Table 3 Model selection results for simulated and real data.

Fitting model	\(\beta =0.2,\alpha =0.05\)		\(\beta =0.2,\alpha =0.1\)		\(\beta =0.3,\alpha =0.05\)		\(\beta =0.3,\alpha =0.1\)
Fitting model	AIC	MLL	AIC	MLL	AIC	MLL	AIC	MLL
(a) Two-dimensional grid model, with connection to \(k=4\) neighbours (i.e. \(d=1\) in (5))
\(\gamma \) and \(\kappa \) free	\(-126.64\)	1.35	\( -61.38\)	0.69	\(-140.59\)	1.49	\(-104.23\)	1.12
\(\gamma \) free \(\kappa =1\)	\(-103.28\)	1.09	\(-43.37\)	0.49	\(-75.48\)	0.81	\(-30.35\)	0.36
\(\gamma =1\) and \(\kappa \) free	7.62	\(-0.02\)	9.05	\(-0.03\)	19.26	\(-0.13\)	21.40	\(-0.15\)
\(\gamma =1\) and \(\kappa =1\)	35.95	\(-0.32\)	24.63	\(-0.21\)	32.19	\(-0.15\)	23.44	\(-0.19\)
(b) Two-dimensional grid model, with connection to \(k=8\) neighbours (i.e. \(d=2\) in (5))
\(\gamma \) and \(\kappa \) free	\(-18.84\)	0.27	\( -62.87\)	0.71	\(-58.93\)	0.67	\(-82.37\)	0.90
\(\gamma \) free \(\kappa =1\)	51.20	\(-0.45\)	14.76	\(-0.09\)	85.41	\(-0.79\)	63.98	\(-0.56\)
\(\gamma =1\) and \(\kappa \) free	54.05	\(-0.48\)	31.90	\(-0.26\)	8.03	\(-0.02\)	21.08	\(-0.15\)
\(\gamma =1\) and \(\kappa =1\)	60.32	\(-0.56\)	32.20	\(-0.29\)	83.58	\(-0.80\)	64.58	\(-0.61\)

Fitting model	\(\beta =0.3,\alpha =0.05\)		\(\beta =0.3,\alpha =0.1\)
Fitting model	AIC	MLL	AIC	MLL
(c) Two-dimensional Gaussian mixture random graph, with connection to \(k=4\) neighbours
\(\gamma \) and \(\kappa \) free	\(-106.28\)	1.14	\(-76.82\)	0.85
\(\gamma \) free \(\kappa =1\)	\(-1.77\)	0.08	\(-1.38\)	0.31
\(\gamma =1\) and \(\kappa \) free	36.68	\(-0.31\)	36.01	\(-0.13\)
\(\gamma =1\) and \(\kappa =1\)	35.80	\(-0.32\)	21.25	\(-0.17\)

Fitting model	Italy		Germany		France		Spain
Fitting model	AIC	MLL	AIC	MLL	AIC	MLL	AIC	MLL
(d) Real data from four different countries
\(\gamma \) free	121.33	\(-0.81\)	156.18	\(-1.01\)	175.54	\(-1.24\)	148.34	\(-1.09\)
\(\gamma =1\)	216.93	\(-1.46\)	203.20	\(-1.31\)	219.55	\(-1.55\)	204.94	\(-1.51\)

For the simulated data, the fitting models are based on (8) with parameters \(c,\gamma ,\kappa ,\sigma \). The models differ in that the exponents \(\gamma \) or \(\kappa \) are fixed or optimally selected (free). For the real data, the size of susceptible population S(t) is assumed to be constant and the exponent \(\kappa \) is ignored. The AIC represents the Akaike information criterion defined according to (7) and MLL denotes the normalized maximum log-likelihood computed by maximizing (9) over free parameters. The reported results for the simulated data are averaged over 100 Monte-Carlo simulations. For the simulated data, the infection parameters \(\alpha \) and \(\beta \) are assumed to take different values as indicated in the top row of the tables.

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