Population models at stochastic times

Orsingher, Enzo; Ricciuti, Costantino; Toaldo, Bruno

doi:10.1017/apr.2016.11

“…The case for stochastic clearing processes is considered by [15]. Relevant also are [16][17][18][19][20]. For references on sample-path analysis, the reader is referred to [1,2,16,[21][22][23][24][25].…”

Section: =0mentioning

confidence: 99%

Asymptotic Time Averages and Frequency Distributions

El-Taha

¹

2016

International Journal of Stochastic Analysis

1

0

View full text Add to dashboard Cite

Consider an arbitrary nonnegative deterministic process (in a stochastic setting {X(t), t≥0} is a fixed realization, i.e., sample-path of the underlying stochastic process) with state space S=(-∞,∞). Using a sample-path approach, we give necessary and sufficient conditions for the long-run time average of a measurable function of process to be equal to the expectation taken with respect to the same measurable function of its long-run frequency distribution. The results are further extended to allow unrestricted parameter (time) space. Examples are provided to show that our condition is not superfluous and that it is weaker than uniform integrability. The case of discrete-time processes is also considered. The relationship to previously known sufficient conditions, usually given in stochastic settings, will also be discussed. Our approach is applied to regenerative processes and an extension of a well-known result is given. For researchers interested in sample-path analysis, our results will give them the choice to work with the time average of a process or its frequency distribution function and go back and forth between the two under a mild condition.

show abstract

“…Saeedian et al [36] showed how another memory functional of the process can lead to replacing the integer derivatives with Caputo fractional derivatives. In this paper, we show how Caputo fractional differential equations follow naturally from fractional stochastic processes like those introduced in [37][38][39][40][41][42][43][44][45][46]. Then we show that for different data sets, FDE models fit the data better for some epidemics whereas ODE models fit better for others.…”

Section: Introductionmentioning

confidence: 99%

Integer Versus Fractional Order SEIR Deterministic and Stochastic Models of Measles

Islam¹,

Peace²,

Medina³

et al. 2020

Preprint

View full text Add to dashboard Cite

In this paper, we compare the performance between systems of ordinary and (Caputo) fractional differential equations depicting the susceptible-exposed-infectious-recovered (SEIR) models of diseases. In order to understand the origins of both approaches as mean-field approximations of integer and fractional stochastic processes, we introduce the fractional differential equations as approximations of some type of fractional nonlinear birth--death processes. Then, we examine validity of the two approaches against empirical courses of epidemics; we fit both of them to case counts of three measles epidemics that occurred during the pre-vaccination era in three different locations. While FDEs appear more flexible in fitting empirical data, our ODEs offered better fits to two out of three data sets. Important differences in transient dynamics between these modeling approaches are discussed.

show abstract

“…Saeedian et al [36] showed how another memory functional of the process can lead to replacing the integer derivatives with Caputo fractional derivatives. In this paper, we show how Caputo fractional differential equations follow naturally from fractional stochastic processes like those introduced in literature [37][38][39][40][41][42][43][44][45][46]. We then compare transient and long term dynamics between the FDE and ODE models while fitting them to three different data sets.…”

Section: Introductionmentioning

confidence: 99%

Integer Versus Fractional Order SEIR Deterministic and Stochastic Models of Measles

Islam

¹

,

Peace

²

,

Medina

³

et al. 2020

IJERPH

View full text Add to dashboard Cite

In this paper, we compare the performance between systems of ordinary and (Caputo) fractional differential equations depicting the susceptible-exposed-infectious-recovered (SEIR) models of diseases. In order to understand the origins of both approaches as mean-field approximations of integer and fractional stochastic processes, we introduce the fractional differential equations (FDEs) as approximations of some type of fractional nonlinear birth and death processes. Then, we examine validity of the two approaches against empirical courses of epidemics; we fit both of them to case counts of three measles epidemics that occurred during the pre-vaccination era in three different locations. While ordinary differential equations (ODEs) are commonly used to model epidemics, FDEs are more flexible in fitting empirical data and theoretically offer improved model predictions. The question arises whether, in practice, the benefits of using FDEs over ODEs outweigh the added computational complexities. While important differences in transient dynamics were observed, the FDE only outperformed the ODE in one of out three data sets. In general, FDE modeling approaches may be worth it in situations with large refined data sets and good numerical algorithms. Laplace transform of a function f (t) is defined as L( f )(s) =f (s) = ∞ 0 e −st f (t)dt.

show abstract

Population models at stochastic times

Cited by 8 publications

References 14 publications

Asymptotic Time Averages and Frequency Distributions

Asymptotic Time Averages and Frequency Distributions

Integer Versus Fractional Order SEIR Deterministic and Stochastic Models of Measles

Integer Versus Fractional Order SEIR Deterministic and Stochastic Models of Measles

Contact Info

Product

Resources

About