Lucia Maddalena Department of Economics University of Foggia

Transcription

1 Lucia Maddalena Department of Economics University of Foggia

2 John von Neumann ( ) Le scienze non cercano di spiegare, a malapena tentano di interpretare, ma fanno soprattutto dei modelli. Per modello s'intende un costrutto matematico che, con l'aggiunta di certe interpretazioni verbali, descrive dei fenomeni osservati. La giustificazione di un siffatto costrutto matematico è soltanto e precisamente che ci si aspetta che funzioni - cioè descriva correttamente i fenomeni in un'area ragionevolmente ampia. Inoltre esso deve soddisfare certi criteri estetici - cioè, in relazione con la quantità di descrizione che fornisce, deve essere piuttosto semplice.

3 Henri Poincaré ( ).fra tutte queste spiegazioni possibili come effettuare una scelta per la quale ci viene a mancare l'ausilio dell'esperienza? Verrà forse il giorno in cui i fisici smetteranno di interessarsi a questioni di questo genere, inaccessibili ai metodi positivi, e le lasceranno ai metafisici. Ma quel giorno non è ancora arrivato: l'uomo non si rassegna così facilmente a ignorare eternamente il fondo delle cose"

4 Galileo Galilei ( )...Il libro della natura è scritto in lingua matematica ed i suoi caratteri sono triangoli, cerchi ed altre figure geometriche, senza i quali mezzi è impossibile a intenderne umanamente parola; senza questi è un aggirarsi vanamente per un oscuro labirinto...

5 E. Malinvaud ( ) Un modello matematico è la rappresentazione formale di idee o conoscenze relative a un fenomeno

6 Problema reale Analisi quantitati va Raccolta dati e osservazione fenomenolo gica Analisi qualitativa Identificazio ne delle variabili Modello matematico

7 Modelli matematici per processi deterministici Equazioni Funzioni Funzionali Equazioni alle differenze Equazioni differenziali ordinarie Equazioni differenziali alle derivate parziali.

8 Models for the diffusion of innovations Asymptotic Behaviour of a nonlinear model for the geographical diffusion of innovations. Dynamic Systems and Applications, 3 (1994), (with V. Capasso,A.Diliddo ) Nonlocal models for the spread of new products in spatially structured market. Dynamic Systems and Applications, 6 (1997), (with A. Diliddo ) La diffusione del fotovoltaico: sistemi dinamici. Proc. Mathematica Italia User Group (2008) (with C. Gallo, M. De Bonis) A mathematical model for renewable technology diffusion. Proc. Mathematica Italia User Group, ISBN: (2011)( with V.Fanelli ) (2011) Diffusion trajectory of self-propagating photovoltaic systems. CIRPAS. Cacucci Editore.Bari( 2012)(with R. Padalino ) A time delay model for the diffusion of a new technology. Non linear analysis: Real World Applications. Vol. 13, pp , (2012).(with V.Fanelli) Investiganting the diffusion of renewable energy technologies in Italy. Advances and Applications in Mathematical Sciences. vol 12, Issue 1, November 2012, pp (with V. Fanelli and S. Musti)

9 Research contracts : "The development of the energy market in Puglia: renewable energy and economic - financial implications" 2012

10 Modelli matematici a confronto per la diffusione di tecnologie per la produzione di energia rinnovabile. (con V. Fanelli,S.Musti) in Un modello per i prezzi forward nel mercato elettrico. (con V. Fanelli,S.Musti) Lo sviluppo delle energie alternative. Il caso Puglia. (edited by L.Maddalena). Franco Angeli Editore. Milano.( 2012)

11 Models for the diffusion of innovations The diffusion of an innovation is the process by which an innovation is communicated through certain channels over time among the members of a social system (Rogers, 2005)

12 The adoption process

13 Assumptions The members of a social system are adopters or non-adopters (the binary diffusion process). The number of potential adopters is known and the model is not expected that the social system can increase or decrease in "amplitude". Adopting an innovation is a discrete event or continuous. Multiple adoptions are not allowed and adoption can not be terminated. Innovation does not change in the diffusion process. Innovation does not depend on other innovations. The geographical boundaries of the social system do not change in the diffusion process. Use the model to make predictions on sales requires the assumption that the past can be used to predict the future. The effects of the interaction between adopters and potential adopters does not take into account the time that elapses between the interaction and the adoption

14 Basic model (exthernal influences)

15 Basic model (internal influences)

16 The Bass model ( 1969)

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18 Delay differential equation

19 Characteristics of DDEs (Books of J.K. Hale, R.Bellman,K. L. Cooke, ) DDEs require an initial function (and not a point) because the derivative depends on the solution at the previous time t τ, it is necessary to provide an initial history function to specify the value of the solution before time t=0. For most problems there is a jump derivative discontinuity at the initial time moreover such derivative jump discontinuity propagates from the initial point t=o along the integration interval and gives rise to subsequent discontinuity points where the solution is smoothed out more and more. The same solution can arise from different initial functions; With the same initial function can have multiple solutions; The solution may cease to exist; The delay can destabilize the solution.

20 Logistic Equation : If x0 < K, the population grows, approaching K asymptotically. If x0 > K, the population decreases, again approaching K asymptotically. If x0 = K, the population remains in time at x = K.

