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    Main Research Areas :
    • Statistical inference and inductive logic; Sampling theory from finite population;
    • Statistical inference in non-linear regression models;

    • Quality and Design of experiments: warning barriers and control charts, D-optimal design for multiresponse models, run orders and statistical dependence among runs in two-levels fractional factorials; design and statistical robust analysis of hierarchic models: split-plot, strip-plot, etc.;
    • Modern basic statistics, covariance decompositions.
    • COVID-19
      Waves Report: Waves-1-2-3-4 24/4/2021 – Italiano.
      Report 10/05/2020 – Italiano.
      Report 10/05/2020 – English.
    • Recent results for Italy
    • Statistical diffusion models of the COVID-19 outbreak
      1. Introduction

    • The Covid-19 outbreak in Italy presented its first technically detected cases, on 20 February 2020 in particular areas of Lombardy, Cologno and other cities in the province of Lodi. Today, there is concrete evidence of the local presence of at least one positive contact without explicit symptoms coming from Munich (Germany), an event that occurred around January 25th. Some recent findings point to a possible “first infection” as early as early January. However, the salient facts are: i) the latency of the phenomenon, ii) its poor knowledge in Italy and Europe, iii) the delay in the scientific and public hygiene acquisition of the Chinese and South Korean experiences.
      The contagion then latently implemented itself for weeks creating a polycentric distribution. This aspect characterized the rapid explosion of the epidemic in Lombardy, as evidenced by the positive cases observed by subsequent tests validated around February 20.
      Also in Veneto there have been the first cases of Covid-19 detected in Vò in the Euganean Hills. This problem was immediately treated by the Veneto Region with a quarantine for all the residents of the town, around 3,300 subjects. Members of this cluster were tested to highlight a possible positivity to Covid-19. The results showed that the contagion activity of positive subjects without symptoms dilates the overall speed of the contagion itself, a speed which is therefore not fueled by symptomatic cases alone.
      The distributional asymmetry of the cases detected in the first three weeks, in Lombardy versus Veneto, has been confirmed empirically and this has determined specific levels in the dynamics parameters, parameters that can be correctly interpreted.
      In that initial logic of diffusion, the Lombardy peak anticipated the corresponding one in Veneto by about two weeks. This is probably due to efficient control of the residents of Vò and subsequent strategies of voluntary isolation and social distancing for all residents of the Veneto region. Depending on national and regional containment policies, this systematic action subsequently produced a distributional contraction of the case curve in Veneto and an absolute decrease.
    • 2. Case dynamics outside mainland China

    • The spread of an epidemic, an information, a fashion, a commercial product, a drug, etc. can be expressed within a complex system consisting of agents. The latter have a strong capacity for self-orientation, decision-making, self-organized sharing which often does not depend on the conditions of the individual profile and does not require centralized guidance.
      Networks of relationships and their thematic formation on various levels make the complex system a macroscopic structure relevant from an evolutionary point of view. Sociologists, physicists, biologists, quantitative marketing experts, statisticians and mathematicians have made significant contributions to the development of these interdisciplinary themes.
      The initial model proposed here for the spread of Covid-19 cases outside China belongs to the family of methodologies for the diffusion of innovations. It is a generalized Bass model, GBM, (Bass et al, 1994) equipped with a control function and improved with the introduction of asymmetric effects due to Bemmaor (Bemmaor, Lee, 2004), effects capable of explaining phenomena of distribution mix at the aggregate level.
      This model assumes a carrying capacity – that is, a potential of the susceptible in the specific case – constant during the evolutionary cycle. This assumption may be an obvious limitation.
      It should be remembered that the Bass models, and their variants in the diffusion of innovations models, differ substantially from the logistic model (Verhulst, 1838) and from the parallel SIR models for a fundamental point concerning the initialization process.
      The initial condition of the logistic equation determines an initialization temporally localized in the process. Bass’s model instead uses a monomolecular sub-model to describe an exogenous initialization of cases distributed over time. In parallel, they both develop the word-of-mouth or contagion mechanism.
    • With reference to cases outside China (data source: Worldometer), the first exponential shock included in the generalized Bass diffusion model, GBM, was identified around February 29 and is absorbable. The initial intervention of the Italian government, for example with the blocking of standard educational activities, can be dated on February 24th. The delay in effects, estimated to be around 14 days, was crucial. The usefulness of these interventions, not only for Italy, can be highlighted and was verifiable around 9 March. However, there is a masking effect in Europe due to the delaying policies of France and Germany. These countries encountered the problem, first in Europe and long before Italy, but introduced some relevant interventions only around March 4th.
      A second growing exponential shock was identified on March 14th. This is an absorbable shock in the spread and is due to the delayed expansion of contagion outside China in countries and regions with different public health policies. A third shock was identified around March 19 probably due to the exponential expansion of the epidemic in the United States.
      In the sequel we have introduced a more flexible model, GGM, qualitatively described in the following Section 3.
    • 3. A dynamic model with variable potential

