A note on perturbation techniques for epidemics

1971 ◽  
Vol 3 (2) ◽  
pp. 214-218
Author(s):  
H. E. Daniels
Keyword(s):  
Difference Equations ◽  
Zero Order ◽  
Similar Type ◽  
Perturbation Techniques ◽  
Epidemic Process ◽  
Stochastic Epidemic ◽  
Explicit Formulae

This note is prompted by the papers of Weiss (this Symposium) and Bailey (1968). Weiss develops a technique for approximation to the moments of an epidemic process by regarding them as expandable in powers of N-1 where N is the size of the population, assumed constant. He first considers the simple stochastic epidemic with no removals and obtains explicit formulae for the terms of order N-1, the zero order terms being the deterministic values. Bailey is concerned with a similar type of approximation and he derives explicit results to the same order. Bailey uses an eigenfunction approach whereas Weiss's method is more direct and perhaps easier to generalise. However, in attempting to extend the method to the case of a closed epidemic with removals Weiss is led to intractable difference equations.

A note on perturbation techniques for epidemics

1971 ◽  
Vol 3 (02) ◽  
pp. 214-218
Author(s):  
H. E. Daniels
Keyword(s):  
Difference Equations ◽  
Zero Order ◽  
Similar Type ◽  
Perturbation Techniques ◽  
Epidemic Process ◽  
Stochastic Epidemic ◽  
Explicit Formulae

This note is prompted by the papers of Weiss (this Symposium) and Bailey (1968). Weiss develops a technique for approximation to the moments of an epidemic process by regarding them as expandable in powers of N-1 where N is the size of the population, assumed constant. He first considers the simple stochastic epidemic with no removals and obtains explicit formulae for the terms of order N -1, the zero order terms being the deterministic values. Bailey is concerned with a similar type of approximation and he derives explicit results to the same order. Bailey uses an eigenfunction approach whereas Weiss's method is more direct and perhaps easier to generalise. However, in attempting to extend the method to the case of a closed epidemic with removals Weiss is led to intractable difference equations.

Two theorems on solutions of differential-difference equations and applications to epidemic theory

1967 ◽  
Vol 4 (02) ◽  
pp. 271-280 ◽  
Author(s):  
Norman C. Severo
Keyword(s):  
Differential Equation ◽  
Difference Equations ◽  
Total Population ◽  
Original System ◽  
Initial Distribution ◽  
Fixed Number ◽  
Total Population Size ◽  
Stochastic Epidemic ◽  
Iterative Solutions ◽  
General Stochastic

We present two theorems that provide simple iterative solutions of special systems of differential-difference equations. We show as examples of the theorems the simple stochastic epidemic (cf. Bailey, 1957, p. 39, and Bailey, 1963) and the general stochastic epidemic (cf. Bailey, 1957; Gani, 1965; and Siskind, 1965), in each of which we let the initial distribution of the number of uninfected susceptibles and the number of infectives be arbitrary but assume the total population size bounded. In all of the references cited above the methods of solution involve solving a corresponding partial differential equation, whereas we deal directly with the original system of ordinary differential-difference equations. Furthermore in the cited references the authors begin at time t = 0 with a population having a fixed number of uninfected susceptibles and a fixed number of infectives. For the simple stochastic epidemic with arbitrary initial distribution we provide solutions not obtainable by the results given by Bailey (1957 or 1963). For the general stochastic epidemic, if we use the results of Gani or Siskind, then the solution of the problem having an arbitrary initial distribution would involve additional steps that would sum proportionally-weighted conditional results.

Bounds on the velocity of spread of infection for a spatially connected epidemic process

1980 ◽  
Vol 17 (3) ◽  
pp. 839-845 ◽  
Author(s):  
M. J. Faddy ◽  
I. H. Slorach
Keyword(s):  
Asymptotic Velocity ◽  
Epidemic Process ◽  
Lower Bounding ◽  
Stochastic Epidemic ◽  
Spread Of Infection

The simple (non-spatial) stochastic epidemic is generalised to allow infected individuals to move forward through a system of spatially connected colonies C1, C2, C3, ·· ·each containing susceptible individuals. Upper and lower bounding processes are considered, to establish bounds on the asymptotic velocity of forward spread of the infection through these spatially connected colonies. These bounds are shown to be asymptotically equivalent under certain conditions, and some simulations reveal other features of the process.

Factorial moments and probabilities for the general stochastic epidemic

1973 ◽  
Vol 10 (02) ◽  
pp. 277-288 ◽  
Author(s):  
L. Billard
Keyword(s):  
Difference Equations ◽  
The State ◽  
Factorial Moments ◽  
Stochastic Epidemic ◽  
The Matrix ◽  
Mean And Variance ◽  
The Mean ◽  
General Stochastic

By an appropriate partitioning of the matrix of coefficients in the system of differential difference equations for the general stochastic epidemic, the nature of the state probabilities is shown to consist of combinations of factorial terms. Further, factorial moments are readily obtained. In particular, the mean and variance of the number of susceptibles are derived.

