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700 Films Gatuits

http://feedproxy.google.com/~r/JordiMirBlogNouvellesTechnologiesWebOutilsPlayersGratuits/~5/u7v2KKF4bgI/cold_sweat.aviplay episode download
13 janvier, par Jordi[ —]

Cet article 700 Films Gatuits est apparu en premier sur Blog de Jordi Mir :.


COVID19 Simulation confinement

Décembre 2020, par Jordi[ —]

Dans l’article précédent nous avons vu un modèle modifié du modèle SIR.http://www.jordimir.com/2020/12/28/covid19-modified-infected-recovered-model/

Dans le modèle SIR les paramètres Beta et Gamma sont constants. Or ils ne le sont pas, en faisant de Beta une variable aléatoire nous pouvons mieux simuler la dynamique du virus. Pour modéliser un confinement nous allons comparer sur ce modèle différentes stratégies. Il suffira de modifier le paramètre Beta en cours de simulation, et voir ainsi l’importance du jour de déclenchement du confinement.

Beta = rand [1,3], puis Beta =rand [1,2]

Si l’on prend des décisions type LockDown au pic des incidences (ici à 123 jours) on n’aplanit pas la courbe, la décision confinement2 est inutile.

Cet article COVID19 Simulation confinement est apparu en premier sur Blog de Jordi Mir :.


COVID19 Modified Infected Recovered Model

Décembre 2020, par Jordi[ —]

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)))

MIR model in red VS Gompertz black curve

<?php
# Model MIR 
#N number people 
$N=1000000;
#Parameters Gamma Beta SIR model
	$Beta = 0.35; 
	 $Gamma = 0.065; 
#I0 First infected 
#Time increment = 1 day 
#Beta transmission  
$iR0=3;
$k=1;
$idays=200;
$NbInfectiondays=7;
$SigmaI = array_fill(0, $idays, 0); 
$S = array_fill(0, $idays, $N); 
$I = array_fill(0, $idays, 1); 
$I[0]=1;
$R = array_fill(0, $idays, 0); 
# Loop days
for ($i = 1; $i < $idays; $i++) { 
#Every infected people can infect randomly
$NewInfected=1;
#Beta calculation
$Beta=rand($k,$iR0)/10;
$im1=$i-1;
	$I[$i]=$I[$i-1]+($S[$i-1]/$N)*($Beta*$I[$i-1])-$Gamma*$I[$i-1]; 
#The number of infected people today (Ii) equals the number who were infected yesterday (Ii-1), PLUS the number of susceptible people who became infected today, MINUS the number of infected yesterday who recovered.
#Every $NbInfectiondays days Infected -> Recovered
if($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 >"; 
} 

?>

Cet article COVID19 Modified Infected Recovered Model est apparu en premier sur Blog de Jordi Mir :.


COVID19 Massive PCR tests a bad idea?

Décembre 2020, par Jordi[ —]

Suppose that you are worried that you might have a rare disease. You decide to get tested, and suppose that the testing methods for this disease are correct 99 percent of the time (in other words, if you have the disease, it shows that you do with 99 percent probability, and if you don’t have the disease, it shows that you do not with 99 percent probability). Suppose this disease is actually quite rare, occurring randomly in the general population in only one of every 10,000 people.

If your test results come back positive, what are your chances that you actually have the disease?

Do you think it is approximately: (a) .99, (b) .90, (c) .10, or (d) .01?

Surprisingly, the answer is (d), less than 1 percent chance that you have the disease!

Presentation Suggestions:
After discussing the reasons for the surprising probability (below), you should see how changing the parameters affects the outcome. Would the result be so surprising if the disease were more common? How would the probability change if you allow the percentage of false positives and false negatives to be different?

The Math Behind the Fact:
This fact may be deduced using something called Bayes’ theorem, which helps us find the probability of event A given event B, written P(A|B), in terms of the probability of B given A, written P(B|A), and the probabilities of A and B: 
P(A|B)=P(A)P(B|A) / P(B)
In this case, event A is the event you have this disease, and event B is the event that you test positive. Thus P(B|not A) is the probability of a “false positive”: that you test positive even though you don’t have the disease.

Here, P(B|A)=.99, P(A)=.0001, and P(B) may be derived by conditioning on whether event A does or does not occur:P(B)=P(B|A)P(A)+P(B|not A)P(not A)or .99*.0001+.01*.9999. Thus the ratio you get from Bayes’ Theorem is less than 1 percent.

