**Abstract**

Objective:

To compare the effectiveness of different models for COVID-19. Dynamical models SIR Like models and empirical functions models.

Proposal of a modified SIR model to avoid lacks found in compartmental models, and obtain results more accurate like emperical models.

In SIR models the transmission term β and the recovery rate γ are constant and this is the principal error of these models. The assumption that the transmission rate parameter is time independent is false. In our work, we have extended the SIR model to a time-varying model, in which the rate of encounters and infection probability between individuals in the population is assumed to be time-varying. This better reflects the reality of our present epidemic where interventions such as stay-at-home have been put in place and relaxed and various times and compliance with recommendations such as wearing masks and maintaining physical density has also been time-varying.

**Keywords: COVID-19; Non-pharmaceutical Interventions; Bayesian SIR Models Gompertz Logistic Richards Rt R0**

Although susceptible-infective-removal (SIR) compartmental model is commonly used to describe the transmission dynamics of an infectious disease, it cannot be used when we consider only the cumulative infected population and capture the temporal variations of an outbreak, such as the turning point that is the point in time at which the rate of accumulation changes from increasing to decreasing. Several models have been proposed to estimate basic reproduction number, turning point, and final size by cumulated cases; some of them are based on purely empirical relationship, while others have a theoretical basis and are realized by differential equations. The simplest and commonly applied model among all the infectious disease models is the Richards model.

The most common approach in infective disease data analyses with simply ODE model is to select one model, usually Richards model, based on the shape of the desired curve and on biological assumptions.

The Richards models constitute a useful family of growth models that amongst a multitude of parameterizations, re-parameterizations and special cases, include familiar models such as the negative exponential, the logistic, the Bertalanffy and the Gompertz. A single wave of infections consisting of a single peak of high incidence, an S-shaped cumulative epidemic curve, and a single turning point of an outbreak can be the best fitting to data using the selected model. Inference and estimation of parameters and their precision are based on the fitted model. Therefore, the interesting questions would be as follows: Can Richards model effectively predict the growth of the cumulative infected population? How to select the best model for fitting the emerging infectious diseases data? Is it possible to predict the turning point and final size and effectively estimate the basic reproduction number which are quite important in the disease control and management?

The basic reproduction number, turning point, and final size are the most important quantities describing the emerging infectious diseases.

The basic reproduction number *R* _{0} can be obtained from the formula *R* _{0} = exp(*rT*) , where *r* denotes the intrinsic growth rate and *T* is the generation time of disease transmission.

Secondly, the turning point (or the inflection point of the cumulative case curve), defined as the time when the rate of case accumulation changes from increasing to decreasing (or vice versa). The turning point plays an important role in determining the rate of change transitions from positive to negative, that is, the moment at which the cases begin to decline. Precisely estimating this point can allow us to determine either the beginning of a new epidemic phase or the peak of the current epidemic phase, representing the point at which disease control activities take effect or the point at which an epidemic begins to wane naturally, defined by Hsieh et al.

During the study of epidemics, one of the most signiﬁcant and also challenging problems is to forecast the future trends, on which all follow-up actions of individuals and governments heavily rely. However, to pick out a reliable predictable model/method is far from simple, a rational evaluation of various possible choices is eagerly needed, especially under the severe threat of COVID-19 epidemics which is spreading world- wide right now.

Through extensive simulations made, we ﬁnd that the inﬂection point plays a crucial role in the choice of the size of dataset in forecasting. Before the inﬂection point, no model considered here could make a reliable prediction. We further notice the Logistic function steadily underestimate the ﬁnal epidemic size, while the Gompertz’s function makes an overestimation in all cases. Since the methods of sequential Bayesian and time-dependent reproduction number take the non-constant nature of the eﬀective reproduction number with the progression of epidemics into consideration, we suggest to employ them especially in the late stage of an epidemic. The transition-like behavior of exponential growth method from underestimation to overestimation with respect to the inﬂection point might be useful for constructing a more reliable forecast. Towards the dynamical models based on ODEs, it is observed that the SEIR-QD and SEIR-PO models generally show a better performance than SIR, SEIR and SEIR- AHQ models on the COVID-19 epidemics, whose success could be attributed to the inclusion of self-protection and quarantine, and a proper trade-oﬀ between model complexity and ﬁtting accuracy.

