A simplified fractional SEIR epidemic model and unique inversion of
the fractional order
YI ZHANG, GONGSHENG LI*
School of Mathematics and Statistics
Shandong University of Technology
No. 266 Xincun Road, Zhangdian District, Zibo Shandong 255000
CHINA
Abstract: - A simplified linear time-fractional SEIR epidemic system is set forth, and an inverse problem of
determining the fractional order is discussed by using the measurement at one given time. By the Laplace
transform the solution to the forward problem is obtained, by which the inverse problem is transformed to a
nonlinear algebraic equation. By choosing suitable model parameters and the measured time, the nonlinear
equation has a unique solution by the monotonicity of the Mittag-Lellfer function. Theoretical testification is
presented to demonstrate the unique solvability of the inverse problem.
Key-Words: - Fractional SEIR model; Laplace transform; Mittag-Lellfer function; inversion of fractional order;
monotonicity; uniqueness
Received: May 22, 2021. Revised: January 25, 2022. Accepted: February 20, 2022. Published: March 23, 2022.
1 Introduction
Coronavirus disease 2019 (COVID-19), a
pneumonia epidemic caused by a new type of
coronavirus 2 (SARS-CoV-2), has become a global
pandemic of the highest priority in the world [1,
2]. It is important to study its origin, its survival
variation characteristics, the spread and infection
laws, as well as the preventive vaccine. However, it
is still of scientific significance for the COVID-19
to give mathematical modelling and perform
optimal prevention and control strategies. There are
quite a lot of mathematical models in describing the
COVID-19 pandemic according to the spreading
rules of human-to-human, most of them are based
on the traditional models of SI, SIR and SEIR, etc.
[3]. We refer to some recent literatures on
mathematical modelling of the COVID-19, such as
the integer-order models [4-6], and the fractional
spreading models [7-10]. Actually, at the initial
stage of the epidemic, the new coronavirus could
survive in a special environment and it can spread
by the infected goods during a relatively long time.
If considering such transmission mode, and
assuming there are only susceptible persons at the
initial stage, such as the elder over 65 years old, and
utilizing the fractional derivative to reveal the
memory effect, a simplified linear SEIR model can
be obtained following the ordinary SEIR model
given as follows:
(1)
where , E, and
denote the number of the susceptible, the latent, the
patient and the recovered people, denotes the
-order Caputo fractional derivatrive of on
, which defined by [11,12]
, (2)
here is the fractional order, is the
latentive rate, is the infected rate, is the
recovered rate.
The system (1) is a linear fractional SEIR model.
The linearity is its advantage as compared with the
known SEIR models. The traditional SEIR models,
including the interger-order and the fractional
models, are nonlinear differential equations. In
addition, the researches on the SEIR models are
almost concentrated on the dynamical analysis with
the spreading mechanism, and numerical
simulations using numerical methods. Although the
known SEIR models are studied under various
conditions, there are no expressions of the solution
in general due to the nonlinearity, and it is difficult
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Yi Zhang, Gongsheng Li
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to give deep researches in mathematics. On the
contrary, we can get the expression of the solution
of the simplified model (1) by the Laplace
transform metod, and we can give more
mathematical analysis on the model.
On the other hand, the fractional order in a
fractional model is a key parameter to characterize
the heavy-tail subdiffusion with memory effect.
However, it is always unknown for real-life
problems which leading to inverse fractional order
problems. The inverse problems of identifying
parameters including fractional orders have been
studied during the last decade [13-24]. It is noted
that in the existing work on inverse fractional order
problems, most of them were studied by using
subdomain measurements or one-point
measurements at , or using subboundary
data also at for arbitrary given . An
interesting problem is whether we can determine the
fractional orders only using limited discretized data.
Exactly speaking, there is one order in
model (1) which is unknown, can we determine it
uniquely only using one measurement?
It is difficult to give an answer in theory for the
above question, but the situation could be changed if
coping with numerical solutions. Here we are
concerned with the inverse problem of determining
the fractional order in (1) using one measurement.
By the Laplace transform method, the solution of
the forward problem is expressed by the Mittag-
Lellfer function, and the inverse problem is
transformed to a nonlinear algebraic equation. By
choosing the measured time, the nonlinear equation
can be solved uniquely by the monotonicity of the
Mittag-Lellfer function. The unique solvability of
the inverse problem is testified by theoretical
examples. The rest of the paper is organized as
follows.
