STATISTICAL FRACTALS WITH CUTOFFS, SHLESINGER-HUGHES RENORMALIZATION,AND THE ONSET OF AN EPIDEMIC

A method for constructing cutoff values of the probability densities a ttached to statistical fractals is introduced for which only the begin ning of the tail of a probability density is slowly decaying and given by an inverse power law; the end of the tail, however, is short, deca ying exponentially or faster. This method is illustrated by an example from the theory of epidemics. The probability density (psi)over tilde >(t)dt of the time interval t within which an infected individual is a ble to spread an epidemic is evaluated based on the following assumpti ons. (1) The spreading of the epidemic depends on the size of the germ load carried by an infected individual; the total germ load is measur ed in large units containing at least 10(5)-10(6) germs. It is assumed that for each unit there is a constant probability alpha of infecting other individuals. (2) The total number of germ units carried by an i ndividual is a random variable; the corresponding distribution is eval uated by assuming that there is a constant probability a: that an infe cted individual carries a unit load of germs. (3) The encounters of an infected individual with healthy individuals susceptible to receiving the illness is a random event occurring with an average time-independ ent contact frequency v. The probability density (psi)over tilde>(t)dt determined by these three assumptions has a long tail characteristic of an ideal statistical fractal behavior: (psi)over tilde>(t)dt simila r to(vt)(-(H+1))Xi(ln(vt))d(vt) as t>>1/v, where H=ln(a)/ln(1-p) is a positive fractal exponent and Xi(ln(vt)) is a periodic function of ln( vt) with a period -ln(1-p). In this case all positive moments [t(q)] o f the infection time with q greater than or equal to H are infinite. A statistical fractal with a cutoff emerges if a fourth hypothesis is a dded: (4) Due to the healing process the bacterial load of an individu al decreases exponentially in time with a rate coefficient b. If the h ealing process is slower than the encounter process of healthy individ uals, v>b, then only the beginning of the tail of (psi)over tilde>(t)d t obeys an inverse power law scaling (psi)over tilde>(t)dt similar to( vt)(-(H+1)))Xi(ln(vt))d(vt) for 1/b>t>1/v; the end of the tail, howeve r, is exponential and determined by the healing process (psi)over tild e>(t)similar to const x exp(-bt) as t>>1/b. Due to the cutoff the mome nts [t(q)] of the infection time, although finite, have an intermitten t behavior characterized by the scaling law [t(q)]similar to const x b (-(q-H)) as b-->0. A generalization of the epidemic model is given by assuming that the size of the germ load carried by an infected individ ual is an arbitrary random function of time with known stochastic prop erties and that the encounters with healthy individuals can be describ ed by a correlated random point process. An analytical expression is d erived for the probability density (psi)over tilde>(t)dt of the infect ion time in terms of the characteristic functionals of the germ load a nd of the encounter process. A comparison is performed between the ons et of an epidemic and the passage over a fluctuating energy barrier wi th dynamical disorder. Some implications of the cutoff of a statistica l fractal for the physics of fractal time are also analyzed.