A STOCHASTIC MODEL FOR EPIDEMICS BASED ON THE RENEWAL EQUATION

2000 ◽  
Vol 08 (01) ◽  
pp. 1-20 ◽  
Author(s):  
J. L. W. GIELEN
Keyword(s):  
Stochastic Model ◽  
Deterministic Model ◽  
Formal Solution ◽  
Infected Individual ◽  
External Source ◽  
Mean Value ◽  
Renewal Equation ◽  
Infection Model ◽  
Final Size ◽  
Infinite Population

A stochastic continuous-infection model is developed that describes the evolution of an infectious disease introduced into an infinite population of susceptibles. The proposed model is the natural stochastic counterpart of the deterministic model for epidemics, based on the renewal equation. As in the deterministic model, the infectivity of an infected individual is a function of his age-of-infection, that is the time elapsed since his own infection. A time-dependent external source of infection is included. The model provides analytical expressions that describe the stochastic infective-age structure of the population at any moment of time. It is shown that the mean value of the number of infectives predicted by the stochastic model satisfies the renewal equation, which furnishes a formal solution of this equation. The model also yields simple expressions for the expected arrival times of infectives, that can be useful for the inverse problem. An explicit expression for the final size distribution is obtained. This leads to a precise quantitative threshold theorem that distinguishes between the possibilities of a minor outbreak or a major build-up of the epidemic.

scholarly journals A renewal equation model to assess roles and limitations of contact tracing for disease outbreak control

2021 ◽  
Vol 8 (4) ◽  
Author(s):  
Francesca Scarabel ◽  
Lorenzo Pellis ◽  
Nicholas H. Ogden ◽  
Jianhong Wu
Keyword(s):  
Deterministic Model ◽  
Infected Individual ◽  
Equation Model ◽  
Renewal Equation ◽  
Contact Tracing ◽  
Calendar Time ◽  
Individual Level ◽  
Total Incidence ◽  
Community Transmission ◽  
The Individual

We propose a deterministic model capturing essential features of contact tracing as part of public health non-pharmaceutical interventions to mitigate an outbreak of an infectious disease. By incorporating a mechanistic formulation of the processes at the individual level, we obtain an integral equation (delayed in calendar time and advanced in time since infection) for the probability that an infected individual is detected and isolated at any point in time. This is then coupled with a renewal equation for the total incidence to form a closed system describing the transmission dynamics involving contact tracing. We define and calculate basic and effective reproduction numbers in terms of pathogen characteristics and contact tracing implementation constraints. When applied to the case of SARS-CoV-2, our results show that only combinations of diagnosis of symptomatic infections and contact tracing that are almost perfect in terms of speed and coverage can attain control, unless additional measures to reduce overall community transmission are in place. Under constraints on the testing or tracing capacity, a temporary interruption of contact tracing may, depending on the overall growth rate and prevalence of the infection, lead to an irreversible loss of control even when the epidemic was previously contained.

scholarly journals A renewal equation model to assess roles and limitations of contact tracing for disease outbreak control

2021 ◽  
Author(s):  
Francesca Scarabel ◽  
Lorenzo Pellis ◽  
Nicholas H Ogden ◽  
Jianhong Wu
Keyword(s):  
Deterministic Model ◽  
Infected Individual ◽  
Equation Model ◽  
Renewal Equation ◽  
Contact Tracing ◽  
Calendar Time ◽  
Individual Level ◽  
Total Incidence ◽  
Community Transmission ◽  
The Individual

We propose a deterministic model capturing essential features of contact tracing as part of public health non-pharmaceutical interventions to mitigate an outbreak of an infectious disease. By incorporating a mechanistic formulation of the processes at the individual level, we obtain an integral equation (delayed in calendar time and advanced in time since infection) for the probability that an infected individual is detected and isolated at any point in time. This is then coupled with a renewal equation for the total incidence to form a closed system describing the transmission dynamics involving contact tracing. We define and calculate basic and effective reproduction numbers in terms of pathogen characteristics and contact tracing implementation constraints. When applied to the case of SARS-CoV-2, our results show that only combinations of diagnosis of symptomatic infections and contact tracing that are almost perfect in terms of speed and coverage can attain control, unless additional measures to reduce overall community transmission are in place. Under constraints on the testing or tracing capacity, a temporary interruption of contact tracing may, depending on the overall growth rate and prevalence of the infection, lead to an irreversible loss of control even when the epidemic was previously contained.