The basic reason we get such a surprising result is because the disease is so rare that the number of false positives greatly outnumbers the people who truly have the disease. This can be seen by thinking about what we can expect in 1 million cases. In those million, about 100 will have the disease, and about 99 of those cases will be correctly diagnosed as having it. Otherwise about 999,900 of the million will not have the disease, but of those cases about 9999 of those will be false positives (test results that are positive because of errors). So, if you test positive, then the likelihood that you actually have the disease is about 99/(99+9999), which gives the same fraction as above, approximately .0098 or less than 1 percent!

Note that you can increase this probability by lowering the false positive rate.

Also note that these calculations wouldn’t hold if the disease were not independently and identically distributed throughout the population (e.g., in the case of cancer due to familial tendency, environmental factor, asbestos exposure, etc.).

Cet article COVID19 Massive PCR tests a bad idea? est apparu en premier sur Blog de Jordi Mir :.


COVID19 Make Your Own SIR Model PHP Code

Décembre 2020, par Jordi[ —]

Medical researchers and mathematicians have developed a series of sophisticated mathematical models to describe the spread of infectious diseases. But even a simple model is useful to predict how long an outbreak of a disease, for example the flu, will last and how many people will be sickened by it.

The oldest and most common model is the SIR model which considers every person in a population to be in one of three conditions:

  • S = Susceptible to becoming infected
  • I = Infected through contact with someone already infected
  • R = Recovered, no longer sick or infected.

Through time a person may move from being susceptible to infected to recovered, so the number of people in each condition changes, but the total of S + I + R is constant. S + I + R = N where N represents the entire population and is identified as closed system population.

This is a compartmental model, with S, I and R being compartments. Every person starts off in a compartment and many move to others over time. Graphically the compartment model looks like the figure below with the rates of movement between compartments given as Greek letters above the arrows indicating direction of movement.

Assumptions & Parameters

This is a steady-state model with no one dying or being born, to change the total number of people. In this model once someone recovers they are immune and can’t be infected again. More sophisticated models allow re-infections. The model also assumes that a disease is passed from person to person. The SIR model can’t be used for diseases that spread other ways, such as by insect bites.

To run this model, you need to know the following:

  • initial population, S (initial number of people who are susceptible),
  • initial number of infected people, I
  • Infection rate, ß (Greek letter beta, the rate (#/day) that susceptible people become infected),
  • recovery rate, γ (Greek letter gamma, the rate that infected people recover),
  • time increment, T (the time interval or steps during which changes occur.) T is usually set at one day and because its value is 1, is often ignored. For rapidly spreading outbreaks T might be one hour or some other short interval.

How do you know the values of these parameters? The number of susceptible people (S) can be the population of a city or town where the outbreak occurs. The number of initially infected people (I) is a guess unless it is known, for example, that a single traveller brought the disease into a community. The infection rate (ß) and the recovery rate (γ) can be selected from rates determined from prior outbreaks, but they often vary for different outbreaks of the same disease.

Equations

With these parameters, the number of people at any time who are Susceptible, Infected, or Recovered can be calculated with these equations:

Sn = Sn-1 – ((Sn-1/S) * (ß * In-1))

In = In-1 + (Sn-1/S) * (ß * In-1) – (In-1 * γ)

Rn = Rn-1 + (In-1 * γ)

Because of subscripts and Greek letters these equations look complicated, but they aren’t really!

These equations calculate the number of people in each condition today (n), based on the number yesterday (n-1) and the rates of change, ß and γ. The subscript n means the number in one time interval, and n-1 means the number in the previous interval. So with a time interval of one day, then the first equation:

Sn = Sn-1 – ((Sn-1/S) * (ß * In-1))

can be understood as:

The number of susceptible people today (Sn) equals the number yesterday (Sn-1), MINUS the percentage of people who become infected today (which is yesterday’s number of susceptible people (Sn-1) divided by the original number (/S)), times their rate of infection (ß), times how many people were infected (In-1) yesterday.

The number of susceptible people today equals the number who were susceptible yesterday minus the number who become infected today. As long as the disease is spreading, the number not yet infected – the remaining susceptibles – decreases every day.

