Dynamics of the COVID-19 Comparison between the Theoretical Predictions and the Real Data, and Predictions about Returning to Normal Life

1. Abstract

A new coronavirus disease, called COVID-19, appeared in the Chinese region of Wuhan at the end of last year; since then the virus spread to other countries, including most of Europe. We propose a differential equation governing the evolution of the COVID-19. This dynamic equation also describes the evolution of the number of infected people for 13 common respiratory viruses (including the SARS-CoV-2). We validate our theoretical predictions with experimental data for Italy, Belgium and Luxembourg, and compare them with the predictions of the logistic model. We find that our predictions are in good agreement with the real world since the beginning of the appearance of the COVID-19; thisis not the case for the logistic model that only applies to the first days. The second part of the work is devoted to modelling the descending phase, i.e. the decrease of the number of people tested positive for COVID-19. Also in this case, we propose a newset of dynamic differential equations that we solved numerically. We use our differential equations parametrised with experimental data to make several predictions, such as the date when Italy, Belgium, and Luxembourg will reach a peak number of SARS-CoV- 2 infected people. The descending curves provide valuable information such as the duration of the COVID-19 epidemic in a given Country and therefore when it will be possible to return to normal life.

2. Key words

Mathematical model; COVID-19;Pneumonia

3. Introduction

Viral infections usually affect the upper or lower respiratory tract. Although respiratory infections can be classified according to the causative agent (e.g. the flu), they are mostly clinically classified according to the type of syndrome (e.g., common cold, bronchiolitis, laryngo-tracheo-bronchitis acute, pneumonia). Although pathogens typically cause characteristic clinical manifestations (e.g., rhinovirus causes the common cold, respiratory syncytial virus [RSV] usually causes bronchiolitis), they can all cause many of the most common respiratory syndromes. illness is more frequent in elderly patients and young children. Morbidity can either directly result from the infecting agent, or may be indirect. The latter case can be due to the exacerbation of an underlying cardiopulmonary disease, or a bacterial superinfection of the lung, paranasal sinuses, or middle ear. The main motivation of this work is to verify, by making theoretical predictions, that political decisions are truly effective to minimise the number of infected people in order to (i) not overload local health services (such as hospitals), and to (ii) gain time to allow research institutes to deliver vaccines or theanti-virals.

4. Comparison with the Real Data for COVID-19 before the Lockdown Measures

From a mathematical point of view, we would like to have R0 = 1 (or, better, R0 < 1), in Eq. (3) instead of R0 > 1. In practical terms, this means reducing the frequency of all involuntary It is understood that the main objective of the lockdown measures established by most European governments and health organisations is to reduce the ability of a virus to spread. contacts with a large number of 2 In this Section we shall follow the definitions and the expressions reported in standard books or thesis dissertation such as, for example, [5, 6]. 3Actually, Eq. (2) applies only if the M outbreaks of the virus are exactly at the same conditions. In general, the correct expression reads 𝑁 = ∑𝑀 𝑅 𝑡⁄𝜇̃𝑖 with 𝜇̃𝑖 indicating the replication time of the virus for the i-th outbreak. 4 In ref. [7], the doubling time is used to calculate R0, by means of the equation R0 = 1 + (γ + ρ) log (2)/µ where γ is the duration of the incubation period, ρ is the duration of the symptomatic period, and µ is the doubling time (see [7]). In this respect, we would also like to mention another excellent work recently produced by G. Steinbrecher [9] (Figure 2 and 3).

5. Modelling the COVID-19 - Virus’ growth

The objective of this section is to determine the coefficients of the evolutionary differential equation for the COVID-19 (see the where τ is linked to the steepness of the curve. Since the environmental condi- tions influence the carrying capacity, as a consequence it can be time-varying, with K(t) > 0, leading to the following mathematical model (see, for example, [12]): 𝑑𝑁 = 𝛼𝑁 (1 − 𝑁 ) (11) forthcoming Eq. (13)). We also determine the generic analytical 𝑑𝑡 𝐾(𝑡) expression for the time-dependent number of infected people through fitting techniques validated by the χ 2 tests. This expression is proposed after having previously analysed 12 respiratory infectious diseases caused by viruses [10], in