21 Hutchinson s Equation or Delay logistic equation:

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25 A time delay model for the diffusion of a new technology. Non linear analysis: R.W.A. Vol. 13, , (2012) (with V.Fanelli)

26 The Model

27 represents the number of adopters that move from the knowledge class to the adoption class, that is those who know about the adoption at time t τ and decide to adopt the technology. The term δa(t) represents the percentage of A that has initially adopted the technology and that after the period of evaluation decides to reverse its decision of adoption.

28 The model

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31 Existence, positivity.asymptotic behaviour solutions. Equilibrium solution

32 Local Stability

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34 Remark. If τ, the term k(τ ) in the delay differential equation (2.1) tends to zero and A (τ ) 0. Numerical simulations in the following section show that any solution of Eq. (2.1) converges to zero. The behavior of the solution A(t) in relation to the delay reflects the behavior of economic individuals. When the period of evaluation about new technology adoption becomes very wide the adopters extinguish since they do not adopt the technology anymore.

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36 Conclusions We have proposed a time delay model with a stage structure that simulates some stages of the adoption process. A good representation of the knowledge and the persuasion stages is given by introducing specific coefficients for the external and internal influences to or within the social system. The time delay characterizes the evaluation and the decision-making stages. Indeed we assume that over the interval [t τ, t] the individual engages in all activities that give him the possibility of knowing the innovation. In this way the individual is able to test the technology and to make a decision about its adoption. We have carried out a qualitative analysis of the delay differential equation and have found the final level of adopters. We have studied the local stability of the equilibrium and through numerical simulations we have highlighted its possible global stability.

37 Investigating the diffusion of renewable energy technologies in Italy. Advances and Applications in Mathematical Sciences. vol 12, Issue 1, November 2012, pp (with V. Fanelli and S. Musti) In this paper we investigate the diffusion process of renewable energy technology in Italy through the study and estimation of different mathematical models proposed in the literature. Basing the estimation on historical data of the installed power, we find that the pioneer of new product diffusion models, the Bass model, is appropriate to represent the photovoltaic technology diffusion process, whereas after a comparison among the most important models discussed in the literature, we conclude that the Non- Uniform Influence (NUI) model describes the wind technology diffusion process in the most accurate way. The NUI model is also used as a prediction instrument for the diffusion dynamics of wind technology. In fact, we fixed the level of installed power to reach at a future data, and simulated the diffusion curve to find how many years are needed to get to the target.

38 We aim at applying the Bass model to real data, in order to study photovoltaic and wind technologies diffusion in Puglia e/o in Italia. Our data set consists of two time series of the annual installed power, measured in Mega Watt (MW), in Puglia, spanning from 1994 to 2009.The source is Terna S.p.A..

39 The Bass Model Applied to RET

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44 We aim at investigating which of the models illustrated better describes the dynamic of the diffusion process of wind technologies in Puglia

45 Bass model drawbacks 1 The Bass model assumes that, for any innovation diffusion, the maximum rate of penetration cannot occur after an innovation has penetrated 50% of its potential market. Even this result is not always found in the realty; 2 The diffusion curve is symmetric with respect to the point of inflection. Actually we expect that earlier adopters behave different from late adopters.

46 Beyond the Bass model We consider five models discussed in the literature that overcome the limitations of the Bass model because they produce: a point of inflection in a range of values; a symmetric or not symmetric diffusion curve

47 Model comparison The Von Bertalanffy model The Sharif Kabir model The Jeuland model Model NSRL Model NUI

48 The Von Bertalanffy model With this model it is possible to have both symmetric and asymmetric curves and the inflection point can take a value between 0 and 1

49 The Sharif Kabir model b represents the coefficient of the internal influence and σ is a positive constant that can vary between zero and one. Diffusion curves are not necessarily symmetric, but all of them have a point of inflection that is in the range 0.33 F 0.5.

50 The Jeuland model The diffusion curve can be symmetric or asymmetric and the point of inflection is always below a 50% of adopters in the social system.

51 Model NSRL (non symmetric responding logistic model) The authors suggest that a coefficient of imitation b constant over time is not consistent with the theory and practice. Empirical evidence shows that the effect of imitation and interpersonal communication changes over time. So it is proposed to measure the internal influence with a timevarying adoption level function.

52 Model NUI( non- uniform influence) Add to the model the external factor a, In both models NSRL and NUI, the point of inflection can assume a value between 0 and 1, the diffusion curves can even be asymmetric.

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57 Conclusions In this paper we have applied the new product diffusion models discussed in the literature for studying the processes of RET adoption. In particular, we have aimed at investigating the diffusion process of technologies for the production of energy from renewable sources, solar and wind power, in Italy. Basing the analysis and estimation of models on historical data of installed power in Italy, we have found that the Bass model, pioneered in the literature of diffusion models, is capable of describing the diffusion of photovoltaic technology in Italy. Instead, to adequately represent the process of adoption and diffusion of wind technology it has been necessary to estimate different models and the comparison among them has shown that the NUI model best describes the diffusion process.

58 A framework for an integro-differential system model for the tumor growth

59 The model

60 p(x; t) = u(x; t) + v(x; t) is the total tumor density, u(x; t)is the density of cancer stem cells CSCs and v(x; t) is the density of tumor cells TCs. We observe that u(x; t) and v(x; t) denote the density in cells per unity cell space, that is the fraction of the interval (x; x + dx) physically occupied by cells. The kernel K(x; y; p(x; t)) redistributes cells only within domain

61 Grazie per l attenzione