The number who become infected today equals yesterday’s number of susceptibles times the rate of infection, but it may seem odd that we also multiply that result by how many were infected yesterday. The reason is that the rate of infection is for each infected person. If 3 people are infected the chance of anyone else becoming infected is 3 times as high as when only 1 person is infected. So any estimate of the rate of spread of the disease requires knowledge of the infection rate and the numbers of initially infected and initially susceptible people.

The 2nd equation says:

In = In-1 + (Sn-1/S) * (ß * In-1) – (In-1 * γ)

The number of infected people today (In) equals the number who were infected yesterday (In-1), PLUS the number of susceptible people who became infected today, MINUS the number of infected yesterday who recovered.

At the beginning of an outbreak the number of people getting infected every day is probably larger than the number recovering, so the number of infected will keep rising until more people recover than get infected.

The 3rd equation says:

Rn = Rn-1 + (In-1 * γ)

The number of recovered people today (Rn) equals the previous number who had recovered, PLUS the number who of infected people yesterday who recovered today.

The number of susceptible people always decreases, but the number of infected and recovered initially rise and then decline, as people get sick and then get better.

Let’s make a simple calculation with these realistic values for a flu outbreak:

initial susceptible population, S = 1000

initial infected people, I = 1

rate of infection, ß = 0.29/day

rate of recovery, γ = 0.15/day

time increment, T = 1 day

Note that the transition rates tell you how many days it takes to double the number of people who are infected or recovered.

Since ß= 0.29/day, the time to double the number of infected people is about 1/0.29 = 3.4 days. And γ = 0.15 implies that it takes 1/0.15 = 6.7 days to recover. Infections quickly outnumber recoveries.

To start your model you set the number of infected people for the previous day, In-1 = 1, and the number of recovered Rn-1 = 0.

Now you calculate numbers of people in each compartment at the end of day 1:

Sn = Sn-1 – ((Sn-1/S) * (ß * In-1))

S1 = 999-((999/1000) * (0.29 1)) = 998.7 are susceptible

I1 = In-1 + (Sn-1/S * ß * In-1) – (In-1 * γ)

    = 1 + (999/1000) * (0.29 * 1) – (1 * 0.15) = 1 + 0.29 – 0.15 = 1.14 people are infected

R1 = Rn-1 + In-1 * γ

     = 0     + 1 * 0.15 = 0 + 0.15 = 0.15 people have recovered

For Day 2 (n = 2):

Sn = Sn-1 – ((Sn-1/S) * (ß * In-1))

S2 = 998.7 – ((998.7/1000) * (0.29 1.14)) = 998.4 are susceptible

I2 = In-1 + (Sn-1/S *ß) – (In-1 * γ)

    = 1.14     + (998.7/1000 0.29) – (1 * 0.15) = 1.14 + .29 – 0.15 = 1.30 people infected

R1 = Rn-1 + In-1 * γ

     = 0.15     + 1 * 0.15 = 0.15 + 0.15 = 0.3 people recovered.

The decrease in numbers of susceptibles, and the increase in numbers of infected and recovered, are all very small on day 1 and day 2. This shows that the disease is not spreading rapidly. But when you plot these data over the course of the outbreak you may be surprised!

Let PHP Do it!

These equations can be calculated by hand for each day of an outbreak, but it’s a lot easier to make a PHPl script to do it. So make one!

< ?php #http://www.pandemsim.com/data/index.php/make-your-own-sir-model/ #N number people #I0 First infected #Time increment = 1 day #Beta transmission #Gamma $N=1000; $I0=1; $Beta = 0.35; $Gamma = 0.035; $idays=200; $idays7=$idays-7; $S = array_fill(0, $N, $N-1); $I = array_fill(0, $N, 1); $R = array_fill(0, $N, 0); for ($i = 1; $i <= $idays; $i++) { $S[$i]=$S[$i-1]-(($S[$i-1]/$N)*$Beta*$I[$i-1]); $I[$i]=$I[$i-1]+($S[$i-1]/$N)*($Beta*$I[$i-1])-($I[$i-1]*$Gamma); $R[$i]=$R[$i-1]+($I[$i-1]*$Gamma); #Print #echo $i.",".$S[$i].",".$I[$i].",".$R[$i].","."« ;
}
for ($i = 1; $i < = $idays7; $i++) { $Rt[$i]=$I[$i+7]/$I[$i]; #Print echo $i.",".$Rt[$i].","."« ;
}

Rt is asymptotic
Infected Cases by days

Cet article COVID19 Make Your Own SIR Model PHP Code est apparu en premier sur Blog de Jordi Mir :.


