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Eigenvalues
And Eigenvectors
Aamir Nazir
Course:- B.Tech 2nd Year (Civil Engineering)
Section:- A
Roll No.:- 120107002
System ID:- 2012018068
Subject:- Mathematics
Subject Code:- MTH-217
Course Code:- CE-107
Teacher Incharge:- Ms. Archana Prasad
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Contents
1. Abstract 3
2. Introduction 3-4
3. Eigenvectors and Eigenvalues of a real matrix 4
a. Characteristic Polynomial 7-8
b. Algebraic Multiplicities 8-9
4. Calculation 9
a. Computing Eigenvalues 9
b. Computing Eigen Vectors 10
5. Applications 10
a. Geology and Glaciology 10-11
b. Vibration Analysis 11-12
c. Tensor of Moment of Inertia 12
d. Stress Tensor 12
e. Basic Reproduction Number. 12
6. Conclusion 13
7. References 13
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Abstract
In abstract linear algebra, these concepts are naturally extended to more general situations, where the set of real scalar factors is complex numbers); the set of Cartesian the continuous functions, the multiplication is replaced by any the derivative from calculus). In such cases, the "vector" in "eigenvector" may be replaced by a more specific term, such as "
This paper is about the various calculations various civil engineering problems like vibrational analysis, stress of inertia, to coop up with real life challenges in the field of engineeri Introduction
An eigenvector of a square matrix is multiplied by , yields a constant multiple of by . That is:
(Because this equation uses post-
The number is called the eigenvalue
In analytic geometry, for example, a three dimensional space starting at the origin. In that case, an eigenvector direction is either preserved or exactly reversed after multiplication by eigenvalue determines how the length of the arrow is changed by the operation, and whether its direction is reversed or not, determined by whether the eigenvalue is negative or posi
The set of all eigenvectors of a matrix (or linear operator), each paired with its corresponding eigenvalue, is called the eigensystem an eigenvector, with the same eigenvalue. An eigenvectors with the same eigenvalue, together with the is any basisfor the set of all vectors that consists of linearly independent eigenvectors of
Not every matrix has an eigenbasis, but every
, these concepts are naturally extended to more general situations, where the set of real scalar factors is replaced by any field of scalars (such as
Cartesian vectors is replaced by any vector space
, the polynomials or the trigonometric series), and matrix multiplication is replaced by any linear operator that maps vectors to vectors (such as
). In such cases, the "vector" in "eigenvector" may be replaced by a more specific term, such as "eigenfunction", "eigenmode", "eigenface", or "eigenstate".
This paper is about the various calculations in the field of eigenvalues and eigen various civil engineering problems like vibrational analysis, stress tensors, tensor with real life challenges in the field of engineering and technology.

square matrix is a non-zero vector that, when the matrix
, yields a constant multiple of , the multiplier being commonly denoted
-multiplication by , it describes a right eigenvector.) eigenvalue of corresponding to .[1]
, for example, a three-element vector may be seen as an arrow in three dimensional space starting at the origin. In that case, an eigenvector is an arrow whose preserved or exactly reversed after multiplication by . The corresponding eigenvalue determines how the length of the arrow is changed by the operation, and whether its direction is reversed or not, determined by whether the eigenvalue is negative or posi
The set of all eigenvectors of a matrix (or linear operator), each paired with its corresponding eigensystem of that matrix.[2] Any multiple of an eigenvector is also an eigenvector, with the same eigenvalue. An eigenspace of a matrix is the set of all eigenvectors with the same eigenvalue, together with the zero vector.[1] An eigenbasis for the set of all vectors that consists of linearly independent eigenvectors of
Not every matrix has an eigenbasis, but every symmetric matrix does.
, these concepts are naturally extended to more general situations,
(such as algebraic or vector space (such as
), and matrix that maps vectors to vectors (such as
). In such cases, the "vector" in "eigenvector" may be replaced by
", or "eigenstate". and eigenvectors and tensor of moment ng and technology.

. that, when the matrix
, the multiplier being commonly denoted right eigenvector.) element vector may be seen as an arrow in three- is an arrow whose
. The corresponding eigenvalue determines how the length of the arrow is changed by the operation, and whether its direction is reversed or not, determined by whether the eigenvalue is negative or positive.