Our main ﬁndings are summarized as follows:

**1-** Sigmoid functions are more suitable for epidemic forecast. Linear,

quadratic, cubic and exponential functions are not suitable for describing epidemic data in general (see SI), while Hill’s, Logistic, Gompertz’s and Richards’ functions can well capture the typical S-shaped curve for the cumulative infected cases. SIR like models can’t well capture the sigmoid shape.

**2-**The inﬂection point is crucial for forecast. The inﬂection point plays in role in forecast. It was suggested by Zhao et al in Zika research that in the period when enough data are collected, typically when the epidemic passes the inﬂection point, predictions on the ﬁnal epidemic size by the sigmoid empirical functions, such as Logistic, Gompertz’s and Richards’ functions, will converge to the true values.

**3-**The dynamical models generally requires more reliable data to achieve

a reliable forecast than empirical functions, since the former usually involves more free parameters and more complicated mathematical structure than the latter. Based on their performance, the dynamical models can be classiﬁed into three groups. The classical SIR model and SEIR model seem to be inadequate to describe the outbreak of COVID-19,

especially the ﬁnal equilibration phase. Contrarily, the SEIR-AHQ model involves too many free parameters as reﬂected through the large AICc value. As a consequence, its robustness is also the poorest among all ﬁve models. The SEIR-QD and SEIR-PO models are two suitable ones for modeling COVID-19 by appropriately incorporating the eﬀects of quarantine and self-protection.

**MIR Model : A Modified Infected Removed Model:**

In all SIR models, β is equal to the probability of an infected person giving the disease to a susceptible person during contact and γ is approximately 1/D with D being the average duration of the disease. By combining the rate of infection and the rate of recovery, we obtain a property of the model known as the epidemiological threshold:

R0 = βS(t)/ γ

When R0 < 1, each infected person will at most infect one other person before dying or recovering. However, if R0 > 1, each infected person will infect 2 or more people causing an exponential rise in the number of people infected. In SIR models β and γ are constant ans this is the principal error of these models.**The assumption that the transmission rate parameter is time independent is false**. So let us take these assumptions:

An average infected individual

• is infectious for a period of 1/γ days and then is recovered

After 7 days an Infected individual is Recovered

• infects β susceptible individuals per day so randomly

β = rand [1, β max]

In the MIR model we found the infected Max point at the inflexion point of the cumulative infected peole, like for example Gompertz function. This is not the case in the SIR model.

The cumulative curve is more sigmoid than the SIR model. So this model can be more precise than a SIR model and very similar with the emperical functions models. Cumulatives Cases follow a Gompertz curve:

G=950000**exp(-exp(-0.058*(*t-123)))

Recovered

if($i>$NbInfectiondays) {

$iInfection=$i-$NbInfectiondays;

$im1=$i-1;

$R[$i]=$R[$im1]+$I[$iInfection];

}

$R[$i]=$R[$i-1]+($I[$i-1]*$Gamma);

$im1=$i-1;

$SigmaI[$i]=$I[$i]+$SigmaI[$im1];

$S[$i]=$S[$im1]-$Beta*$S[$im1]*$I[$im1]/$N;

#$S[$i]=$N-$I[$i]-$R[$i];

$G=950000*exp(-exp(-0.058*($i-123)));

if ($S[$i]<=0) $S[$i]=0;

#test stop

if ($I[$i]<=0) exit;

#Print

#echo $i. », ».$S[$i]. », ».$SigmaI[$i]. », ».$I[$i]. », ».$R[$i]. », ». »</br > »;

#echo $i. », ».$S[$i]. », ».$I[$i]. », ».$R[$i]. », ». »</br > »;

echo $i. », ».$G. », ».$R[$i]. », ». »</br > »;

}

?>

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