5.1 General Background Letting N represent population size and t represent time, the Logistic model model is formalised by the differential equation below: 𝑑𝑁 𝑑𝑡 = 𝛼𝑁 (1 − 𝑁) (7) 𝐾 Figure 4: Number of infected people in Italy on the 10th of March 2020 (before the adoption of lockdown measures). The blue line corresponds to the theoretical predictions and the black dots correspond to experimental data. The values of the parameters 𝑟𝑙𝑇and 𝜇𝑙𝑇 are 𝑟𝑙𝑇 ≃ 3.8 days and 𝜇𝑙𝑇 ≃ 2.6 days, respectively. where α > 0 defines the grow rate and K > 0 is the carrying capacity. In this equation, the early, unimpeded growth rate is modelled by the first term +αN. The value of the rate α representsthe proportional increase of the population N in one unit of time. Later, if the system is closed (i.e. the system is isolated and, hence, not in contact with a reservoir allowing the system to exchange individuals), as the population grows the modulus of the second term, α N 2 /K, becomes almost as large as the first, until to saturating the exponential growth. This antagonistic effect is called the bottleneck, and is modelled by the value of the parameter K. The competition diminishes the combined growth rate, until the value of N ceases to grow (this is called maturity of the population). The solution of Eq. (7) is

Determination of the Carrying Capacity and the O.D.E. [10] 𝑑𝑁 = 𝛼𝑁 (1 − 𝑑𝑡̂ 𝑁 𝐾𝑁 2 ) − ( 𝛼𝑡 −1 ) 𝑁 with 𝑡̂> 1⁄𝛼1⁄2 𝑡̂ (14) Lockdown Coefficient for the COVID-19 According to ref. [15]6 Respiratory viruses remain quiet for where we have introduced the dimensionless time ^t ≡ t/t0. The coefficient months, inactive but viable, within living cells. Then suddenly they activate and become virulent as they say, the infectious 𝑐(𝑡) ≡ ( 𝛼𝑡̂2−1 ) with 𝑡̂> 1⁄𝛼1⁄2 𝑡̂ (15) capacity grows to a maximum, after which it decreases. The time duration is about of 2 or 3 months. So we can expect that the epidemic will soon die out in Italy too. So, there is no valid reason to think that this coronavirus behaves differently from others [15]. The present work starts from the following hypothesis: the SARS-CoV-2 behaves like the other viruses that cause respiratory diseases. As a consequence, for the COVID-19 case, functions K(t) and c(t) are determined by performing several fittings on the growth rate-trends of infection capacity of the viruses that mainly affect the respiratory system. More specifically, we considered the following 13 different diseases caused by 12 different viruses: Whooping Cough (Pertussis), Swine Flu (H1N1), Bird Flu (Avian Flu H5N1), Enterovirus, Flu in Children, Flu in Adults, Bacterial Pneumonia, Viral Pneumonia, Bronchitis, Common Cold (Head Cold), Severe acute respiratory syndrome (SARS), and MERS (Middle East Respiratory Syndrome). In all the examined cases, we took into account the fact that the therapy-induced death rate is greater than is referred to as the average therapy-induced death rate. In our case the term c(t)N in the dynamic equation represents the lockdown measures. The lockdown is mainly based on the isolation of the susceptible individuals, eventually with the removal of infected people by hospitalisation7 . In the idealised case, for 𝛼𝑡̂2 > 1, c(t) may be modelled as a linear function of 𝑡̂, by getting 𝑐(𝑡̂) = 𝛼𝑡̂ (16) As for the epidemiological explanations relating to the various modelling of c(t) (constant, linear in time etc.), we refer the reader to the well-known and extensive literature on the subject (see, for example, Ref [25] or to the references cited in [26]). Here, we limit ourselves to give a very intuitive explanation on the physical meaning of this contribution Immediately after the lockdown measures have been adopted, i.e. during the very first initial

pComparison between the Theoretical Predictions and Experimental Data For Italy and Belgium one observes two distinct phases related to the dynamics of the COVID-19, which we classify as before the adoption of the lockdown measures and after some days after the adoption of the lockdown measures. The question therefore naturally arises, of whether these two types of regime are 𝑡 ≡ 𝑡𝑀𝑎𝑥 = 1 + 1 (19) separated by a well-defined transition. We shall see that this is 𝑀𝑎𝑥 𝑡0 𝛼1⁄2 2 indeed the case. We may identify three different periods, which may be classified as follows: Notice that α islinked to µ. Indeed, as shown in Section 1, during the exponential period the COVID-19 grows according to the law (see Eq. (6)): 1. The exponential period. As seen in Section 1, before the adoption of lockdown measures, the exponential trend is the intrinsic behaviour of the grow rate of the COVID-19