COVID19 Rt R0 un mauvais critère

Décembre 2020, par Jordi[ —]

L’outil de base de suivi d’une épidémie est la courbe épidémique. Celle-ci est construite en reportant sur un graphique l’incidence des cas, c’est-à-dire le nombre de nouveaux cas détectés, en fonction du temps, mesuré avec l’unité la plus appropriée (heure, jour, semaine…). Cette courbe fournit une description simple, visuelle, du déroulement de l’épidémie. Elle permet, par exemple, de connaître le « temps de doublement » de l’épidémie (c’est-à-dire le temps nécessaire pour que le nombre des cas soit multiplié par deux) et de constater (mais a posteriori…) que le pic épidémique a été atteint.

Une description dite « compartimentale » de la maladie transmissible, en trois phases, chez son hôte : une personne est, successivement, « susceptible » (avant d’être infectée), puis « infectieuse » (lorsqu’elle est infectée et devient contagieuse) et, enfin, « retirée » de la chaîne de transmission de l’épidémie (lors de sa guérison, de sa mise en isolement ou son décès). Le nom donné aux trois ‘compartiments’ successifs a consacré l’appellation « S-I-R » désignant ce genre de modélisation. Deux auteurs britanniques,
Kermack et McKendrik, ont fourni (dès les années 1930) un cadre mathématique précis au traitement de ces modèles par la formulation d’équations différentielles ordinaires. En résumé, l’incidence de la maladie (fonction dérivée du nombre des ‘susceptibles’) est décrite par une loi d’action de masse, où les « espèces » en inter-réaction sont, d’un côté les ‘susceptibles’ et, de l’autre, les ‘infectieux’. En effet, c’est au cours de leurs contacts, supposés proportionnels à leurs effectifs, que la maladie sera transmise et que de nouveaux cas apparaîtront. La maladie est supposée guérir à un taux constant dans le déroulement du temps, ce qui permet de compléter le modèle : on s’intéresse dans la suite de l’évolution de l’épidémie plutôt à des cas aigus qu’à des malades qui resteraient ‘infectieux’ pendant toute leur vie.

Ces trois équations ont eu un retentissement très important, car elles ont permis de progresser dans la compréhension de la dynamique d’une épidémie. En premier lieu, elles permettent, à partir d’hypothèses et
de paramètres très peu nombreux, de produire des courbes épidémiques (la valeur de I(t), en fonction du temps), qui présente une forme « en cloche», caractéristique des épidémies observées. Mais ce qui a assuré leur succès, c’est le fait qu’elles fournissent un outil très simple permettant de mesurer le potentiel épidémique : le ratio de reproduction.

Le ratio de reproduction
Dans la dynamique précoce d’une maladie transmissible, un paramètre essentiel est le ratio de reproduction, noté « R0 » . La valeur de ce paramètre permet de classer les maladies par potentiel épidémique. De plus, il est d’une interprétation fort simple. R0 correspond, en effet, au « nombre de cas secondaires directement infectés par une unique personne infectieuse, placée dans une population totalement susceptible à la maladie ». L’intérêt du paramètre R0 est immédiat (comme l’illustre la Figure ) : si R0 est supérieur à 1 (R0>1) chaque individu infecté va être capable de « se reproduire » en infectant plus qu’un seul autre individu, ce qui permettra à la maladie de se répandre dans la population, causant une épidémie. En revanche, si R0 est inférieur à 1 (R0<1), un individu infecté aura (en moyenne) moins d’un descendant : il n’y aura donc pas d’épidémie.

Rt est-il un bon critère de suivi d’épidémie?

R0 est la valeur du taux de reproduction à t=0, ensuite il varie en fonction du temps.

Dans les articles précédents, nous avons vu que d’autres modèles existent pour étudier les épidémies:

L’équation de Gompertz permet d’obtenir le nombre d’incidences en fonction du temps.

Si nous reprenons l’équation de Gompertz pour I(t+7) et I(t) nous pouvons étudier l’évolution de Rt7 selon Gompertz:

Rt= It7/It

RT7 GOMPERTZ

Au max des incidences t=60 nous avons Rt=1 puis une décroissance asymptotique vers 0.8.

Il est donc illusoire de vouloir des taux de reproduction inférieurs à cette valeur. Les incidences obtenus par test PCR ne donnent pas une valeur exacte du nombre de personnes infectées. Le nombre de test varie sur une population plus ou moins ciblée. Le taux de reproduction est donc un critère de suivi épidémique à prendre avec circonspection.