The set of all eigenvectors of a matrix (or linear operator), each paired with its corresponding multiple of an eigenvector is also is the set of all eigenbasis for for the set of all vectors that consists of linearly independent eigenvectors of .
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The terms characteristic vector used for these concepts. The prefix or "unique to", "peculiar to", or "belonging to"
Eigenvalues and eigenvectors have many applications in both pure and applied mathemati
They are used in matrix factorization, in
Eigenvectors and eigenvalues of a real matrix
In many contexts, a vector can be assumed to be a list of real numbers (called elements), written vertically with brackets around the entire list, such as the vectors u and v below. Two vectors are said to be scalar multiples
(also called parallel or collinear) if they have the same number of elements, and if every element of one vector is obtained by multiplying each corresponding element in the other vector by the same number (known as a scaling factor, or a scalar). For example, the vectors are scalar multiples of each other, because each element of element of .
A vector with three elements, like space, relative to some Cartesian coordinate system tip of an arrow whose tail is at the origin of the coordinate system. In this case, the condition
" is parallel to " means that the two arrows lie on the same straight line, and may differ only in length and direction along that l
If we multiply any square matrix result will be another vector characteristic vector
,
characteristic value, and characteristic space used for these concepts. The prefix eigen- is adopted from the German word eigen or "unique to", "peculiar to", or "belonging to"
Eigenvalues and eigenvectors have many applications in both pure and applied mathemati matrix factorization, in quantum mechanics, and in many other areas.
Eigenvectors and eigenvalues of a real matrix
In many contexts, a vector can be assumed to be elements), written brackets around the entire list, below. Two vectors scalar multiples of each other
) if they have the same number of elements, and if every element of one vector is obtained by ltiplying each corresponding element in the other vector by the same number (known as scalar). For example, the and are scalar multiples of each other, because each element of is −20 times the corresponding
A vector with three elements, like or above, may represent a point in three
Cartesian coordinate system. It helps to think of such a vector as the tip of an arrow whose tail is at the origin of the coordinate system. In this case, the condition
" means that the two arrows lie on the same straight line, and may differ only in length and direction along that line. any square matrix with rows and columns by such a vector
, also with rows and one column. That is, characteristic space are also eigen for "self-"
Eigenvalues and eigenvectors have many applications in both pure and applied mathematics. quantum mechanics, and in many other areas.
−20 times the corresponding above, may represent a point in three-dimensional of such a vector as the tip of an arrow whose tail is at the origin of the coordinate system. In this case, the condition
" means that the two arrows lie on the same straight line, and may differ columns by such a vector , the rows and one column. That is,
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is mapped to where, for each index ,
In general, if is not all zeros, the vectors they are parallel (that is, when there is some real number that is an eigenvector of the eigenvalue corresponding to that eigenvector.
In particular, multiplication by a 3×3 matrix magnitude of an arrow in three with eigenvalue , the operation may only change its length, and either keep its direction or flip it (make the arrow point in the exact opposite direction). Specifically, the length of the arrow will increase if , remain the same if
Moreover, the direction will be precisely the same if
If , then the length of the arrow becomes zero.
An example
For the transformation matrix the vector is an eigenvector with eigenvalue 2. Indeed, is not all zeros, the vectors and will not be parallel. When parallel (that is, when there is some real number such that
. In that case, the scale factor is said to be corresponding to that eigenvector.
In particular, multiplication by a 3×3 matrix may change both the direction and the in three-dimensional space. However, if is an eigenvector of
, the operation may only change its length, and either keep its direction it (make the arrow point in the exact opposite direction). Specifically, the length of the
, remain the same if , and decrease it if
Moreover, the direction will be precisely the same if , and flipped if
, then the length of the arrow becomes zero. is an eigenvector with eigenvalue 2. Indeed, will not be parallel. When
) we say is said to be may change both the direction and the is an eigenvector of
, the operation may only change its length, and either keep its direction it (make the arrow point in the exact opposite direction). Specifically, the length of the
, and decrease it if .
, and flipped if .
