scholarly journals A Review of Matrix SIR Arino Epidemic Models

Mathematics ◽  
2021 ◽  
Vol 9 (13) ◽  
pp. 1513
Author(s):  
Florin Avram ◽  
Rim Adenane ◽  
David I. Ketcheson
Keyword(s):  
Simple Formula ◽  
Reproduction Number ◽  
Integral Formula ◽  
First Integral ◽  
Compartmental Models ◽  
Probabilistic Interpretation ◽  
Mathematical Epidemiology ◽  
Type Distribution ◽  
Scalar Quantity ◽  
Approximate Control

Many of the models used nowadays in mathematical epidemiology, in particular in COVID-19 research, belong to a certain subclass of compartmental models whose classes may be divided into three “(x,y,z)” groups, which we will call respectively “susceptible/entrance, diseased, and output” (in the classic SIR case, there is only one class of each type). Roughly, the ODE dynamics of these models contains only linear terms, with the exception of products between x and y terms. It has long been noticed that the reproduction number R has a very simple Formula in terms of the matrices which define the model, and an explicit first integral Formula is also available. These results can be traced back at least to Arino, Brauer, van den Driessche, Watmough, and Wu (2007) and to Feng (2007), respectively, and may be viewed as the “basic laws of SIR-type epidemics”. However, many papers continue to reprove them in particular instances. This motivated us to redraw attention to these basic laws and provide a self-contained reference of related formulas for (x,y,z) models. For the case of one susceptible class, we propose to use the name SIR-PH, due to a simple probabilistic interpretation as SIR models where the exponential infection time has been replaced by a PH-type distribution. Note that to each SIR-PH model, one may associate a scalar quantity Y(t) which satisfies “classic SIR relations”, which may be useful to obtain approximate control policies.

scholarly journals A random forest model for forecasting regional COVID-19 cases utilizing reproduction number estimates and demographic data

2021 ◽  
Author(s):  
Joseph Galasso ◽  
Duy M. Cao ◽  
Robert Hochberg
Keyword(s):  
Random Forest ◽  
Human Error ◽  
Compartmental Model ◽  
Reproduction Number ◽  
Local Level ◽  
Demographic Data ◽  
Compartmental Models ◽  
Random Forest Model ◽  
Public Health Response ◽  
Forest Model

During the COVID-19 pandemic, predicting case spikes at the local level is important for a precise, targeted public health response and is generally done with compartmental models. The performance of compartmental models is highly dependent on the accuracy of their assumptions about disease dynamics within a population; thus, such models are susceptible to human error, unexpected events, or unknown characteristics of a novel infectious agent like COVID-19. We present a relatively non-parametric random forest model that forecasts the number of COVID-19 cases at the U.S. county level. Its most prioritized training features are derived from easily accessible, standard epidemiological data (i.e., regional test positivity rate) and the effective reproduction number R(t) from compartmental models. A novel input training feature is case projections generated by aligning estimated effective reproduction number from a compartmental model with real time testing data until maximally correlated, helping our model fit better to the epidemic's trajectory ascertained by traditional models. Any poor reliability of R(t) due to flaws in the compartmental model are mitigated with dynamic population mobility and prevalence and mortality of non-COVID-19 diseases to gauge population disease susceptibility. The model was used to generate forecasts for 1, 2, 3, and 4 weeks into the future for each reference week within 11/01/2020 - 01/10/2021 for 3068 counties. Over this time period, it maintained a mean absolute error (MAE) of less than 300 weekly cases/100,000 and consistently outperformed or performed comparably with gold-standard compartmental models. Furthermore, it holds great potential in ensemble modeling due to its potential for a more expansive training feature set while maintaining good performance and limited resource utilization.

scholarly journals Modeling the role of clusters and diffusion in the evolution of COVID-19 infections during lock-down

