A note on the recurrence of yellow fever epidemics in urban populations

Abstract

An approximate expression is found for the probability density of the recurrence time between major outbreaks of a host-vector disease such as yellow fever, in which the infectious period of a host is short and the lifetime of a vector is short compared with that of a host. The typical interval between epidemics based on the modal value of this distribution is obtained. These results are the analogues of those obtained by Bartlett (1956), (1966) for the periodicity of measles epidemics in small communities.

Local and global approaches to the problem of Poincaré recurrences. Applications in nonlinear dynamics

“…Comparing (14) and (23) we can obtain that ν(ω) = α C in the considered case. This implies that the AP dimension coincides with the approximation rate of the irrational rotation number ω.…”

“…The map (40), which is a bijection of a unit square on the (x, y) plane into itself, belongs to the class of hyperbolic maps. Relations (14) and (15) have been proven for such systems. For δ < 1/(2π), the map (40) is dissipative and has a chaotic attractor with a positive Lyapunov exponent λ 1 > 0.…”

“…14b), the dependences τ inf (ε) do not lie on a straight line, but represent exponentially decreasing functions. It is interesting to note that if these exponential dependences are used and results are presented in the coordinates (ln τ inf (ε) , ln ε), we can obtain straight lines which correspond to (14). However, these results are erroneous and are related to the insufficient number of initial points or returns into a cell.…”

“…It is known that the Lozi map is characterized by a quasihyperbolic chaotic attractor [70] and suits the test experiments in the best way, because it has ergodic properties which were used in rigorous proofs of (14), (15). The results of computations performed for a chaotic attractor in (41) are shown in Fig.…”

“…The idea of recurrence of complex systems in a neighborhood of a certain state over time becomes one of the fundamental concepts of modern science. This concept allows a unified consideration of such evolutionary processes as the change of climate [4], of solar activity [5,6], of population dynamics [7,8,9,10], changes in financial markets [11,12,13], the spread of epidemics [7,14,15,16] and references therein, and many others [17]. Long-term observations have revealed that many evolutionary processes in nature and society are characterized by the property of recurrence.…”