Further Notes on the Basic Reproduction Number

Driessche, P. van den; Watmough, James

doi:10.1007/978-3-540-78911-6_6

Cited by 319 publications

(291 citation statements)

References 20 publications

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“…0 E is unstable. Now using the approach of Next Generation Matrix Operator described in [6], [7], [8], [9], [10] and [11], we obtained the reproduction number.…”

Section: Mathematical Analysis Of the Modelmentioning

confidence: 99%

The Dynamics of Poverty, Drug Addiction and Snatching In Sylhet, Bangladesh

Islam¹,

Sakib²,

Shahrear³

et al. 2017

IOSR JM

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“…0 E is unstable. Now using the approach of Next Generation Matrix Operator described in [6], [7], [8], [9], [10] and [11], we obtained the reproduction number.…”

Section: Mathematical Analysis Of the Modelmentioning

confidence: 99%

The Dynamics of Poverty, Drug Addiction and Snatching In Sylhet, Bangladesh

Islam¹,

Sakib²,

Shahrear³

et al. 2017

IOSR JM

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“…To find the basic reproduction number which is very important in the qualitative analysis of the model, we use the method of next generation matrix discussed in [6,20]. For this, we consider the vector function…”

Section: Equilibrium Points and Their Stability Analysismentioning

confidence: 99%

Analysis and dynamically consistent nonstandard discretization for a rabies model in humans and dogs

Chapwanya

Lubuma

Terefe

2015

RACSAM

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Rabies is a fatal disease in dogs as well as in humans. A possible model to represent rabies transmission dynamics in human and dog populations is presented. The next generation matrix operator is used to determine the threshold parameter R 0 , that is the average number of new infective individuals produced by one infective individual introduced into a completely susceptible population. If R 0 < 1, the disease-free equilibrium is globally asymptotically stable, while it is unstable and there exists a locally asymptotically stable endemic equilibrium when R 0 > 1. A nonstandard finite difference scheme that replicates the dynamics of the continuous model is proposed. Numerical tests to support the theoretical analysis are provided.

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“…Furthermore, as the number of equations in the system gets large, it becomes difficult to compute the eigenvalues of A. Therefore, in my analysis, I apply the next generation matrix approach [65,97] (see also Section 6.3 of [204]) to compute R 0 and use Theorem 1 of [204] to establish local stability of the DFE of the ODE studied in this thesis. R 0 is defined as the expected number of secondary infections produced by an index case in a completely susceptible population [5, page 17], [65, page 4], [66], [97].…”

Section: Disease-free and Endemic Equilibria And Their Stabilitymentioning

confidence: 99%

“…On the contrary, if R 0 > 1, then the number of infected individuals will increase with each generation and the disease will spread [204]. Before presenting the next generation method and Theorem 1 of [204], I will first establish some definitions and notations which will be used in the rest of the thesis.…”

Section: Disease-free and Endemic Equilibria And Their Stabilitymentioning

confidence: 99%

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A model for the spread of an SIS epidemic in a human population

Shausan¹

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The aim of this work is to understand how infectious diseases spread through human populations.Attention is given to those diseases which follow the Susceptible-Infective-Susceptible (SIS) pattern.When modelling diseases spread in a human population, it is important to consider the social and spatial structure of the population. Humans usually live in groups such as work places, households, towns and cities. However, an individual's membership of a particular group is not fixed. Rather, it changes over time. This structure determines two paths for a disease to spread through the population.Disease is spread between individuals in the same group by contact between infected and susceptible individuals, and is spread from one group to another by the migration of infected individuals. This type of population structure can be modelled by a metapopulation network. I develop a continuoustime Markov chain (CTMC) model that describes the spread of an SIS epidemic in a metapopulation network.I establish an ordinary differential equations (ODE) and a Gaussian diffusion analogue of the stochastic process by applying, respectively, the theory of differential equation approximations for Markov chains, and the theory of density dependent Markov chains. I use the ODE model to derive analytic expressions for various epidemiological quantities of interest. In particular, I obtain expressions for two threshold quantities; the basic reproduction number, and a quantity called T 0 which is greater than the basic reproduction number. If the basic reproduction number is above 1, then the disease persists and if the basic reproduction number is below 1, then the disease-free equilibrium (DFE) is locally attractive. However, if T 0 is less than or equal to 1, then the DFE is globally attractive. Using the theory of cooperative differential equations and the theory of asymptotically autonomous differential equations, I show the existence and global stability of a unique endemic equilibrium (EE) and the global stability of the DFE in terms of the basic reproduction number, provided that the migration rates of susceptible and infected individuals are equal. Numerical examples indicate that a unique stable EE exists when the condition on the migration rates is relaxed. The approximating Gaussian diffusion shows that the distribution of the population at the endemic level has an approximate multivariate normal distribution whose mean is centered at the endemic equilibrium of the ODE model. iThe results of this study can serve as a basic framework on how to formulate and analyse a more realistic stochastic model for the spread of an SIS epidemic in a metapopulation which accounts for births, deaths, age, risk, and level of infectivities.Assuming that the model presented here accurately describes the spread of an SIS epidemic in a metapopulation, another question which I address is how to control the spread of the disease. Since most control strategies such as vaccination, treatment and public awareness require a high cost for their imp...

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Further Notes on the Basic Reproduction Number

Cited by 319 publications

References 20 publications

The Dynamics of Poverty, Drug Addiction and Snatching In Sylhet, Bangladesh

The Dynamics of Poverty, Drug Addiction and Snatching In Sylhet, Bangladesh

Analysis and dynamically consistent nonstandard discretization for a rabies model in humans and dogs

A model for the spread of an SIS epidemic in a human population

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