In Section 2, some preliminaries on the Mittag-
Lellfer function, and the forward problem are
introduced, and its solution is deduced by the
Laplace transform. In Section 3, the inverse problem
of identifying the fractional order using one
measurement is given, which is transformed to
solving of a nonlinear equation by the additional
data, and the unique solvability of the inverse
problem is proved by the monotonicity of the
Mittag-Lellfer function, and numerical testification
is presented. Conclusion is given in Section 4.
2 The forward problem and its
solution
2.1 The forward problem and preliminaries
Assume the epidemic has occurred in a region, and
the elder are the susceptible and high-risk groups of
infectious diseases. At the initial stage, there are
only the susceptible people, and some of them
become latent persons, some of the latent persons
become patients with the pandemic spreading.
Therefor we give the initial condition for the model
(1) , . (3)
As a result the froward problem on the simplified
SEIR model (1) is composed by the model (1), with
the initial condition (3). When the parameters in the
model are known in advance, the analytical solution
of the above forward problem can be obtained by
the Laplace transform with the aids of the Mittag-
Lellfer function. Firstly we give some basic facts on
the Mittag-Leffler function and its properties and
the Laplace transform [25].
For real numbers and complex number
, the two-parametric Mittag-Leffler function is
defined as:
, (4)
where denotes the Gamma function, and there
is the so-called one-parametric Mittag-Leffler
function as . On the Mittag-
Leffler function, there holds the estimate in general
,, (5)
where is a constant.
For the real-valued Mittag-Leffler function, there
holds the complete monotonicity [25].
Definition 1. A function is called
completely monotonic if it posseses derivatives
of any order , and the derivatives
are alternative in sign, i.e.,
, . (6)
Lemma 1[25]. (i) The Mittag-Leffler function
is completely monotonic on for
all .
(ii) The two-parametric Mittag-Leffler function
is also completely monotonic on for
all .
Corollary 1. For , the function is
strictly decreasing on , and the function
is strictly increasing on .
The function of complex variable defined by
, (7)
is called the Laplace transform of , where
satisfies the growth condition as
, and are positive constants.
On performing Laplace transform for a fractional
derivative function, some regularity is needed for
the performed function, see [26] for detailed
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analysis. In this work we mainly focus on the
inverse fractional order problem, and we assume
that the solution to the forward problem has the
regularity such that the Laplace transform on the
Caputo fractional derivative of the solution can be
performed. For the fractional derivative (
), there holds:
, (8)
where denotes the Laplace transform of .
In addition, the following formula plays a key role
in solving fractional differential equations by the
Laplace transform:
, (9)
where , and be constant.
2.2 The solution of the forward problem
By performing the Laplace transform on the first
equation in model (1), we have
.
Noting , there holds
.
By utilizing the formula (9), we get the expression
of : . (10)
Next, by performing the Laplace transform on the
second equation in (1), there holds
,
and thanks to , we have
.
Let . Then by using (9) again we get:
. (11)
Similarly, if , then by the Laplace
transform method, we can obtain the expressions of
and given as follows:
. (12)
. (13)
Theorem 1. Assume the parameters and are
positive constants and satisfy the condition:
or . Then for any given ,
the forward problem (1), (3) has a unique,
nonnegative solution
expressed by (10)-(13), and the solution has the
asymptotic behaviors ,
as , where is a constant.
Proof. Obviously the solution is positive, and
strictly decreasing on due to the monotonicity
of the Mittag-Lellfer function given in Corollary 1.
Next, if , by the monocinity of the Mittag-
Lellfer function, the solution is positive too,
whether or , and there holds
by the estimate of
as .
Thirdly we are to prove the nonnegativity and
monotonicity of and . Rewrite as
, (14)
For given and , define a function
, . (15)
By Corollary 1 there holds
and , . (16)
Then by the meanvalue theorem we get
, (17)
where is between and , is between and ,
and is between and . According to the
conditions of the parameters, we deduce that
has the same sign with , and thus there must
have by (16). Noting the expression of
is of a similar form as , and the Mittag-
Lellfer function has the same
properties as , we can get its nonnegative
as done in the above, as well as its asymptotic
behavior based on (5). The proof is over.
