S

A

H

E

EqSq

R

I𝝀Sq

𝜹qEq (𝜸H+𝜶)H

(𝜸I+𝜶)I

𝜸AA

𝜹II

𝝈𝝆E

𝝈(1-𝝆)E

𝜷c(1-q)S(I+𝜽A)

𝜷cqS(I+𝜽A)(1-𝜷)cqS(I+𝜽A)

To obtain the variation over time of each compartment we need to add inbound fluxes and subtract outbound ones. Here is the procedure for S’. derivative of S with respect to time:

S’ = 𝝀Sq- 𝜷c(1-q)S(I+𝜽A) - 𝜷cqS(I+𝜽A) - (1-𝜷)cqS(I+𝜽A)

= 𝝀Sq- (𝜷c - 𝜷cq + 𝜷cq + cq - 𝜷cq)S(I+𝜽A)

= 𝝀Sq- (𝜷c + cq - 𝜷cq)S(I+𝜽A)

= 𝝀Sq- (𝜷c + cq(1 - 𝜷))S(I+𝜽A)

The model is therefore represented by the following system of eight differential equations:

S’ = - (𝜷c + cq(1 - 𝜷))S(I + 𝜽A) + 𝝀Sq

E’ = 𝜷c(1 - q)S(I + 𝜽A) - 𝝈E

I’ = 𝝈𝝆E - (𝜹I + 𝜶 + 𝜸I)I

A’ = 𝝈(1 - 𝝆)E - 𝜸AA

Sq’ = (1 - 𝜷)cqS(I + 𝜽A) - 𝝀Sq

Eq’ = 𝜷cqS(I + 𝜽A) - 𝜹qEq

H’ = 𝜹II + 𝜹qEq - (𝜶 + 𝜸H)H

R’ = (𝜸I + 𝜶)I + 𝜸AA + (𝜸H+𝜶)H

S’ + E’ + I’ + A’ + Sq’ + Eq’ + H’ + R’ = 0. as we do not consider newborns. deaths. immigration and emigration.

The dataset used is the one provided by Italian Civil Protection site. The time independent model is run to fit the total active positive cases from 24 Feb to 8 March.

H = we fit on the number of active known cases

We used the Markov Chain Monte Carlo (MCMC) method to fit the model and adopted an adaptive Metropolis–Hastings (M-H) algorithm to carry out the MCMC procedure.

The algorithm is used for 4 concatenated runs with the following number of iterations: 100.000-50.000-25.000-10.000.

The procedure is run in Matlab with the MCMC toolbox for Matlab.

Burn-in is not used as there is no theoretical justification for it. see Burn-In is Unnecessary.

MCMC for time independent model

Dataset for MCMC Time Independent model

Date Day H24/02/2020 0 22125/02/2020 1 31126/02/2020 2 38527/02/2020 3 58828/02/2020 4 82129/02/2020 5 104901/03/2020 6 157702/03/2020 7 183503/03/2020 8 226304/03/2020 9 270605/03/2020 10 329606/03/2020 11 391607/03/2020 12 506108/03/2020 13 6387

Parameters obtained with fit MCMC (4 concatenated chains) for dataset 24/2 - 8/3 total Italy - Time Independent

Mean Std Dev MC_err tau gewekealfa 1.2197E-05 3.00E-08 6.39E-09 1142.2 0.99798beta 1.4410E-08 3.36E-11 7.24E-12 779.16 0.99538c 4.5248 0.0203 0.0043452 1052.1 0.99351gammaA 0.1177 0.0002 0.00004965 973.1 0.99943gammaH 0.0515 0.0002 0.000037676 972.97 0.99821gammaI 0.1704 0.0005 0.00010929 1049.6 0.99940deltaI 0.1086 0.0001 1.77E-05 1134.3 0.99907deltaq 0.1471 0.0001 1.10E-05 692.94 0.99980rho 0.2344 0.0006 0.00012594 985.68 0.99881q 1.1550E-07 3.86E-10 8.08E-11 1051.9 0.99940Rc 7.2870 0.0021 0.00043247 674.12 0.99986teta 0.0840 0.0001 0.000027318 783.62 0.99859E0 21.308.00 89.945 18.539 790.43 0.99950I0 201.03 0.15402 0.033963 791.21 0.99884A0 493.03 0.56176 0.11666 779.57 0.99926Eq0 522.36 1.3152 0.2746 694.47 0.99743

Time independent model versus data from 24/02 to 05/04. log and normal plot.