1. The transient period. The transient period starts after having applied the severe lockdown measures. In this period, we observe a sort of oscillations (or fluctuations) of µ versus time. In this case the time variation of µ(t) reflects the behaviour of the time effective reproduction number, R(t), defined as the number of cases generated in the current state of a population, which does not have to be the uninfected state. Fig. 6 and Fig. 7 show the behaviour of the parameter µ versus time for Italy and Belgium, respectively. The transient period ends when the last step of the exponential trend fits real data as good as the linear trend9 . 2. The bell-shaped period (or the post-transient period). In the bell-shaped period parameter µ is a (typical) function of time obtained by using Eq. (14). Several theoretical models can be used to study the post-transient period (e.g., by using Gompertz’s law [28]). Here, we use two mathematical models: the solution of the differential equation (14) and the logistic model (see, for example, Ref. [16]), and we compare these two theoretical models with real data for Italy and Belgium. (Figures 8, 9, 16) (see Appendix) compare the predictions of our model (blue lines) against the logistic model (red lines) for Italy, Belgium, and Luxembourg, respectively. Notice that the number of free parameters of these two models are exactly the same, since α and τ cannot be chosen arbitrarily. More specifically, a) The logistic model possesses two free parameters: K and t0L. Notice that parameter τ is not free since it is linked to the doubling time µ; b) Also our model possesses two free parameters: KN and t0. Notice that parameter α is linked to the doubling time µ (see Eqs (14) and (18)). (Figure 8, 9) compare the theoretical predictions, with the experimental data for Italy and Belgium updated to the 15th of May 2020. The values of the parameters τ , KN, and t0L for Eq. (14) and the parameters τ , t0L and K for the logistic function are reported in the figure captions. As we can see, for both Countries Eq. (14) fits well all the real data from the initial days, while the logistic model applies only to the first data. The curves reach the plateau at the time tMax given byEq. (19). By inserting the values of the parameters, we get

6. Modelling the COVID-19

The Descending Phase Here, for the descending phase is intended the phase where the number of the positive cases starts to decrease10. So, our model cannot be used for describing also the descending phase since 𝑁𝑡 is the number of the total cases and, during the descending phase, 𝑁𝑡 tendsto reach the plateau. The objective of thissection is to determine the trend of the curve of positive people during the descending phase. This task is accomplished by establishing the appropriate equations for the recovered people and the deceased people for COVID-19. During the descent phase the logistic model applies only to the first data. The values of the parameters of Eq. (14) and the logistic function (10) are: 𝑟𝐵𝐸 ≃ 5.3 days (𝜇𝐵𝐸 = 3.7 days), 𝐾𝐵𝐸 ≃42626, and 𝑡 ≃53.4 days for Eq. (14), and 𝑟 ≃5.3 days (𝜇 = 3.7 number of active people over time must satisfy a conservation 𝑁 0𝐵𝐸 𝐵𝐸 𝐵𝐸 equation. This allows determining the time-evolution for the positive people. In the sequel, we denote with r , d , and n the days), 𝐾𝐵𝐸 = 111000, 𝑡0𝐿𝐵𝐸 = 39.5 days for the Logistic function, respectively. The zone I corresponds to the period before the adoption of the lockdown t t t number of people released from the hospital at the time t, the total deaths, and the number of positive individuals at time t, respectively