Based on the above solutions, the trends of the
pandemic with the time are plotted in Fig.1, where
the fractional order is chosen as , and other
model parameters are given as 25, ,
and . From Fig.1 it can be seen
that, the number of the susceptible is strictly
decreasing, the number of the latent is going up
firstly and then decreasing as well as the patient, and
the number of the recovered is increasing, which
basically coincide with the spreading trends of the
pandemic at its initial stage.
Fig1. The spreading trends of the pandemic with time
3 Inverse fractional order problem
3.1 The inverse problem
Although we get the solution to the forward problem
given by (10)-(13), it cannot be put into practice if
there are unknown parameters in the model.
Actually, the fractional order in (1) is an important
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index characterizing the slower spreading with
memory effect in time, but it is always unknown
which leads to inverse fractional order problems.
Noting that the recovered persons in the pandemic
are known every day, so we can get some additional
data on at given time :
, , (18)
where . The inverse fractional order problem
is to identify the order by using the data
(18) based on the forward problem, where the
parameters are known and different each
other.
An interesting problem for the above inverse
problem is:
Can we determine uniquely only using
one measurement of at given ?
Generally speaking, it is very difficult to answer the
above question in theory. Nevertheless, we can deal
with such problem in some special cases. Denote
at as the data. Then noting the
solution's expression (13), we get a nonlinear
algebraic equation on :
(19)
As a result the inverse fractional order problem is
transformed to solving of the nonlinear equation
(19).
3.2 Unique solvability
As said in the above, it is very difficult to prove the
uniqueness of a nonlinear algebra equation in theory.
However, it is possible and feasible to get a unique
solution of the nonlinear equation (19) under
suitable conditions.
Theorem 2. Assume the parameters and are
positive constants ranged in , and satisfy the
condition of Theorem 1. Then the nonlinear
equation (19) has a unique solution for suitable
.
Proof. We are to prove the function defined
by (19) is monotonic increasing on the fractional
order, i.e., on . As done in the
proof of Theorem 1, define three functions on
by
(20)
and rewrite as
, (21)
where
.
Then we have
. (22)
With a simlar method as usd in the proof of
Theorem 1, we can deduce that in the
case of or , as well as
. So by suitably choosing the measured time
, there holds
which implies that the equation (19) has a unique
solution. The proof is completed.
3.2.1 Example 1
Let the exact fractional order in model (1) be
, and other parameters be and
, , and the measured time be
. By (13) we get the additional data
. By solving the nonlinear equation (13), the
fractional order can be reconstructed. In order to see
the unique solvability of the equation, we plot the
functions , and on in
Fig.2.
It can be seen clearly that the function is
strictly monotonic on , and there exists
unique solution to the equation , which
demonstrates the uniqueness of the inverse problem.
Fig.2 The pictures of and in Ex.1
3.2.2 Example 2
Let the exact fractional order be in this
example, and other parameters be
and , , and the measured time be
. Also by (13) the additional data is given as
. As done in Example 1, the functions
and on are plotted in Fig.3.
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Fig.3 The pictures of and in Ex.2
From Fig.3 it can be seen again that the function
is also strictly monotonic on , and
the inverse problem is of uniqueness too.
4 Conclusion
A linear time-fractional SEIR epidemic model and
the inverse fractional order problem using one
measurement are investigated in mathematics.
Based on the expression of the solution to the
forward problem, the inverse problem is reduced to
a nonlinear algebraic equation on the fractional
order, and the unique solvability can be obtained by
the complete monotonicity of the Mittag-Lellfer
function of real variable under suitable order
conditions for the parameters. Theoretical examples
are presented to illustrate the uniqueness of the
inverse problem. It is noted that the derivative of the
function on can be computed by
(22), and some gradient-type iterative algorithms
can be applied to solve the nonlinear equation for
which we will give details in the near future.
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Contribution of individual authors to
the creation of a scientific article
(ghostwriting policy)
Yi Zhang, deduced the solution of the model, and
carried out the numerical algorithm and
computations of Figures 1-3.
Gongsheng Li, as the corresponding author, put
forward the fractional SEIR model, and finished the
proofs of Theorems 1-2.
Sources of funding for research
presented in a scientific article or
scientific article itself
This work is supported by the National Natural
Science Foundation of China (no. 11871313), and
the Natural Science Foundation of Shandong
Province, China (no. ZR2019MA021).
Creative Commons Attribution
License 4.0 (Attribution 4.0
International , CC BY 4.0)
This article is published under the terms of the
Creative Commons Attribution License 4.0
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