The basic reproduction number

2012 ◽  
Author(s):  
Odo Diekmann ◽  
Hans Heesterbeek ◽  
Tom Britton
Keyword(s):  
Infectious Disease ◽  
Basic Reproduction Number ◽  
Reproduction Number ◽  
Infected Individual ◽  
Vital Role ◽  
Point Of View ◽  
Epidemiological Models ◽  
Disease Epidemiology ◽  
Infectious Disease Epidemiology ◽  
The Basic Reproduction Number

The basic reproduction number (or ratio) R₀ is arguably the most important quantity in infectious disease epidemiology. It is among the quantities most urgently estimated for infectious diseases in outbreak situations, and its value provides insight when designing control interventions for established infections. From a theoretical point of view R₀ plays a vital role in the analysis of, and consequent insight from, infectious disease models. There is hardly a paper on dynamic epidemiological models in the literature where R₀ does not play a role. R₀ is defined as the average number of new cases of an infection caused by one typical infected individual, in a population consisting of susceptibles only. This chapter shows how R₀ can be characterized mathematically and provides detailed examples of its calculation in terms of parameters of epidemiological models, culminating in a set of algorithms (or “recipes”) for the calculation for compartmental epidemic systems.

scholarly journals Threshold parameters for a model of epidemic spread among households and workplaces

2009 ◽  
Vol 6 (40) ◽  
pp. 979-987 ◽  
Author(s):  
L. Pellis ◽  
N. M. Ferguson ◽  
C. Fraser
Keyword(s):  
Infectious Disease ◽  
Basic Reproduction Number ◽  
Reproduction Number ◽  
Model Parameters ◽  
Epidemic Spread ◽  
Threshold Parameter ◽  
Disease Epidemiology ◽  
Infectious Disease Epidemiology ◽  
Complex Models ◽  
The Basic Reproduction Number

The basic reproduction number R 0 is one of the most important concepts in modern infectious disease epidemiology. However, for more realistic and more complex models than those assuming homogeneous mixing in the population, other threshold quantities can be defined that are sometimes more useful and easily derived in terms of model parameters. In this paper, we present a model for the spread of a permanently immunizing infection in a population socially structured into households and workplaces/schools, and we propose and discuss a new household-to-household reproduction number R H for it. We show how R H overcomes some of the limitations of a previously proposed threshold parameter, and we highlight its relationship with the effort required to control an epidemic when interventions are targeted at randomly selected households.

scholarly journals The construction of next-generation matrices for compartmental epidemic models

2009 ◽  
Vol 7 (47) ◽  
pp. 873-885 ◽  
Author(s):  
O. Diekmann ◽  
J. A. P. Heesterbeek ◽  
M. G. Roberts
Keyword(s):  
Basic Reproduction Number ◽  
Reproduction Number ◽  
Dominant Eigenvalue ◽  
Next Generation ◽  
Complete Proof ◽  
Malthusian Parameter ◽  
Disease Epidemiology ◽  
Infectious Disease Epidemiology ◽  
Basic Model ◽  
The Basic Reproduction Number

The basic reproduction number ℛ 0 is arguably the most important quantity in infectious disease epidemiology. The next-generation matrix (NGM) is the natural basis for the definition and calculation of ℛ 0 where finitely many different categories of individuals are recognized. We clear up confusion that has been around in the literature concerning the construction of this matrix, specifically for the most frequently used so-called compartmental models. We present a detailed easy recipe for the construction of the NGM from basic ingredients derived directly from the specifications of the model. We show that two related matrices exist which we define to be the NGM with large domain and the NGM with small domain . The three matrices together reflect the range of possibilities encountered in the literature for the characterization of ℛ 0 . We show how they are connected and how their construction follows from the basic model ingredients, and establish that they have the same non-zero eigenvalues, the largest of which is the basic reproduction number ℛ 0 . Although we present formal recipes based on linear algebra, we encourage the construction of the NGM by way of direct epidemiological reasoning, using the clear interpretation of the elements of the NGM and of the model ingredients. We present a selection of examples as a practical guide to our methods. In the appendix we present an elementary but complete proof that ℛ 0 defined as the dominant eigenvalue of the NGM for compartmental systems and the Malthusian parameter r , the real-time exponential growth rate in the early phase of an outbreak, are connected by the properties that ℛ 0 > 1 if and only if r > 0, and ℛ 0 = 1 if and only if r = 0.