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On the other hand the vector is not an eigenvector, since and this vector is not a multiple of the original vector
Another example
For the matrix we have
Therefore, the vectors corresponding to the eigenvalues 1, 3, and 2, indicates matrix transposition, in this case turning the row vectors into column vectors.) and this vector is not a multiple of the original vector .
, and are eigenvectors of corresponding to the eigenvalues 1, 3, and 2, respectively. (Here the symbol
, in this case turning the row vectors into column vectors.) are eigenvectors of respectively. (Here the symbol
, in this case turning the row vectors into column vectors.)
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The transformation matrix and (in violet). The points that lie on the line through the origin, parallel to an eigenvector, remain on the line after the transformation. The vectors in red are not eigenvectors, therefore their direction is altered by the transformation.
Characteristic polynomial
The eigenvalue equation for a matrix which is equivalent to where is the identity matrix. It is a fundamental result of linear algebra that an equation has a non-zero solution the matrix is zero. It follows that the eigenvalues of that satisfy the equation
The left-hand side of this equation can be seen (using a polynomial function of the variable matrix. Its coefficients depend on the entries of always . This polynomial is called the above equation is called thecharacteristic equat preserves the direction of vectors parallel to
(in violet). The points that lie on the line through the origin, parallel to an eigenvector, remain on the line after the transformation. The vectors in red are not eigenvectors, therefore their direction is altered by the transformation.
The eigenvalue equation for a matrix is identity matrix. It is a fundamental result of linear algebra that an zero solution if, and only if, the determinant zero. It follows that the eigenvalues of are precisely the real numbers hand side of this equation can be seen (using Leibniz' rule for the determinant) to be function of the variable . The degree of this polynomial is , the order of the depend on the entries of , except that its term of degree
. This polynomial is called the characteristic polynomial of above equation is called thecharacteristic equation (or, less often, the secular equation) of (in blue)
(in violet). The points that lie on the line through the origin, parallel to an eigenvector, remain on the line after the transformation. The vectors in red are not identity matrix. It is a fundamental result of linear algebra that an determinant of are precisely the real numbers for the determinant) to be
, the order of the
, except that its term of degree is of ; and the secular equation) of .
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For example, let be the matrix
The characteristic polynomial of which is
The roots of this polynomial are 2, 1, and 11. Indeed these are the only three eigenvalues of , corresponding to the eigenvectors zero multiples thereof).
Algebraic multiplicities
Let be an eigenvalue of an its multiplicity as a root of the characteristic polynomial, that is, the largest integer that divides evenly that polynomial.
Like the geometric multiplicity
; and the sum of complex eigenvalues are considered,
It can be proved that the geometric multiplicity algebraic multiplicity . Therefore,
If , then
Example
For the matrix: be the matrix
The characteristic polynomial of is
The roots of this polynomial are 2, 1, and 11. Indeed these are the only three eigenvalues eigenvectors and matrix . The algebraic multiplicity of the characteristic polynomial, that is, the largest integer that polynomial.
, the algebraic multiplicity is an integer between 1 over all distinct eigenvalues also cannot exceed complex eigenvalues are considered, is exactly.
It can be proved that the geometric multiplicity of an eigenvalue never exceeds its
. Therefore, is at most. is said to be a semisimple eigenvalue.
The roots of this polynomial are 2, 1, and 11. Indeed these are the only three eigenvalues (or any non- of is of the characteristic polynomial, that is, the largest integer such
, the algebraic multiplicity is an integer between 1 and eigenvalues also cannot exceed . If of an eigenvalue never exceeds its
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the characteristic polynomial of is being the product of the diagonal with a lower triangular matrix.
The roots of this polynomial, and hence the eigenvalues, are 2 and 3. The multiplicity of each eigenvalue is 2; in other words they are both double roots. On the other hand, the geometric multiplicity spanned by the vector multiplicity of the eigenvalue 3 is 1 because its eigenspace is spanned by the total algebraic multiplicity of A, denoted by 4 matrix. The geometric multiplicity which has two distinct eigenvalues.
Calculation
Computing the eigenvalue
The eigenvalues of a matrix polynomial. Explicit algebraic formulas degree is 4 or less. 5.