We run MCMC procedure with Tang time dependent equations for c(t) and 𝜹I(t) and a different q with multiple dataset from 08/03/2020. the day when social measures were widely adopted in Italy.

We also introduced an exponential decay for q with an expression similar to c(t).

No satisfactory fit was obtained. probably due to the dishomogeneity of situations all around Italy.

Anyway. the data suggest the existence of at least three periods and we manually fit the data to the curve varying c and q.

Time dependent model

From 24/02 to 08/03 c0 and q0 from MCMC

From 08/03 to 29/03 c1 = 0.5c0 and q1 = 5.5q2 Strong reduction of contact rate and strong increase of quarantine rate

From 29/03 to 06/04 c2 = 1.4c1 and q2 = 2q1 Slight recover of contact rate and further increase of quarantine rate

Time independent model with manual fit versus data from 24/02 to 06/04. log and normal plot.

The reproduction number obtained with Tang formula does not to go below 1 with the parameters of manual fit for Italy total data.

R0

Parameters obtained with fit MCMC (3 concatenated chains: 10,000/10,000/30,000 iterations) for dataset 24/2 - 8/3 Lombardy - Time Independent

Mean Std Dev MC_err tau gewekealfa 1,7304E-05 1,33E-07 2,67E-08 2028 0,98303beta 3,6806E-08 7,55E-11 1,31E-11 2949,8 0,99782c 9,5777 0,0292 0,0053672 2536,6 0,99786gammaA 0,1503 0,0007 0,00013968 3544,5 0,99497gammaH 0,1134 0,0005 0,00010152 3399,3 0,99254gammaI 0,3738 0,0008 0,00016612 3093,2 0,99990deltaI 0,1359 0,0008 1,71E-04 2296,7 0,99426deltaq 0,0301 0,0003 6,00E-05 2619,5 0,97722rho 0,4900 0,0010 0,00018131 1747 0,99658q 1,7409E-08 1,33E-10 2,71E-11 3393,7 0,99880Rc 2,5229 0,0125 0,0023939 3400,7 0,99389teta 0,0242 0,0008 0,00016753 2150,4 0,96392E0 6.560,10 26,142 5,0533 3484,9 0,99875I0 122,59 0,13962 0,025129 2676,7 0,99794A0 613,81 2,4313 0,46995 2995,5 0,99178Eq0 1.223,10 2,1097 0,38125 1590,2 0,99881

Time independent model for Lombardy versus data from 24/02 to 07/04. log and normal plot.

Parameters obtained with fit MCMC (3 concatenated chains: 10,000/10,000/30,000 iterations) for dataset 9/3 - 20/3 Lombardy - Time Dependent

Note that, while the contact rate decrease, the diagnostic time slightly increase.

Mean Std Dev MC_err tau gewekecb 4,3685 4,35E-03 7,74E-04 2386,6 0,99891r1 0,5668 1,08E-03 2,08E-04 1903,9 0,99929deltaIf 0,1020 1,87E-05 3,43E-06 1259,8 0,99980r2 20,5790 1,15E-01 1,74E-02 1458,7 0,99594q 2,71E-07 1,50E-09 2,26E-10 1452,3 0,99603

Time dependent graph for Lombardy - Dataset 09/03-20/3 (3 MCMC runs 10,000/10,000/30,000 iterations)

Parameters obtained with fit MCMC (3 concatenated chains: 10,000/10,000/30,000 iterations) for dataset 21/3 - 04/04 Lombardy - Time Dependent

Note a further decrease in contact rate and a recover of diagnostic time

Mean Std Dev MC_err tau gewekecb 2,4918 8,61E-03 1,88E-03 3282,6 0,99511r1 0,4172 7,61E-03 1,72E-03 2793,4 0,97378deltaIf 0,1564 2,52E-04 4,58E-05 1386,8 0,99905r2 0,3300 1,48E-03 2,41E-04 2622,8 0,99686q 3,25E-08 3,63E-10 8,20E-11 2924,2 0,98422

From 24/02 to 08/03Light measures in place, R0 = 3.7502;

From 09/02 to 20/03Measures in place, contact rate reduction, reduced diagnostic ability, R0 = 1.8202;

From 20/03Measures in place, further contact rate reduction, recovered diagnostic ability, R0 = 0.9419;

Model for Lombardy with one time independent phase and two time dependent phases, one beginning on 09/03 and the other on 21/03