Incidences Rt7=(exp(-exp(-0.03($i-53))))(0.03exp(-0.03($i-53)))*1700000

Cet article COVID19 Rt R0 un mauvais critère est apparu en premier sur Blog de Jordi Mir :.


COVID19 Modélisation d’une épidémie et mesures

Novembre 2020, par Jordi[ —]

La modélisation d’une épidémie est importante pour prendre de bonnes décisions. Ne pas comprendre la dynamique de l’épidémie est susceptible d’ajouter du malheur au malheur.

Le modèle SIR

Le modèle SIR est un exemple de modèle à compartiments, c’est à dire que l’on divise la population en plusieurs groupes.
Pour une population donnée, on étudie la taille de trois sous-populations au cours du temps t : 

S(t) représente les personnes saines (susceptible en anglais) au temps t,

 I(t) les personnes infectées (infected),

et R(t) les personnes retirées (removed)  ;

N=S(t)+I(t)+R(t) représente alors la population constante totale au cours du temps. Il convient de bien différencier les personnes saines des personnes retirées : les personnes saines n’ont pas encore été touchées par le virus, alors que les personnes retirées sont guéries, et donc immunisées. Autrement dit, les personnes retirées ne sont plus prises en compte. 

β représente le taux de transmission, c’est à dire le taux de personnes saines qui deviennent infectées et γ le taux de guérison, c’est à dire le taux de personnes infectées qui deviennent retirées

Système d’équations

    \[ \begin{cases} \frac{dS(t)}{dt} &=& -\beta S(t)I(t)&(1.1)\\ \frac{dI(t)}{dt} &=& \beta S(t)I(t)-\gamma I(t)&(1.2)\\ \frac{dR(t)}{dt} &=& \gamma I(t)&(1.3) \end{cases} \]

Le terme S(t)I(t) représente le nombre de contacts entre des personnes saines et des personnes infectées. β étant le taux de transmission, il y a dès lors βS(t)I(t) personnes nouvellement infectées. Celles-ci se soustraient des personnes saines (1.1), et s’ajoutent aux personnes infectées (1.2). De même, parmi les personnes infectées, certaines vont guérir : γ étant le taux de guérison, il a γI(t) personnes nouvellement guéries qui s’enlèvent des personnes infectées (1.2) et s’ajoutent aux personnes retirées (1.3).


Taux de reproduction

Le taux de reproduction R0 est le nombre moyen de cas secondaires produits par un individu infectieux au cours de sa période d’infection.

Au début de l’épidémie, l’expression de R0 est β/γ puisque 1/γ représente la durée moyenne de la maladie et qu’au début, les personnes rencontrées sont presque toutes saines.

Si R0>1, alors I(t) croît, atteint son maximum puis décroît vers 0 quand t tend vers +∞ : c’est une épidémie.

Sinon, I(t) décroît directement vers 0 quand t tend vers +∞ : il n’y a pas d’épidémie. C’est sur ce théorème que se basent les scientifiques et les politiques lorsqu’ils disent, pour l’épidémie de Covid-19, qu’il faut à tout prix réduire R0 pour le rendre le plus proche possible de 1

Efficacité des mesures prises et effets sur la courbe

Modification du taux de transmission

On peut essayer de modifier le taux de transmission par des mesures comme le couvre feu ou le confinement partiel ou total.


Comparaison du pic en modifiant le taux de transmission
Le taux de transmission est de 0,9 à gauche et de 0,2 à droite. Le taux de guérison est fixé à 0,1.

On observe très clairement que le pic de courbe de I (en orange) est moins haut (0,8 contre 0,6) et que la courbe est plus étalée. De plus, la courbe de S (en bleu) décroit beaucoup plus rapidement à gauche qu’à droite.

Quand le taux de transmission baisse on obtient un Max plus faible mais la courbe s’allonge dans le temps.

Le couvre feu est-il efficace? En Espagne certaines provinces ont mises en place un couvre feu, d’autres non. Un graphique des incidences entre 2 provinces, montre que la province qui a établi un couvre feu a un Max plus élevé que celle qui ne l’a pas fait. Le taux de transmission a été plus faible dans la province qui n’a pas fait de couvre feu.

Le couvre feu n’a pas eu d’efficacité!