It turns out that any polynomial with degree some companion matrix of order eigenvalues and eigenvectors cannot be obtained by an explicit algebraic formula, and must therefore be computed by approximate
In theory, the coefficients of the characteristic polynomial can be computed exactly, since they are sums of products of matrix elements; and there are algorithms that can find all the roots of a polynomial of arbitrary degree to any required is not viable in practice because the coefficients would be contaminated by unavoidable round-off errors, and the roots of a polynomial can be an extremely sensitive function of the coefficients being the product of the diagonal with a lower triangular matrix.
The roots of this polynomial, and hence the eigenvalues, are 2 and 3. The of each eigenvalue is 2; in other words they are both double roots. On the other geometric multiplicity of the eigenvalue 2 is only 1, because its eigenspace is
, and is therefore 1 dimensional. Similarly, the geometric multiplicity of the eigenvalue 3 is 1 because its eigenspace is spanned by otal algebraic multiplicity of A, denoted , is 4, which is the most it could be for a 4 by 4 matrix. The geometric multiplicity is 2, which is the smallest it could be for a matrix which has two distinct eigenvalues.
Computing the eigenvalues can be determined by finding the roots of the characteristic algebraic formulas for the roots of a polynomial exist only if the
It turns out that any polynomial with degree is the characteristic polynomial of of order . Therefore, for matrices of order 5 or more, the eigenvalues and eigenvectors cannot be obtained by an explicit algebraic formula, and must therefore be computed by approximate numerical methods.
In theory, the coefficients of the characteristic polynomial can be computed exactly, since they are sums of products of matrix elements; and there are algorithms that can find all the roots of a polynomial of arbitrary degree to any required accuracy.[3] However, this approach is not viable in practice because the coefficients would be contaminated by off errors, and the roots of a polynomial can be an extremely sensitive
,
The roots of this polynomial, and hence the eigenvalues, are 2 and 3. The algebraic of each eigenvalue is 2; in other words they are both double roots. On the other of the eigenvalue 2 is only 1, because its eigenspace is
, and is therefore 1 dimensional. Similarly, the geometric
. Hence,
, is 4, which is the most it could be for a 4 is 2, which is the smallest it could be for a matrix can be determined by finding the roots of the characteristic for the roots of a polynomial exist only if the the characteristic polynomial of
. Therefore, for matrices of order 5 or more, the eigenvalues and eigenvectors cannot be obtained by an explicit algebraic formula, and must
In theory, the coefficients of the characteristic polynomial can be computed exactly, since they are sums of products of matrix elements; and there are algorithms that can find all the
However, this approach is not viable in practice because the coefficients would be contaminated by off errors, and the roots of a polynomial can be an extremely sensitive
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Computing the eigenvectors
Once the (exact) value of an eigenvalue is known, the corresponding eigenvectors can be found by finding non-zero solutions of the eigenvalue equation, that becomes a linear equations with known coefficients. For example, once it is known that 6 is an eigenvalue of the matrix we can find its eigenvectors by solving the equation
This matrix equation is equivalent to two
Both equations reduce to the single linear equation form , for any non eigenvalue .
The matrix above has another eigenvalue corresponding eigenvectors are the non the form , for any non-
Some numeric methods that compute the eigenvalues of a matrix also determine a set of corresponding eigenvectors as a by
Applications
Geology and glaciology
In geology, especially in the study of method by which a mass of information of a clast fabric's constituents' orientation and dip can be summarized in a 3-D space by six numbers. In the field, a geologist may collect such data for hundreds or thousands of clasts e the (exact) value of an eigenvalue is known, the corresponding eigenvectors can be zero solutions of the eigenvalue equation, that becomes a with known coefficients. For example, once it is known that 6 is an we can find its eigenvectors by solving the equation , that is
This matrix equation is equivalent to two linear equations that is reduce to the single linear equation . Therefore, any vector of the
, for any non-zero real number , is an eigenvector of above has another eigenvalue . A similar calculation shows that the tors are the non-zero solutions of , that is, any vector of
-zero real number .