scholarly journals The Basic Reproduction Number of Plant Pathogens: Matrix Approaches to Complex Dynamics

2008 ◽  
Vol 98 (2) ◽  
pp. 239-249 ◽  
Author(s):  
F. van den Bosch ◽  
N. McRoberts ◽  
F. van den Berg ◽  
L. V. Madden
Keyword(s):  
Basic Reproduction Number ◽  
Plant Pathogens ◽  
Complex Dynamics ◽  
Reproduction Number ◽  
Infected Individual ◽  
Host Population ◽  
Matrix Population Models ◽  
Wide Range ◽  
Simple Systems ◽  
The Basic Reproduction Number

The basic reproduction number, R0, is defined as the total number of infections arising from one newly infected individual introduced into a healthy (disease-free) host population. R0 is widely used in ecology and animal and human epidemiology, but has received far less attention in the plant pathology literature. Although the calculation of R0 in simple systems is straightforward, the calculation in complex situations is challenging. A very generic framework exists in the mathematical and biomathematical literature, which is difficult to interpret and apply in specific cases. In this paper we describe a special case of this general framework involving the use of matrix population models. Leading by example, we explain the existing mathematical literature on this subject in such a way that plant pathologists can apply the method for a wide range of pathosystems.

scholarly journals The long-term dynamics of tuberculosis and other diseases with long serial intervals: implications of and for changing reproduction numbers

1998 ◽  
Vol 121 (2) ◽  
pp. 309-324 ◽  
Author(s):  
E. VYNNYCKY ◽  
P. E. M. FINE
Keyword(s):  
Infectious Disease ◽  
Basic Reproduction Number ◽  
Reproduction Number ◽  
England And Wales ◽  
Transmission Potential ◽  
Reproduction Numbers ◽  
Serial Interval ◽  
Net Reproduction ◽  
Epidemiological Parameters ◽  
The Basic Reproduction Number

The net and basic reproduction numbers are among the most widely-applied concepts in infectious disease epidemiology. A net reproduction number (the average number of secondary infectious cases resulting from each case in a given population) of above 1 is conventionally associated with an increase in incidence; the basic reproduction number (defined analogously for a ‘totally susceptible’ population) provides a standard measure of the ‘transmission potential’ of an infection. Using a model of the epidemiology of tuberculosis in England and Wales since 1900, we demonstrate that these measures are difficult to apply if disease can follow reinfection, and that they lose their conventional interpretations if important epidemiological parameters, such as the rate of contact between individuals, change over the time interval between successive cases in a chain of transmission (the serial interval).The net reproduction number for tuberculosis in England and Wales appears to have been approximately 1 from 1900 until 1950, despite concurrent declines in morbidity and mortality rates, and it declined rapidly in the second half of this century. The basic reproduction number declined from about 3 in 1900, reached 2 by 1950, and first fell below 1 in about 1960. Reductions in effective contact between individuals over this period, measured in terms of the average number of individuals to whom each case could transmit the infection, meant that the conventional basic reproduction number measure (which does not consider subsequent changes in epidemiological parameters) for a given year failed to reflect the ‘actual transmission potential’ of the infection. This latter property is better described by a variant of the conventional measure which takes secular trends in contact into account. These results are relevant for the interpretation of trends in any infectious disease for which epidemiological parameters change over time periods comparable to the infectious period, incubation period or serial interval.

scholarly journals Graphs with specified degree distributions, simple epidemics, and local vaccination strategies

2007 ◽  
Vol 39 (04) ◽  
pp. 922-948 ◽  
Author(s):  
Tom Britton ◽  
Svante Janson ◽  
Anders Martin-Löf
Keyword(s):  
Infectious Disease ◽  
Social Structure ◽  
Basic Reproduction Number ◽  
Degree Distribution ◽  
Reproduction Number ◽  
Final Size ◽  
Vaccination Strategies ◽  
The Social ◽  
Major Outbreak ◽  
The Basic Reproduction Number

