, Clinical Infectious Diseases 62, 477 (2015). Pauline Vetter et al. , “Daily viral kinetics and innate and adaptiveimmune response assessment in COVID-19: a case series”, medRxiv[doi:https://doi.org/10.1101/2020.07.02.20143271] (2020).69