Some numeric methods that compute the eigenvalues of a matrix also determine a set of corresponding eigenvectors as a by-product of the computation. geology, especially in the study of glacial till, eigenvectors and eigenvalues are used as a method by which a mass of information of a clast fabric's constituents' orientation and dip can by six numbers. In the field, a geologist may collect such data clasts in a soil sample. e the (exact) value of an eigenvalue is known, the corresponding eigenvectors can be zero solutions of the eigenvalue equation, that becomes a system of with known coefficients. For example, once it is known that 6 is an
. Therefore, any vector of the
, is an eigenvector of with
. A similar calculation shows that the
, that is, any vector of
Some numeric methods that compute the eigenvalues of a matrix also determine a set of glacial till, eigenvectors and eigenvalues are used as a method by which a mass of information of a clast fabric's constituents' orientation and dip can by six numbers. In the field, a geologist may collect such data
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The output for the orientation tensor is in the three orthogonal (perpendicu
The three eigenvectors are ordered then is the primary orientation/dip of clast, terms of strength. The clast orientation is defined as the direction of the a compass rose of 360°. Dip is measured as the eigenvalue, the modulus of the tensor: this is valued from 0° (no dip) to 90° (vertical). The relative values of by the nature of the sediment's fabric. If
If , the fabric is said to be planar. If be linear.[5]
Vibration analysis
Eigenvalue problems occur naturally in the vibration analysis of mechanical structures with many degrees of freedom. The eigenvalues are used to determine the natural frequencies
(or eigenfrequencies) of vibration, and the eigenvectors determine the shapes of these vibrational modes. In particular, undamped vibration is governed by or that is, acceleration is proportional to position (i.e., we expect
In dimensions, becomes a are then a linear combination of solutions to the where is the eigenvalue and modes are different from the principal compliance modes, which are the eigenvectors of alone. Furthermore, damped vibration
The output for the orientation tensor is in the three orthogonal (perpendicular) axes of space.
The three eigenvectors are ordered by their eigenvalues then is the primary orientation/dip of clast, is the secondary and is the tertiary, in terms of strength. The clast orientation is defined as the direction of the eigenvector, on
360°. Dip is measured as the eigenvalue, the modulus of the tensor: this is valued from 0° (no dip) to 90° (vertical). The relative values of , , and by the nature of the sediment's fabric. If , the fabric is said to be isotropic.
, the fabric is said to be planar. If , the fabric is said to
Eigenvalue problems occur naturally in the vibration analysis of mechanical structures with
The
eigenvalues are used to determine the natural frequencies
) of vibration, and the eigenvectors determine the shapes of these vibrational modes. In particular, undamped vibration is governed by proportional to position (i.e., we expect to be sinusoidal in time). becomes a mass matrix and a stiffness matrix. Admissible solutions are then a linear combination of solutions to the generalized eigenvalue problem is the angular frequency. Note that the principal vibration modes are different from the principal compliance modes, which are the eigenvectors of damped vibration, governed by lar) axes of space.
;[4]
is the tertiary, in eigenvector, on
360°. Dip is measured as the eigenvalue, the modulus of the tensor: this is are dictated c is said to be isotropic.
, the fabric is said to and the eigenvectors determine the shapes of these vibrational modes. In particular, to be sinusoidal in time).
. Admissible solutions generalized eigenvalue problem
. Note that the principal vibration modes are different from the principal compliance modes, which are the eigenvectors of
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leads to what is called a so-called
This can be reduced to a generalized eigenvalue problem by of solving a larger system.
The orthogonality properties of the eigenvectors allow decoupling of the differential equations so that the system can be represented as linear summation of the eigenvectors. The eigenvalue problem of complex structures is often solved using neatly generalize the solution to scalar
Tensor of moment of inertia
In mechanics, the eigenvectors of the a rigid body. The tensor of moment of rotation of a rigid body around its
Stress tensor
In solid mechanics, the stress tensor is symmetric and so can be decomposed into a diagonal tensor with the eigenvalues on the diagonal and eigenvectors as a basis. Because it is diagonal, in this orientation, the stress tensor has no does have are the principal components.
Basic reproduction number
The basic reproduction number ( diseases spread. If one infectious person is put into a population of completely susceptible people, then is the average number of people that one typical infectious infect. The generation time of an infection is the time, infected to the next person becoming infected. In a heterogeneous population, the next generation matrix defines how many people in the population will become time has passed. is then the largest eigenvalue of the next generation matrix.