    • An alternative model, based on a dynamic potential – function of a latent network of growing relationships -, GGM (Guseo, Guidolin, 2009), can be taken into consideration to assess the presence, outside China, of new waves of cases detected in the Covid-19 epidemic. This model is based on two drivers, one latent which describes a dynamic potential of increasing nature, based on a network of interpersonal relationships, and a second, nested in the previous one, which implements the observed contagion events. This model is characterized by distributed initialization and can have one or two modal peaks depending on the level of involved parameters: two parameters for controlling the dynamic potential (assumed to be increasing here), two parameters for controlling the dynamics of the contagions and one for scale.
      These new directions of investigation could be recognized or not after March 15 in the presence of sufficient information on the empirical characteristics of the cycles observed.
      As is well known, South Korea, Iran and Italy were the first three countries in which the epidemic emerged, probably due to the intense commercial and tourist activities directed with China. The first country to reach the unimodal peak of new Covid-19 cases in the first week of March was South Korea which implemented innovative policies for detecting symptomless vectors. As of March 14, South Korea itself has been characterized by a low stationary level of Covid-19. This endemic effect is easily identified with the integration, in the main cycle, of a specific local diffusion model, a Bass sub-model. In the sequel this endemic behaviour is automatically included in the GGM model without extensions.< br/> Iran had a first local peak in the first week of March. Based on the data of 3/12 and using a GGM model, a second peak could have appeared in the third week of March. This prediction was not confirmed on March 24th. In fact, four large successive increments of new cases postponed the event for a week. The current GGM model in enriched with a simple shock that identify a relevant change of regime, GGM + e1P. The corresponding description of the last 25 days is practically without errors.
    • 4. Statistical modeling of COVID-19 cases in Italy, Lombardy, Veneto, Padua province, Piedmont, Puglia and other countries

    • The initial model proposed for the Italian cases of Covid-19 – based on data from the Ministry of Health – was a Bass-like diffusion model with the inclusion of the effects of Bemmaor (Bemmaor, Lee, 2004). In this case, the longer right tail is due to an effect of the policies implemented to counter the spread. This therefore implies a distributed delay.
      Due to a large increase in cases in Lombardy, the Italian government introduced a new law decree on March 8 with strong restrictions on individual mobility regarding Lombardy and 14 other provinces. A subsequent decree of March 9 extended the restrictions to all regions of Italy until April 3. Further measures have tightened the constraints with increasing controls on the territory. Around March 16, Italy could have reached its peak on the basis of previous dynamics not yet subject to the effects of containment initiatives. This hypothesis could be confirmed on March 11-13 if the previous restrictions had been systematically applied by the population. Unfortunately, as of March 16, there has been evidence of the geographical spread outside the regions of Lombardy and Veneto with a dynamic expansion of the national potential.
      This new situation has suggested, from a technical point of view, the introduction of a GGM with two exponential shocks to recover the significant regime changes due to the main containment policy actions. In the same period, Lombardy achieved very high levels of case growth especially in Bergamo, Brescia and other connected cities.
    • The GGM distribution of cases over time (Guseo, Guidolin, 2011) has two distribution factors that act jointly on the diffusion process: “Communication”, C and “Adoption”, A. The corresponding time peaks are indicated with tC and tA, respectively.
      In the Covid-19 epidemic, “Adoption” mainly represents the individual ability to develop pathology, infectivity. The “Communication” indicates the predisposition to contagion, the transmissibility. The previous peak times are independent of the absolute scale of the phenomenon. On March 22, the estimated times for Veneto (in days from the origin of the data) were tA = 18 and tC = 46. Lombardy had a very different behavior: tA = 23 and tC = 34. In other words, Veneto has a lower resistance to infectivity (higher fraction of the elderly population?), however it has developed an effective resistance to infection with a longer time tC than in Lombardy.
      On March 30, the full effect of government and regional measures becomes evident in the modeling of data at national level and in the regions of northern Italy. The average peaks of the cases in Lombardy and Veneto are well highlighted around 3/23. In particular, Veneto obtains a contraction in the temporal distribution of cases with a parallel reduction, in absolute number, of events. At the national level, the dynamics of the epidemic are more complex and bimodal. These are the mixture of two main general events attributable, on the one hand, to expansion in northern Italy with the peak of March 22nd, and on the other, to the subsequent expansion in the rest of Italy, with the peak of March 28th. The current model for Italian Covid-19 daily cases is a simple GGM. The deviations around it, stochastic and seasonal (weekly) effects, behave in a compensating manner.
    • 4. Mortality in Italy