Cet article COVID19 Modélisation d’une épidémie et mesures est apparu en premier sur Blog de Jordi Mir :.


COVID19 Haro sur la Suède

Novembre 2020, par Jordi[ —]

Drapeau de la Suède

#COVID19 -19 : Suède

Flèche vers la droite

Aucun couvre feu

Aucun confinement

Restaurants, Bars et Discothèques ouverts

Aucun port de masque obligatoire

Mortalité par million d’habitants

Drapeau de la France

724/million d’habitants

Drapeau de la Suède

640/million d’habitants

Pour justifier le confinement, 400 000 morts si l’on ne faisait rien, il est de bon ton de dénigrer la gestion covid19 de la Suède.

La tactique de ne rien faire ou du moins pas grand chose, choisie par la Suède est vilipendée par tous les médias des pays qui ont adopté le confinement.

La Suède n’a pas su géré ses EHPAD et a eu de nombreux décès. Elle n’a pas fait beaucoup de tests au printemps, elle est donc loin de la première place de bon élève par rapport aux pays asiatiques.

C’est pourtant, un exemple en vrai grandeur à suivre pour analyser la dynamique du Sars Cov2.

Seconde vague :

La Suède a essayé la méthode de l’immunité de groupe. Les incidences ont augmenté fortement cet automne. Mais au printemps la Suède n’a pas fait de tests massifs. Difficile donc de comparer. Si l’on regarde la mortalité sur 2020 on ne voit pas de pic du au covid19. Si cette tendance se confirme, la Suède n’aura pas eu de seconde vague.

Z score
Cumulative Deaths
Incidences/jour

Cet article COVID19 Haro sur la Suède est apparu en premier sur Blog de Jordi Mir :.


COVID19 An energetic analogy

Novembre 2020, par Jordi[ —]

The Total Mechanical Energy

The mechanical energy of an object can be the result of its motion (i.e., kinetic energy) and/or the result of its stored energy of position (i.e., potential energy). The total amount of mechanical energy is merely the sum of the potential energy and the kinetic energy. This sum is simply referred to as the total mechanical energy (abbreviated TME)

TME = PE + KE

Analogical models are a method of representing a phenomenon of the world, often called the « target system » by another, more understandable or analysable system. They are also called dynamical analogies.

Kinetic energy KE= 1/2* IA14* Rt*Rt

Incidence IA14 (cases / 100,000 inhab. In the last 14 days)

Speed Rt = ( IA14(t+1)-IA14(t))/t

Potential energy KP=1/2* K* t*t

K is the stiffness to resist to the pandemia

Kinetic energy

    \[    \boxed{KE=\int_0^{\infty}\! (IA14*Ro7) \mathrm{d}t} \]

Example : Spain source https://biocomsc.upc.edu/en/covid-19/Worldcovid-en

IA14 Spain
Risc EPG=IA14*Ro7

Rt
Kinetic energy Phase 2

First days Stiffness K increases till 40 days. After the stiffness decreases, the max is at 40 days. For Spain so the phase2 will be finished at 120 days, first days of December.

Cet article COVID19 An energetic analogy est apparu en premier sur Blog de Jordi Mir :.


COVID19 Diagramme de risque

http://www.jordimir.com/2020/11/20/covid19-diagramme-de-risque/play episode download
Novembre 2020, par Jordi[ —]

En Catalogne l’université polytechnique de Barcelona a proposé un diagramme de risque pour suivre la pandémie de Sars Cov2.

Pour visualiser l’évolution du virus elle a mis en ligne un diagramme de risque :

En abscisse IA14 le nombre d’incidences sur 14 jours pour 100 000 habitants

En ordonnée ρ7 la vitesse de contagion

Le Risque EPG = IA14 * ρ7

Une zone verte pour EPG=30 et une zone rouge pour EPG=100

Cette visualisation est intéressante pour suivre la pandémie.

Pour mieux prendre en compte la vitesse de contagion et par analogie à une énergie cinétique du virus je préfère calculer un risque EPGJ :

EPGJ = IA14 * ρ7 * ρ7

Diagramme de risque EPGJ
Bleu France Rouge Espagne

Pour suivre interactivement sur une carte du monde BIOCOMSC

https://biocomsc.upc.edu/en/covid-19/Worldcovid-en

Cet article COVID19 Diagramme de risque est apparu en premier sur Blog de Jordi Mir :.


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