The final outcome of an epidemic model with several different types of infective in a large population

1995 ◽  
Vol 32 (3) ◽  
pp. 579-590 ◽  
Author(s):  
Frank Ball ◽  
Damian Clancy
Keyword(s):  
Stochastic Model ◽  
Size Distribution ◽  
Epidemic Model ◽  
Infected Individual ◽  
Large Population ◽  
Asymptotic Distributions ◽  
Initial Population ◽  
Group Model ◽  
Final Size ◽  
Different Types

We consider a stochastic model for the spread of an epidemic amongst a closed homogeneously mixing population, in which there are several different types of infective, each newly infected individual choosing its type at random from those available. The model is based on the carrier-borne model of Downton (1968), as extended by Picard and Lefèvre (1990). The asymptotic distributions of final size and area under the trajectory of infectives are derived as the initial population becomes large, using arguments based on those of Scalia-Tomba (1985), (1990). We then use our limiting results to compare the asymptotic final size distribution of our model with that of a related multi-group model, in which the type of each infective is assigned deterministically.

The final outcome of an epidemic model with several different types of infective in a large population

1995 ◽  
Vol 32 (03) ◽  
pp. 579-590 ◽  
Author(s):  
Frank Ball ◽  
Damian Clancy
Keyword(s):  
Stochastic Model ◽  
Size Distribution ◽  
Epidemic Model ◽  
Infected Individual ◽  
Large Population ◽  
Asymptotic Distributions ◽  
Initial Population ◽  
Group Model ◽  
Final Size ◽  
Different Types

We consider a stochastic model for the spread of an epidemic amongst a closed homogeneously mixing population, in which there are several different types of infective, each newly infected individual choosing its type at random from those available. The model is based on the carrier-borne model of Downton (1968), as extended by Picard and Lefèvre (1990). The asymptotic distributions of final size and area under the trajectory of infectives are derived as the initial population becomes large, using arguments based on those of Scalia-Tomba (1985), (1990). We then use our limiting results to compare the asymptotic final size distribution of our model with that of a related multi-group model, in which the type of each infective is assigned deterministically.

The stationary distribution and extinction of a double thresholds HTLV-I infection model with nonlinear CTL immune response disturbed by white noise

2019 ◽  
Vol 12 (05) ◽  
pp. 1950058 ◽  
Author(s):  
Kai Qi ◽  
Daqing Jiang ◽  
Tasawar Hayat ◽  
Ahmed Alsaedi
Keyword(s):  
Immune Response ◽  
Stochastic Model ◽  
White Noise ◽  
Stationary Distribution ◽  
Lyapunov Functions ◽  
Deterministic Model ◽  
Dynamical Behavior ◽  
Infection Model ◽  
Reproduction Numbers ◽  
Ctl Immune Response

This paper investigates the stochastic HTLV-I infection model with CTL immune response, and the corresponding deterministic model has two basic reproduction numbers. We consider the nonlinear CTL immune response for the interaction between the virus and the CTL immune cells. Firstly, for the theoretical needs of system dynamical behavior, we prove that the stochastic model solution is positive and global. In addition, we obtain the existence of ergodic stationary distribution by stochastic Lyapunov functions. Meanwhile, sufficient condition for the extinction of the stochastic system is acquired. Reasonably, the dynamical behavior of deterministic model is included in our result of stochastic model when the white noise disappears.

scholarly journals Stochastic and deterministic mathematical model of cholera disease dynamics with direct transmission

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Getachew Teshome Tilahun ◽  
Woldegebriel Assefa Woldegerima ◽  
Aychew Wondifraw
Keyword(s):  
Mathematical Model ◽  
Stochastic Model ◽  
Deterministic Model ◽  
Equilibrium Points ◽  
Invariant Solution ◽  
Stochastic Approach ◽  
Disease Dynamics ◽  
Treatment Rate ◽  
Invariant Region ◽  
Contact Transmission

AbstractIn this paper we develop a stochastic mathematical model of cholera disease dynamics by considering direct contact transmission pathway. The model considers four compartments, namely susceptible humans, infectious humans, treated humans, and recovered humans. Firstly, we develop a deterministic mathematical model of cholera. Since the deterministic model does not consider the randomness process or environmental factors, we converted it to a stochastic model. Then, for both types of models, the qualitative behaviors, such as the invariant region, the existence of a positive invariant solution, the two equilibrium points (disease-free and endemic equilibrium), and their stabilities (local as well as global stability) of the model are studied. Moreover, the basic reproduction numbers are obtained for both models and compared. From the comparison, we obtained that the basic reproduction number of the stochastic model is much smaller than that of the deterministic one, which means that the stochastic approach is more realistic. Finally, we performed sensitivity analysis and numerical simulations. The numerical simulation results show that reducing contact rate, improving treatment rate, and environmental sanitation are the most crucial activities to eradicate cholera disease from the community.