    • The process that describes mortality over time from the Covid-19 epidemic is, by its nature, more regular and depends on the temporal cycle of the cases detected ex-ante and ex-post. This dynamic is naturally a function of the current capacity of healthcare facilities and the characteristics of the population. Data collection is certainly more precise and documented.
      The initial evolutionary model proposed here for the Italian deaths of Covid-19 was a Bass model with the correction of Bemmaor and exploits the data of the Ministry of Health. The dynamic evolution was fairly regular where natural dynamics, contrast policies and the presence – at the aggregate level – of a mixture of specific Bass-like subpopulations at the territorial level were combined. A reasonable stability of the estimates may have been reached around March 18-20.
      The present approach is based on a more flexible and efficient GGM model.
    • 5. Some modeling of the Covid-19 epidemic in the most affected countries

    • The modeling of the cases detected in China was carried out using a GBM adjusted with the Bemmaor effect to take into account the mixture of subpopulations. The data used were obtained from Worldometer. Obviously, only the official data is used whose quality depends on the supplier country.
      In this case, the evident change of technical case detection regime produced a surge accounting that was suitably isolated, by means of a calibrated exponential shock, allowing a purified evaluation of the dynamics.
      An ARMAX model – not necessary for the long term assessment – was subsequently applied to locally refine the forecasts. At this moment, China has an endemic low-level stationary queue. On average 50 new cases per day.
      Australia, Austria, Belgium, the Netherlands, Spain, Turkey, France, Spain, Germany and Switzerland have been modeled with a GGM and often present the need for the introduction of a locally exponential regime change, e1P, to take into account systematic changes in the average evolution due to the policies gradually introduced.
      Then, in the various countries, the local policies for defining the detected Covid-19 cases change. These sources of variability determine local fluctuations of the residuals, sometimes not stationary.
      UK, Canada and the USA did not produce timely containment interventions and the evolution is now explosive as evidenced by the GGM models applied to the series of detected cases.
    • 5. Current situation: May 10