Number of the Recovered People We start with the recovered people previously hospitalised. Let us suppose that a hospital has 50 patients in intensive therapy, corresponding to its maximum availability capacity. If the hospital is unable to heal any patient, the growth rate of healed people is necessarily equal to zero. On the other hand, if the hospital is able to heal a certain number of people, the places previously occupied by the sick people will free and other patients affected by COVID-19 will be able to be hospitalized. In the latter case, the growth rate of the healed people will rise thanksto the fact that the hospital is able to heal more and more patients. This initial phase may be modelled by introducing into the dynamic equation the term +γrt, with rt indicating the number of the recovered people at the time t, previously hospitalized

6.1.1. Approximated O.D.E. for the Recovered People Previously Hospitalised 6.2. Equation for the Deceased People The rate of deceased people per unit time is modelled by the following dimensionless equation We assume that all the infected people entering in the hospitals will heal. So 𝑑 𝑑𝑡̂ 𝑡 = 𝛼𝑑𝑛(𝑡−𝑡𝑑) − 𝛽𝑑𝑛2 (33) 𝐷𝑟(𝑡 + 𝖯1) ≈ 0 hence 𝐼𝑟(𝑡 − 𝖯) ≃ 𝑟(𝑡) (30) The meaning of Eq. (31) is the following. Manifestly, the rate of deaths is proportional to the number of active people

6.3 Equation for the Positive People 𝑑𝑡̂ 𝑡 𝑟 𝑡 𝐾𝑟 𝑡=0 𝑑 𝑑 = 𝛼 𝑛 − 𝛽 𝑛2 with 𝑑 = 0 Of course, during the descent phase, the number of active 𝑑𝑡̂ 𝑡 𝑑 (𝑡−𝑡𝑑) 𝑑 (𝑡−𝑡𝑑) 𝑡=0 people nt satisfies a simple law of conservation: If we are in the situation where there are no longer new cases of people tested 𝑛𝑡 = 𝑁𝑡 − ℎ𝑡 − 𝑑𝑡 with 𝑛𝑡=0 = 0 positive for COVID-19 and if we assume that the active people 𝑑 𝑁 = 𝛼𝑁 (1 − 𝑁𝑡 𝛼𝑡 ̂2−1 ) 𝑁 cannot leave their country of origin (or else, if they do, they will 𝑑𝑡̂ 𝑡 𝑡 𝐾𝑁 𝑡̂ 𝑡 ℎ = ∑ 𝑛=𝑡/∆𝑡 𝑟 with ∆𝑡 ≃ 1 day be rejected by the host Country), then the number of infected people cannot but decrease either because some people are deceased or because others have been recovered. In mathematical terms 𝑛𝑡 = 𝑛0 − (ℎ𝑡 − ℎ0) − (𝑑𝑡 − 𝑑0) = 𝑁𝑀𝑎𝑥 − ℎ𝑡 − 𝑑𝑡 (36) with h0, d0 and t0 denoting the values of ht, dt and nt evaluated

6.4. Theoretical Predictions for the Descending Phase In this subsection, we report the numerical solutions of Eqs (37)-(38) for Italy and Belgium. The solution for Luxembourg can be found in the Appendix. Fig. (10) and (11) concern the Italian situation. They show the numerical solution of Eqs (37)- (38) for the number of recovered people and deaths, respectively. These theoretical predictions are plotted against the experimental data reported in the (Table 1). According to the theoretical predictions, for Italy we get tIT = 12 days. (Figure Figure 13: Belgian situation. Theoretical predictions (blue line) against the experimental data (black circles) for the recovered people. Figure 14: Belgian situation. Theoretical predictions (blue line) against the experimental data (black circles) for the deceased people. 12), illustrates the descendant-phase for Italy.