Conclusion
If we consider matrix as a transformation then in simple terms eigenvalue is the strength of that transformation in a particular dir called quadratic eigenvalue problem,
This can be reduced to a generalized eigenvalue problem by clever use of algebra
The orthogonality properties of the eigenvectors allow decoupling of the differential equations so that the system can be represented as linear summation of the eigenvectors. The blem of complex structures is often solved using finite element analysis neatly generalize the solution to scalar-valued vibration problems. mechanics, the eigenvectors of the moment of inertia tensor define the principal axes of moment of inertia is a key quantity required to determine the rotation of a rigid body around its center of mass. tensor is symmetric and so can be decomposed into tensor with the eigenvalues on the diagonal and eigenvectors as a basis. Because it is diagonal, in this orientation, the stress tensor has no shear components; the components it does have are the principal components. he basic reproduction number ( ) is a fundamental number in the study of how infectious diseases spread. If one infectious person is put into a population of completely susceptible is the average number of people that one typical infectious infect. The generation time of an infection is the time, , from one person becoming infected to the next person becoming infected. In a heterogeneous population, the next generation matrix defines how many people in the population will become is then the largest eigenvalue of the next generation matrix.
If we consider matrix as a transformation then in simple terms eigenvalue is the strength of that transformation in a particular direction known as eigenvector. Eigenvectors and clever use of algebra at the cost
The orthogonality properties of the eigenvectors allow decoupling of the differential equations so that the system can be represented as linear summation of the eigenvectors. The finite element analysis, but principal axes of is a key quantity required to determine the tensor is symmetric and so can be decomposed into tensor with the eigenvalues on the diagonal and eigenvectors as a basis. Because it components; the components it
) is a fundamental number in the study of how infectious diseases spread. If one infectious person is put into a population of completely susceptible is the average number of people that one typical infectious person will
, from one person becoming infected to the next person becoming infected. In a heterogeneous population, the next infected after is then the largest eigenvalue of the next generation matrix.[6] [7]
If we consider matrix as a transformation then in simple terms eigenvalue is the strength of
Eigenvectors and
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eigenvalues are used widely in science and engineering. They have many applications, particularly in physics. Consider rigid physical bodies. Rigid physical bodies have a preferred direction of rotation, about which they can rotate freely. For example, if someone were to throw a football it would rotate around its axis while flying prettily through the air. If someone were to hit the ball in the air, the ball would be likely to flop in a less simple way.
Although this may seem like common sense, even rigid bodies with a more complicated shape will have preferred directions of rotation. These are called axes of inertia, and they are calculated by finding the eigenvectors of a matrix called the inertia tensor. The eigenvalues, also important, are called moments of inertia. In civil engineering there is a necessity of vibrational analysis of materials, effects of tensile, compressive and shear stresses, which is made easy by the knowledge of eigenvalues and eigenvectors.
Refrences
1. Wolfram Research, Inc. (2010) Eigenvector. Accessed on 2010-01-29.
2. William H. Press, Saul A. Teukolsky, William T. Vetterling, Brian P. Flannery
(2007), Numerical Recipes: The Art of Scientific Computing, Chapter 11: Eigensystems., pages=563–597. Third edition, Cambridge University Press. ISBN 9780521880688
3. Trefethen, Lloyd N.; Bau, David (1997), Numerical Linear Algebra, SIAM
4. Stereo32 software
5. Benn, D.; Evans, D. (2004), A Practical Guide to the study of Glacial Sediments, London:
Arnold, pp. 103–107
6. Diekmann O, Heesterbeek JAP, Metz JAJ (1990), "On the definition and the computation of the basic reproduction ratio R0 in models for infectious diseases in heterogeneous populations",Journal of Mathematical Biology 28 (4): 365–
382, doi:10.1007/BF00178324,PMID 2117040
7. Odo Diekmann and J. A. P. Heesterbeek (2000), Mathematical epidemiology of infectious diseases, Wiley series in mathematical and computational biology, West Sussex, England:
John Wiley & Sons
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