STOCHASTIC PREDATION MODEL WITH DEPLETION

1979 ◽  
Vol 111 (4) ◽  
pp. 465-470 ◽  
Author(s):  
Guy L. Curry ◽  
Richard M. Feldman
Keyword(s):  
Stochastic Model ◽  
Deterministic Model ◽  
Expected Number ◽  
Numerical Comparisons ◽  
Prey Depletion

AbstractA stochastic model is developed for the expected number of prey taken by a single predator when prey depletion is apparent. The so-called “random predator equation” with prey exploitation of Royama and Rogers is compared with the stochastic model. The numerical comparisons illustrate situations where the deterministic model provides adequate and inadequate approximations.

The dynamics of a delayed deterministic and stochastic epidemic SIR models with a saturated incidence rate

2018 ◽  
Vol 26 (4) ◽  
pp. 235-245 ◽  
Author(s):  
Modeste N’zi ◽  
Ilimidi Yattara
Keyword(s):  
Stochastic Model ◽  
White Noise ◽  
Incidence Rate ◽  
Deterministic Model ◽  
Contact Rate ◽  
Almost Sure Stability ◽  
Saturated Incidence Rate ◽  
Saturated Incidence ◽  
Global Positive Solution ◽  
Disease Free

AbstractWe treat a delayed SIR (susceptible, infected, recovered) epidemic model with a saturated incidence rate and its perturbation through the contact rate using a white noise. We start with a deterministic model and then add a perturbation on the contact rate using a white noise to obtain a stochastic model. We prove the existence and uniqueness of the global positive solution for both deterministic and stochastic delayed differential equations. Under suitable conditions on the parameters, we study the global asymptotic stability of the disease-free equilibrium of the deterministic model and the almost sure stability of the disease-free equilibrium of the stochastic model.

Gray-Box Approach for Fault Detection of Dynamical Systems

2003 ◽  
Vol 125 (3) ◽  
pp. 451-454 ◽  
Author(s):  
Han G. Park ◽  
Michail Zak
Keyword(s):  
Neural Network ◽  
Time Delay ◽  
Fault Detection ◽  
Stochastic Model ◽  
Deterministic Model ◽  
Feed Forward Neural Network ◽  
The Neural Network ◽  
Simulated System ◽  
Auto Regressive ◽  
Nonlinear Components

We present a fault detection method called the gray-box. The term “gray-box” refers to the approach wherein a deterministic model of system, i.e., “white box,” is used to filter the data and generate a residual, while a stochastic model, i.e., “black-box” is used to describe the residual. The residual is described by a three-tier stochastic model. An auto-regressive process, and a time-delay feed-forward neural network describe the linear and nonlinear components of the residual, respectively. The last component, the noise, is characterized by its moments. Faults are detected by monitoring the parameters of the auto-regressive model, the weights of the neural network, and the moments of noise. This method is demonstrated on a simulated system of a gas turbine with time delay feedback actuator.

A Dynamic Generalization of Zipf's Rank-Size Rule

1982 ◽  
Vol 14 (11) ◽  
pp. 1449-1467 ◽  
Author(s):  
B Roehner ◽  
K E Wiese
Keyword(s):  
Stochastic Model ◽  
Size Distribution ◽  
Urban Growth ◽  
Simple Form ◽  
Deterministic Model ◽  
City Size ◽  
Zipf’S Law ◽  
Zipf's Law ◽  
City Size Distribution ◽  
Zero Growth

A dynamic deterministic model of urban growth is proposed, which in its most simple form yields Zipf's law for city-size distribution, and in its general form may account for distributions that deviate strongly from Zipf's law. The qualitative consequences of the model are examined, and a corresponding stochastic model is introduced, which permits, in particular, the study of zero-growth situations.

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