On the asymptotic distribution of the size of a stochastic epidemic

1983 ◽  
Vol 20 (2) ◽  
pp. 390-394 ◽  
Author(s):  
Thomas Sellke
Keyword(s):  
Population Size ◽  
Asymptotic Distribution ◽  
Epidemic Process ◽  
Stochastic Epidemic ◽  
Original Number ◽  
Intuitive Proof

For a stochastic epidemic of the type considered by Bailey [1] and Kendall [3], Daniels [2] showed that ‘when the threshold is large but the population size is much larger, the distribution of the number remaining uninfected in a large epidemic has approximately the Poisson form.' A simple, intuitive proof is given for this result without use of Daniels's assumption that the original number of infectives is ‘small'. The proof is based on a construction of the epidemic process which is more explicit than the usual description.

Two theorems on solutions of differential-difference equations and applications to epidemic theory

1967 ◽  
Vol 4 (2) ◽  
pp. 271-280 ◽  
Author(s):  
Norman C. Severo
Keyword(s):  
Differential Equation ◽  
Difference Equations ◽  
Total Population ◽  
Original System ◽  
Initial Distribution ◽  
Fixed Number ◽  
Total Population Size ◽  
Stochastic Epidemic ◽  
Iterative Solutions ◽  
General Stochastic

We present two theorems that provide simple iterative solutions of special systems of differential-difference equations. We show as examples of the theorems the simple stochastic epidemic (cf. Bailey, 1957, p. 39, and Bailey, 1963) and the general stochastic epidemic (cf. Bailey, 1957; Gani, 1965; and Siskind, 1965), in each of which we let the initial distribution of the number of uninfected susceptibles and the number of infectives be arbitrary but assume the total population size bounded. In all of the references cited above the methods of solution involve solving a corresponding partial differential equation, whereas we deal directly with the original system of ordinary differential-difference equations. Furthermore in the cited references the authors begin at time t = 0 with a population having a fixed number of uninfected susceptibles and a fixed number of infectives. For the simple stochastic epidemic with arbitrary initial distribution we provide solutions not obtainable by the results given by Bailey (1957 or 1963). For the general stochastic epidemic, if we use the results of Gani or Siskind, then the solution of the problem having an arbitrary initial distribution would involve additional steps that would sum proportionally-weighted conditional results.

Factorial moments and probabilities for the general stochastic epidemic

1973 ◽  
Vol 10 (2) ◽  
pp. 277-288 ◽  
Author(s):  
L. Billard
Keyword(s):  
Difference Equations ◽  
The State ◽  
Factorial Moments ◽  
Stochastic Epidemic ◽  
The Matrix ◽  
Mean And Variance ◽  
The Mean ◽  
General Stochastic

By an appropriate partitioning of the matrix of coefficients in the system of differential difference equations for the general stochastic epidemic, the nature of the state probabilities is shown to consist of combinations of factorial terms. Further, factorial moments are readily obtained. In particular, the mean and variance of the number of susceptibles are derived.

Some results for complex partial q-difference equations in ℂ n \mathbb{C}^{n}

2019 ◽  
Vol 26 (3) ◽  
pp. 471-481
Author(s):  
Yue Wang
Keyword(s):  
Difference Equations ◽  
Nevanlinna Theory ◽  
Meromorphic Functions ◽  
Zero Order ◽  
Value Distribution ◽  
Meromorphic Solutions ◽  
Difference Polynomials

Abstract Using the Nevanlinna theory of the value distribution of meromorphic functions, the value distribution of complex partial q-difference polynomials of meromorphic functions of zero order is investigated. The existence of meromorphic solutions of some types of systems of complex partial q-difference equations in {\mathbb{C}^{n}} is also investigated. Improvements and extensions of some results in the literature are presented. Some examples show that our results are, in a certain sense, the best possible.

scholarly journals Existence of zero-order meromorphic solutions of certain q-difference equations

2018 ◽  
Vol 2018 (1) ◽  
Author(s):  
Yunfei Du ◽  
Zongsheng Gao ◽  
Jilong Zhang ◽  
Ming Zhao
Keyword(s):  
Difference Equations ◽  
Zero Order ◽  
Meromorphic Solutions

A unified approach to the distribution of total size and total area under the trajectory of infectives in epidemic models

1986 ◽  
Vol 18 (2) ◽  
pp. 289-310 ◽  
Author(s):  
Frank Ball
Keyword(s):  
Probabilistic Approach ◽  
Heterogeneous Population ◽  
Epidemic Models ◽  
Infectious Period ◽  
Unified Approach ◽  
Total Size ◽  
Epidemic Process ◽  
Stochastic Epidemic ◽  
General Stochastic

We provide a unified probabilistic approach to the distribution of total size and total area under the trajectory of infectives for a general stochastic epidemic with any specified distribution of the infectious period. The key tool is a Wald&s identity for the epidemic process. The generalisation of our results to epidemics spreading amongst a heterogeneous population is straightforward.

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