scholarly journals On discrete time epidemic models in Kermack-McKendrick form

2021 ◽  
Author(s):  
Odo Diekmann ◽  
Hans G. Othmer ◽  
Robert Planque ◽  
Martin CJ Bootsma
Keyword(s):  
Infectious Diseases ◽  
Discrete Time ◽  
Epidemic Model ◽  
Continuous Time ◽  
Reproduction Number ◽  
Compartmental Models ◽  
Initial Speed ◽  
Fixed Duration ◽  
Epidemic Spread ◽  
Special Cases

Surprisingly, the discrete-time version of the general 1927 Kermack-McKendrick epidemic model has, to our knowledge, not been formulated in the literature, and we rectify this omission here. The discrete time version is as general and flexible as its continuous-time counterpart, and contains numerous compartmental models as special cases. In contrast to the continuous time version, the discrete time version of the model is very easy to implement computationally, and thus promises to become a powerful tool for exploring control scenarios for specific infectious diseases. To demonstrate the potential, we investigate numerically how the incidence-peak size depends on model ingredients. We find that, with the same reproduction number and initial speed of epidemic spread, compartmental models systematically predict lower peak sizes than models that use a fixed duration for the latent and infectious periods.

The discrete-time Kermack–McKendrick model: A versatile and computationally attractive framework for modeling epidemics

2021 ◽  
Vol 118 (39) ◽  
pp. e2106332118
Author(s):  
Odo Diekmann ◽  
Hans G. Othmer ◽  
Robert Planqué ◽  
Martin C. J. Bootsma
Keyword(s):  
Discrete Time ◽  
Continuous Time ◽  
Generation Time ◽  
Time Distribution ◽  
Reproduction Number ◽  
Compartmental Models ◽  
Infectious Period ◽  
Initial Growth ◽  
Fixed Duration ◽  
Spread Of Infection

The COVID-19 pandemic has led to numerous mathematical models for the spread of infection, the majority of which are large compartmental models that implicitly constrain the generation-time distribution. On the other hand, the continuous-time Kermack–McKendrick epidemic model of 1927 (KM27) allows an arbitrary generation-time distribution, but it suffers from the drawback that its numerical implementation is rather cumbersome. Here, we introduce a discrete-time version of KM27 that is as general and flexible, and yet is very easy to implement computationally. Thus, it promises to become a very powerful tool for exploring control scenarios for specific infectious diseases such as COVID-19. To demonstrate this potential, we investigate numerically how the incidence-peak size depends on model ingredients. We find that, with the same reproduction number and the same initial growth rate, compartmental models systematically predict lower peak sizes than models in which the latent and the infectious period have fixed duration.

scholarly journals When will the Covid-19 epidemic fade out?

2020 ◽  
Author(s):  
Ilaria Renna
Keyword(s):  
Infectious Diseases ◽  
Discrete Time ◽  
Epidemic Model ◽  
Small World ◽  
Small World Network ◽  
Sirs Model ◽  
Gaussian Fit

AbstractA discrete-time deterministic epidemic model is proposed with the aim of reproducing the behaviour observed in the incidence of real infectious diseases. For this purpose, we analyse a SIRS model under the framework of a small world network formulation. Using this model, we make predictions about the peak of the Covid-19 epidemic in Italy. A Gaussian fit is also performed, to make a similar prediction.

scholarly journals Dynamic Programming and Hamilton–Jacobi–Bellman Equations on Time Scales

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-11
Author(s):  
Yingjun Zhu ◽  
Guangyan Jia
Keyword(s):  
Dynamic Programming ◽  
Dynamic System ◽  
Time Scales ◽  
Discrete Time ◽  
Continuous Time ◽  
Stochastic Dynamic ◽  
Optimality Principle ◽  
Bellman Equations ◽  
Special Cases ◽  
Hamilton Jacobi Bellman

Bellman optimality principle for the stochastic dynamic system on time scales is derived, which includes the continuous time and discrete time as special cases. At the same time, the Hamilton–Jacobi–Bellman (HJB) equation on time scales is obtained. Finally, an example is employed to illustrate our main results.

scholarly journals The Dynamics of Epidemic Model with Two Types of Infectious Diseases and Vertical Transmission

2016 ◽  
Vol 2016 ◽  
pp. 1-16 ◽  
Author(s):  
Raid Kamel Naji ◽  
Reem Mudar Hussien
Keyword(s):  
Infectious Diseases ◽  
Vertical Transmission ◽  
Bifurcation Analysis ◽  
Epidemic Model ◽  
Reproduction Number ◽  
Equilibrium Points ◽  
Host Population ◽  
Local Bifurcation ◽  
Local Bifurcation Analysis ◽  
Different Types

An epidemic model that describes the dynamics of the spread of infectious diseases is proposed. Two different types of infectious diseases that spread through both horizontal and vertical transmission in the host population are considered. The basic reproduction numberR0is determined. The local and the global stability of all possible equilibrium points are achieved. The local bifurcation analysis and Hopf bifurcation analysis for the four-dimensional epidemic model are studied. Numerical simulations are used to confirm our obtained analytical results.