Consider a random graph, having a prespecified degree distribution F, but other than that being uniformly distributed, describing the social structure (friendship) in a large community. Suppose that one individual in the community is externally infected by an infectious disease and that the disease has its course by assuming that infected individuals infect their not yet infected friends independently with probability p. For this situation, we determine the values of R 0, the basic reproduction number, and τ0, the asymptotic final size in the case of a major outbreak. Furthermore, we examine some different local vaccination strategies, where individuals are chosen randomly and vaccinated, or friends of the selected individuals are vaccinated, prior to the introduction of the disease. For the studied vaccination strategies, we determine R v , the reproduction number, and τ v , the asymptotic final proportion infected in the case of a major outbreak, after vaccinating a fraction v.

scholarly journals Estimation of Hospital Potential Capacity and Basic Reproduction Number

2014 ◽  
Vol 2014 ◽  
pp. 1-5
Author(s):  
Fei Wang ◽  
Linhua Wang ◽  
Peng Wang
Keyword(s):  
Infectious Disease ◽  
Decision Making ◽  
Resource Allocation ◽  
Basic Reproduction Number ◽  
Reproduction Number ◽  
Medical Resource ◽  
Population Dynamic Model ◽  
Potential Capacity ◽  
And Control ◽  
The Basic Reproduction Number

In order to reflect the population covered by institutional medical services, the concept of hospital potential capacity is proposed and a formula for its estimation is developed based on a population dynamic model. Using the collected data on hospital outpatient and inpatient services and the demographical information on Chongqing as an example, the demand for medical resource allocation in Chongqing is dynamically estimated. Moreover, the proposed formula is also useful in the estimation of the basic reproduction number in epidemiology. The results can be contributed to the improvement of decision-making in the allocation of medical resources and the evaluation of the interventions and control efforts of the infectious disease.

scholarly journals ANALISIS MODEL MATEMATIKA PENYEBARAN PENYAKIT H1N1 DENGAN TINGKAT KEJADIAN TERSATURASI

2018 ◽  
Vol 1 (April) ◽  
pp. 29-33
Author(s):  
M. Ivan Ariful Fathoni
Keyword(s):  
Mathematical Model ◽  
Equilibrium Point ◽  
Basic Reproduction Number ◽  
Reproduction Number ◽  
Equilibrium Points ◽  
Infected Individual ◽  
Swine Flu ◽  
Occurrence Rate ◽  
Model Formation ◽  
The Basic Reproduction Number

Swine flu is an acute respiratory infection that attacks the body's organs especially the lungs. The disease is caused by Influenza Virus Type A, type H1N1. In this article constructed mathematical model of the spread of H1N1 disease. Mathematical model that created the model Susceptible, Exposed, Infective, and Treatment. The existence of behavior change and influence of infected individual density become the reason of model formation with saturation occurrence rate. From the dynamic analysis, the model has two equilibrium points, that is, a stable equilibrium free equilibrium point when the basic reproduction number is less or equal to one, and an endemic equilibrium point that exists and is stable when the basic reproduction number is greater than one. Finally, the results of the analysis prove the control of the spread of disease into a disease-free state.   Flu babi adalah infeksi saluran pernapasan akut yang menyerang organ tubuh terutama paru-paru. Penyakit ini disebabkan oleh Virus Influenza tipe A, jenis H1N1. Pada artikel ini dikonstruksi model matematika penyebaran penyakit H1N1. Model matematika yang dibuat yaitu model Susceptible, Exposed, Infective, dan Treatment. Adanya perubahan perilaku dan pengaruh kepadatan individu terinfeksi menjadi alasan pembentukan model dengan tingkat kejadian tersaturasi. Dari hasil analisis dinamik, model memiliki dua titik kesetimbangan, yaitu titik kesetimbangan bebas penyakit yang bersifat stabil saat bilangan reproduksi dasar bernilai lebih kecil atau sama dengan satu, dan titik kesetimbangan endemi yang eksis dan bersifat stabil saat bilangan reproduksi dasar bernilai lebih besar dari satu. Pada akhirnya, hasil analisis membuktikan adanya kontrol penyebaran penyakit menjadi keadaan bebas penyakit.

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