    • Cases outside China have probably reached a peak around April 24.
      Italy has attained a good position with containment policies. The 95% of cumulative cases will be reached on May 13. The 99% is predicted on June 4. Current reproduction number of GGM model for new cases is R(t)=0.067.
      Lombardy and Veneto have different behaviour with R(t) equal to 0.124 and 0.030, respectively. Veneto has reached  the 95% of cumulative cases on May 4  and will reach the 99% on May 21. Lombardy has a longer right tail with a delay of four weeks, due to a more complex situation.
      Emilia-Romagna is approaching a good position with R(t)=0.070.
      States that have an optimal control with minor endemic cases are China, S. Korea, Australia, Austria, and Switzerland. Within two weeks, Germany will be in the same position: R(t)=0.048.
      France has not a stable administrative data collection with frequent revisions of public information. This may affect statistical assessments.
      UK, Canada, Spain, and the USA are around their peaks but, in some situations, very far from a stationary endemic behaviour.
    • Some references
    • Bass, F.M. (1969). A new product growth model for consumer durables, em Management Science, 15, 215-227.
      Bass, F.M., Krishnan, T.V., Jain, D.C. (1994). Why the Bass model fits without decision variables, Marketing Science, 13, 203-223.
      Bemmaor, A.C., Lee, J. (2002). The Impact of Heterogeneity and Ill-Conditioning on Diffusion Model Parameter Estimates. Marketing Science, 21(2), 209-220.
      Boccara, N. (2004). Modeling Complex Systems, Springer-Verlag, New York.
      Fibich, G., Gibori, R. (2010). Aggregate Diffusion Dynamics in Agent-Based Models with a Spatial Structure. Operations Research, 58(5), 1450-1468.
      Guidolin, M., Guseo, R. (2014). Modelling Seasonality in Innovation Diffusion. Technological Forecasting and Social Change, 86, 33-40.
      Guseo, R. (2016). Diffusion of innovations dynamics, biological growth and catenary function. Physica A: Statistical Mechanics and its Applications, 464, 1-10.
      Guseo, R., Guidolin, M. (2008). Cellular Automata and Riccati Equation Models for Diffusion of Innovations. Statistical Methods and Applications, 17(3), 291-308.
      Guseo, R., Guidolin, M. (2009). Modelling a Dynamic Market Potential: A Class of Automata Networks for Diffusion of Innovations. Technological Forecasting and Social Change, 76(6), 806-820.
      Guseo, R., Guidolin, M. (2010). Cellular Automata with Network Incubation in Information Technology Diffusion. Physica A: Statistical Mechanics and its Applications, 389(12), 2422-2433.
      Guseo, R., Guidolin, M. (2011). Market potential dynamics in innovation diffusion: modelling the synergy between two driving forces. Technological Forecasting and Social Change, 78(1), 13-24.
      Guseo, R., Mortarino, C. (2012). Sequential Market Entries and Competition Modelling in Multi-Innovation Diffusions. European Journal of Operational Research, 216, 658-667.
      Guseo, R., Mortarino, C. (2015). Modeling competition between two pharmaceutical drugs using innovation diffusion models, The Annals of Applied Statistics, 9(4), 2073-2089.
      Meade, N., Islam, T. (2006). Modelling and forecasting the diffusion of innovation – a 25-year review. Int. Journal of Forecasting, 22(3), 519-545.
      Norton, J.A., Bass, F.M. (1987). A diffusion theory model adoption and substitution for successive generations of high technology products, Management Science, 33, 1069-86.
      Peres, R., Muller, E., Mahajan, V. (2010). Innovation diffusion and new product growth models: a critical review and research directions, Inter. Journal of Research in Marketing, 27(2), 91-106.
      Verhulst, P.F. (1838). Notice sur la loi qui la population suit dans son accroissement, Corres. Math. et Physique, 10, 113-121.
      Volterra, V. (1926). Fluctuations in the abundance of a species considered mathematically, Nature, 118(2972), 558-60.

      COVID-19: daily Reproduction Number for GGM, R(t)
        Piemonte Lombardia Veneto Emilia-R. Puglia Italy Germany USA Switzerland
      4/27     0.110       0.085   0.029
      4/28     0.098       0.088   0.026
      4/29   0.298 0.088     0.162 0.090 0.263 0.024
      4/30   0.250 0.081     0.149 0.083 0.278 0.021
      5/1 0.124 0.237 0.075 0.181 0.096 0.141 0.083 0.321 0.019
      5/2 0.128 0.205 0.072 0.164 0.088 0.133 0.076 0.308 0.017
      5/3 0.112 0.182 0.063 0.147 0.074 0.120 0.068 0.272 0.011
      5/4 0.091 0.180 0.053 0.110 0.069 0.104 0.068 0.270 0.011
      5/5 0.082 0.171 0.048 0.103 0.063 0.097 0.067 0.266 0.011
      5/6 0.074 0.164 0.044 0.097 0.058 0.091 0.063 0.245 0.010
      5/7 0.066 0.156 0.040 0.091 0.053 0.085 0.059 0.225 0.010
      5/8 0.060 0.137 0.037 0.080 0.050 0.077 0.056 0.206 0.009
      5/9 0.054 0.130 0.033 0.075 0.046 0.072 0.052 0.189 0.008
      5/10 0.049 0.124 0.030 0.070 0.042 0.067 0.048 0.172 0.007


Dipartimento di Scienze Statistiche | Università degli studi di Padova