2020 ◽  
Author(s):  
W. J. T. Bos ◽  
J.-P. Bertoglio ◽  
L. Gostiaux
Keyword(s):  
Compartmental Model ◽  
Reproduction Number ◽  
Sir Model ◽  
Compartmental Models ◽  
European Countries ◽  
Epidemic Spreading ◽  
Decay Exponent ◽  
Local Reproduction ◽  
And Diffusion

Epidemics such as the spreading of the SARS-CoV-2 virus are highly non linear, and therefore difficult to predict. In the present pandemic as time evolves, it appears more and more clearly that a clustered dynamics is a key element of description. This means that the disease rapidly evolves within spatially localized networks, that diffuse and eventually create new clusters. We improve upon the simplest possible compartmental model, the SIR model, by adding an additional compartment associated with the clustered individuals. This sophistication is compatible with more advanced compartmental models and allows, at the lowest level of complexity, to leverage the well-mixedness assumption. The so-obtained SBIR model takes into account the effect of inhomogeneity on epidemic spreading, and compares satisfactorily with results on the pandemic propagation in a number of European countries, during and immediately after lock-down. Especially, the decay exponent of the number of new cases after the first peak of the epidemic is captured without the need to vary the coefficients of the model with time. We show that this decay exponent is directly determined by the diffusion of the ensemble of clustered individuals and can be related to a global reproduction number, that overrides the classical, local reproduction number.

scholarly journals A stochastic SEIHR model for COVID-19 data fluctuations

2021 ◽  
Author(s):  
Ruiwu Niu ◽  
Yin-Chi Chan ◽  
Eric W. M. Wong ◽  
Michaël Antonie van Wyk ◽  
Guanrong Chen
Keyword(s):  
Basic Reproduction Number ◽  
Stochastic Stability ◽  
Reproduction Number ◽  
General Trend ◽  
Sufficient Conditions ◽  
Compartmental Models ◽  
Threshold Behavior ◽  
Random Fluctuations ◽  
Parameter Values ◽  
Disease Free

Abstract Although deterministic compartmental models are useful for predicting the general trend of a disease's spread, they are unable to describe the random daily fluctuations in the number of new infections and hospitalizations, which is crucial in determining the necessary healthcare capacity for a specified level of risk. In this paper, we propose a stochastic SEIHR (sSEIHR) model to describe such random fluctuations and provide sufficient conditions for stochastic stability of the disease-free equilibrium, based on the basic reproduction number that we estimated. Our extensive numerical results demonstrate strong threshold behavior near the estimated basic reproduction number, suggesting that the necessary conditions for stochastic stability are close to the sufficient conditions derived. Furthermore, we found that increasing the noise level slightly reduces the final proportion of infected individuals. In addition, we analyze COVID-19 data from various regions worldwide and demonstrate that by changing only a few parameter values, our sSEIHR model can accurately describe both the general trend and the random fluctuations in the number of daily new cases in each region, allowing governments and hospitals to make more accurate caseload predictions using fewer compartments and parameters than other comparable stochastic compartmental models.

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 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.

scholarly journals DINAMIKA LOKAL MODEL EPIDEMI SVIR DENGAN IMIGRASI PADA KOMPARTEMEN VAKSINASI

2020 ◽  
Vol 14 (2) ◽  
pp. 297-304
Author(s):  
Joko Harianto ◽  
Titik Suparwati ◽  
Inda Puspita Sari
Keyword(s):  
Equilibrium Point ◽  
Basic Reproduction Number ◽  
Reproduction Number ◽  
Equilibrium Points ◽  
Asymptotically Stable ◽  
Mathematical Epidemiology ◽  
Unique Equilibrium ◽  
The Stability ◽  
Disease Free ◽  
The Basic Reproduction Number