7. Conclusions

In this work we studied the spread of SARS-CoV-2 until when the strict measures have been adopted (i.e. until 16th May 200). The dynamics of COVID-19 when the population is under less restrictive lockdown measures will be subject of future studies. Through fitting techniques previously performed, caused by viruses, including SARS-CoV-2. The solution of Eq. (14) provides the number of the total case in time. Successively, we compared the theoretical predictions, provided by the solution of Eq. (14) and by the logistic model (see Eq. (7)), with the real data for Italy and Belgium (for Luxembourg see Appendix). We saw that the solution of Eq. (14) is in good agreement with the experimental data since the beginning of the appearance of the COVID-19; this is not the case for the logistic model which applies only to the few easily checked that, for large values of KN, the value of tˆf lex satisfies, approximatively, the equation Belgium by parametrising the solution of Eq. (41) with 3 𝑓𝑙𝑒𝑥 2 𝑓𝑙𝑒𝑥 + (𝛼 − 3)𝑡𝑓 ̂ 𝑙𝑒𝑥 + 2 ≃ 0 with 𝑡𝑓 ̂ 𝑙𝑒𝑥 ≡ experimental data: we get, tMaxIT 21 April 2020 and tMaxBE 2 May 2020 for Italy and Belgium, respectively. 𝑡𝑓𝑙𝑒𝑥 𝑡0 (41) We also noted, empirically, that the infection process caused by SARS-CoV-2 may be divided into three qualitatively different periods; i.e., the exponential period, the transient period and the bell-shaped period (or the post-transient period). The solution of Eq. (14) allows defining more precisely these three periods. Indeed, we may classify the above periods as follows The exponential period for 0 ≤ 𝑡̂≤ 𝑡̂𝐿𝑀 (40) The transient period for 𝑡̂𝐿𝑀 < 𝑡̂≤ 𝑡𝑓 ̂ 𝑙𝑒𝑥 The bell-shaped period for 𝑡̂> 𝑡𝑓 ̂ 𝑙𝑒𝑥 With tLM indicating the dimensionless time when the lockdown measures are applied and tflex the dimensionless inflection point of the solution of Eq. (14), respectively. It is http://www.acmcasereports.com/ Hence, according to Eq. (41), the transient period ended on 31 March 2020 for Italy and on 7 April 2020 for Belgium, respectively. The second part of the work is devoted to modelling the descending phase, i.e. the decrease of the number of people tested positive for COVID-19. Also in this case, we proposed a new set of dynamic differential equations that we solved numerically. The solution of Eqs (37) (and Eq. (38)) provided valuable information such as the duration of the COVID-19 epidemic in a given Country and therefore when it will be possible to return to normal life.

8. Acknowledgments

I am very grateful to Alberto Sonnino from Facebook Calibra and University College London for comments on late manuscript, and to Ing. Alessandro Leone from the Italian

9. Appendix: Comparison between the Theoretical Predictions of Eq. (14) and Experimental Data for Luxembourg

We have stressed the main difference between the closed systems and the open systems. Luxembourg, due to the particularly severe lockdown measures adopted by the government, may be considered, with good approximation, as a closed system (628108 inhabitants, most of them concentrated in only one town). Indeed, right from the start, the city of Luxembourg was literally closed and citizens were unable to enter and leave the city freely (people who had to enter in the city for working reasons were obliged to undergo each time the test that, of course, had to result negative). Italy, on the other hand may be considered, with a god approximation, as an open system (60317116 inhabitants dislocated in all the Country). In Italy, especially during the Table 4: Situation in Belgium on 15 May 2020. Columns report the number of active people (currently infected by SARS-CoV-2), the number of recovered people, and the number of deceased people. initial phase, the citizens of northern Italy moved freely to the south of Italy, by train, by plans or by car. Only in a second time the Italian government decided to introduce much more restrictive measures concerning the movement of citizens from one region to another. For the reason mentioned above, it is our opinion that it is very interesting to analyse these two Countries, Luxembourg and Italy, which are so different with each other. In this Appendix we report the comparison between the theoretical predictions of the COVID-19 model (14) and the real data for Luxembourg update to 15 May 2020 (see (Figure 16)). In the columns of (table 5) we can find the number of active people (currently infected by SARS-CoV-2), the number of recovered people, and the number of deceased people, respectively. The experimental data have been found in the databases [31, 32]. Luxembourg reached its peak on 12 April 2020.

The Descending Phase for Luxembourg (Figures 17, 18) refer to the Luxembourg situation. The figures illustrate the numerical solutions of Eqs (37) -(38) for the number of recovered people and deaths, respectively. The theoretical predictions are plotted

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Tang X. Dynamics of the COVID-19 Comparison between the Theoretical Predictions and the Real Data, and Predictions about Returning to Normal Life . Annals of Clinical and Medical Case Reports 2020