scholarly journals A Stochastic Model for Malaria Transmission Dynamics

2018 ◽  
Vol 2018 ◽  
pp. 1-13 ◽  
Author(s):  
Rachel Waema Mbogo ◽  
Livingstone S. Luboobi ◽  
John W. Odhiambo
Keyword(s):  
Infectious Diseases ◽  
Stochastic Model ◽  
Malaria Transmission ◽  
Stochastic Models ◽  
Continuous Time ◽  
Reproduction Number ◽  
Transmission Dynamics ◽  
Infected Mosquito ◽  
Deterministic Models ◽  
Discrete State

Malaria is one of the three most dangerous infectious diseases worldwide (along with HIV/AIDS and tuberculosis). In this paper we compare the disease dynamics of the deterministic and stochastic models in order to determine the effect of randomness in malaria transmission dynamics. Relationships between the basic reproduction number for malaria transmission dynamics between humans and mosquitoes and the extinction thresholds of corresponding continuous-time Markov chain models are derived under certain assumptions. The stochastic model is formulated using the continuous-time discrete state Galton-Watson branching process (CTDSGWbp). The reproduction number of deterministic models is an essential quantity to predict whether an epidemic will spread or die out. Thresholds for disease extinction from stochastic models contribute crucial knowledge on disease control and elimination and mitigation of infectious diseases. Analytical and numerical results show some significant differences in model predictions between the stochastic and deterministic models. In particular, we find that malaria outbreak is more likely if the disease is introduced by infected mosquitoes as opposed to infected humans. These insights demonstrate the importance of a policy or intervention focusing on controlling the infected mosquito population if the control of malaria is to be realized.

scholarly journals A Nonlinear Observer to Estimate the Effective Reproduction Number of Infectious Diseases

2021 ◽  
Vol 4 (1) ◽  
pp. 39-45
Author(s):  
Agus Hasan
Keyword(s):  
Kalman Filter ◽  
Infectious Diseases ◽  
Extended Kalman Filter ◽  
Discrete Time ◽  
Reproduction Number ◽  
Epidemiological Data ◽  
Sir Model ◽  
Nonlinear Observer ◽  
Linear Matrix ◽  
Matrix Inequality

In this paper, we design a Nonlinear Observer (NLO) to estimate the effective reproduction number (Rt) of infectious diseases. The NLO is designed from a discrete-time augmented Susceptible-Infectious-Removed (SIR) model. The observer gain is obtained by solving a Linear Matrix Inequality (LMI). The method is used to estimate Rt in Jakarta using epidemiological data during COVID-19 pandemic. If the observer gain is tuned properly, this approach produces similar result compared to existing approach such as Extended Kalman filter (EKF).

scholarly journals Stability of the equilibria in a discrete-time sivs epidemic model with standard incidence

Filomat ◽  
2019 ◽  
Vol 33 (8) ◽  
pp. 2393-2408 ◽  
Author(s):  
Mahmood Parsamanesh ◽  
Saeed Mehrshad
Keyword(s):  
Discrete Time ◽  
Basic Reproduction Number ◽  
Epidemic Model ◽  
Total Population ◽  
Reproduction Number ◽  
Flip Bifurcation ◽  
Sis Epidemic Model ◽  
Fold Bifurcation ◽  
The Stability ◽  
Conditions Of Stability

A discrete-time SIS epidemic model with vaccination is presented and studied. The model includes deaths due to disease and the total population size is variable. First, existence and positivity of the solutions are discussed and equilibria of the model and basic reproduction number are obtained. Next, the stability of the equilibria is studied and conditions of stability are obtained in terms of the basic reproduction number R0. Also, occurrence of the fold bifurcation, the flip bifurcation, and the Neimark-Sacker bifurcation is investigated at equilibria. In addition, obtained results are numerically discussed and some diagrams for bifurcations, Lyapunov exponents, and solutions of the model are presented.

scholarly journals Dynamics of infectious diseases: A review of the main biological aspects and their mathematical translation

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Deccy Y. Trejos ◽  
Jose C. Valverde ◽  
Ezio Venturino
Keyword(s):  
Mathematical Model ◽  
Infectious Diseases ◽  
Discrete Time ◽  
Continuous Time ◽  
Control Strategies ◽  
Transmission Dynamics ◽  
Model Parameters ◽  
Simulation Techniques ◽  
Biological Aspects ◽  
Tools And Methods

Abstract In this paper, the main biological aspects of infectious diseases and their mathematical translation for modeling their transmission dynamics are revised. In particular, some heterogeneity factors which could influence the fitting of the model to reality are pointed out. Mathematical tools and methods needed to qualitatively analyze deterministic continuous-time models, formulated by ordinary differential equations, are also introduced, while its discrete-time counterparts are properly referenced. In addition, some simulation techniques to validate a mathematical model and to estimate the model parameters are shown. Finally, we present some control strategies usually considered to prevent epidemic outbreaks and their implementation in the model.

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