Abstrak Artikel ini termasuk dalam ruang lingkup matematika epidemiologi. Tujuan ditulisnya artikel ini untuk mendeskripsikan dinamika lokal penyebaran suatu penyakit dengan beberapa asumsi yang diberikan. Dalam pembahasan, dianalisis titik ekuilibrium model epidemi SVIR dengan adanya imigrasi pada kompartemen vaksinasi. Dengan langkah pertama, model SVIR diformulasikan, kemudian titik ekuilibriumnya ditentukan, selanjutnya, bilangan reproduksi dasar ditentukan. Pada akhirnya, kestabilan titik ekuilibirum yang bergantung pada bilangan reproduksi dasar ditentukan secara eksplisit. Hasilnya adalah jika bilangan reproduksi dasar kurang dari satu maka terdapat satu titik ekuilbirum dan titik ekuilbrium tersebut stabil asimtotik lokal. Hal ini berarti bahwa dalam kondisi tersebut penyakit akan cenderung menghilang dalam populasi. Sebaliknya, jika bilangan reproduksi dasar lebih dari satu, maka terdapat dua titik ekuilibrium. Dalam kondisi ini, titik ekuilibrium endemik stabil asimtotik lokal dan titik ekuilibrium bebas penyakit tidak stabil. Hal ini berarti bahwa dalam kondisi tersebut penyakit akan tetap ada dalam populasi. Kata Kunci : Model SVIR, Stabil Asimtotik Lokal Abstract This article is included in the scope of mathematical epidemiology. The purpose of this article is to describe the dynamics of the spread of disease with some assumptions given. In this paper, we present an epidemic SVIR model with the presence of immigration in the vaccine compartment. First, we formulate the SVIR model, then the equilibrium point is determined, furthermore, the basic reproduction number is determined. In the end, the stability of the equilibrium point is determined depending on the number of basic reproduction. The result is that if the basic reproduction number is less than one then there is a unique equilibrium point and the equilibrium point is locally asymptotically stable. This means that in those conditions the disease will tend to disappear in the population. Conversely, if the basic reproduction number is more than one, then there are two equilibrium points. The endemic equilibrium point is locally asymptotically stable and the disease-free equilibrium point is unstable. This means that in those conditions the disease will remain in the population. Keywords: SVIR Model, Locally Asymptotically stable.

scholarly journals An Introduction to The Basic Reproduction Number in Mathematical Epidemiology

2018 ◽  
Vol 62 ◽  
pp. 123-138 ◽  
Author(s):  
Antoine Perasso
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Basic Reproduction Number ◽  
Matrix Method ◽  
Reproduction Number ◽  
Mathematical Epidemiology ◽  
Pde Model ◽  
Mathematical Definition ◽  
Next Generation Matrix ◽  
The Basic Reproduction Number

This article introduces the notion of basic reproduction number R0 in mathematical epi-demiology. After an historic reminder describing the steps leading to the statement of its mathematical definition, we explain the next-generation matrix method allowing its calculation in the case of epidemic models described by ordinary differential equations (ODEs). The article then focuses, through four ODEs examples and an infection load structured PDE model, on the usefulness of the R0 to address biological as well mathematical issues.

scholarly journals TW-SIR: time-window based SIR for COVID-19 forecasts

2020 ◽  
Vol 10 (1) ◽  
Author(s):  
Zhifang Liao ◽  
Peng Lan ◽  
Zhining Liao ◽  
Yan Zhang ◽  
Shengzong Liu
Keyword(s):  
Prediction Model ◽  
Reproduction Number ◽  
Time Window ◽  
Machine Learning Method ◽  
Exponential Growth Rate ◽  
Learning Method ◽  
Mathematical Epidemiology ◽  
Dynamic Data ◽  
Time Changes ◽  
The Basic Reproduction Number

AbstractSince the outbreak of COVID-19, many COVID-19 research studies have proposed different models for predicting the trend of COVID-19. Among them, the prediction model based on mathematical epidemiology (SIR) is the most widely used, but most of these models are adapted in special situations based on various assumptions. In this study, a general adapted time-window based SIR prediction model is proposed, which is characterized by introducing a time window mechanism for dynamic data analysis and using machine learning method predicts the basic reproduction number and the exponential growth rate of the epidemic. We analyzed COVID-19 data from February to July 2020 in seven countries–––China, South Korea, Italy, Spain, Brazil, Germany and France, and the numerical results showed that the framework can effectively measure the real-time changes of the parameters during the epidemic, and error rate of predicting the number of COVID-19 infections in a single day is within 5%.

scholarly journals Are college campuses superspreaders? A data-driven modeling study

2020 ◽  
Author(s):  
Hannah Lu ◽  
Cortney Weintz ◽  
Joseph Pace ◽  
Dhiraj Indana ◽  
Kevin Linka ◽  
...  
Keyword(s):  
Bayesian Learning ◽  
Reproduction Number ◽  
Case Reports ◽  
College Campuses ◽  
Data Driven ◽  
Mathematical Epidemiology ◽  
Ongoing Debate ◽  
Risk Period ◽  
Order Of Magnitude ◽  
Data Driven Modeling

ABSTRACTThe COVID-19 pandemic continues to present enormous challenges for colleges and universities and strategies for save reopening remain a topic of ongoing debate. Many institutions that reopened cautiously in the fall experienced a massive wave of infections and colleges were soon declared as the new hotspots of the pandemic. However, the precise effects of college outbreaks on their immediate neighborhood remain largely unknown. Here we show that the first two weeks of instruction present a high-risk period for campus outbreaks and that these outbreaks tend to spread into the neighboring communities. By integrating a classical mathematical epidemiology model and Bayesian learning, we learned the dynamic reproduction number for 30 colleges from their daily case reports. Of these 30 institutions, 14 displayed a spike of infections within the first two weeks of class, with peak seven-day incidences well above 1,000 per 100,000, an order of magnitude larger than the nation-wide peaks of 70 and 150 during the first and second waves of the pandemic. While most colleges were able to rapidly reduce the number of new infections, many failed to control the spread of the virus beyond their own campus: Within only two weeks, 17 campus outbreaks translated directly into peaks of infection within their home counties. These findings suggests that college campuses are at risk to develop an extreme incidence of COVID-19 and become superspreaders for neighboring communities. We anticipate that tight test-trace-quarantine strategies, flexible transition to online instruction, and–most importantly–compliance with local regulations will be critical to ensure a safe campus reopening after the winter break.

scholarly journals Monitoring and Analysis of COVID-19 Pandemic: The Need for an Empirical Approach

2021 ◽  
Vol 9 ◽  
Author(s):  
Martí Català ◽  
Miquel Marchena ◽  
David Conesa ◽  
Pablo Palacios ◽  
Tomas Urdiales ◽  
...  
Keyword(s):  
Reproduction Number ◽  
Regional Level ◽  
Empirical Approach ◽  
European Countries ◽  
New Disease ◽  
Actual Situation ◽  
Mathematical Epidemiology ◽  
The Real ◽  
Real Risk ◽  
Current Risk

The current worldwide pandemic produced by coronavirus disease 2019 (COVID-19) has changed the paradigm of mathematical epidemiology due to the high number of unknowns of this new disease. Thus, the empirical approach has emerged as a robust tool to analyze the actual situation carried by the countries and also allows us to predict the incoming scenarios. In this paper, we propose three empirical indexes to estimate the state of the pandemic. These indexes quantify both the propagation and the number of estimated cases, allowing us to accurately determine the real risk of a country. We have calculated these indexes' evolution for several European countries. Risk diagrams are introduced as a tool to visualize the evolution of a country and evaluate its current risk as a function of the number of contagious individuals and the empiric reproduction number. Risk diagrams at the regional level are useful to observe heterogeneity on COVID-19 penetration and spreading in some countries, which is essential during deconfinement processes. During the pandemic, there have been significant differences seen in countries reporting case criterion and detection capacity. Therefore, we have introduced estimations about the real number of infectious cases that allows us to have a broader view and to better estimate the risk. These diagrams and indexes have been successfully used for the monitoring of European countries and regions during the COVID-19 pandemic.

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