Global stability and asymptotic profiles of a partially degenerate reaction diffusion Cholera model with asymptomatic individuals
-
Shengfu Wang
Abstract
Considering the prevalence of asymptomatic individuals during the spread of disease, this article develops a model of degenerate reaction diffusion Cholera with asymptomatic individuals. First, the well-posedness of model is studied, including the global existence of solutions and the existence of attractor. Second, the basic reproduction number
1 Introduction
Cholera is an acute diarrheal infection caused by eating or drinking food and water contaminated with Vibrio cholerae. It is typical symptoms are diarrhea and vomiting, dehydration, muscle cramps, etc. [22]. When medical care is inadequate and treatment is not timely, the mortality rate is high. Although health care is improving in many countries around the world, the highly contagious cholera still occurs after natural disasters such as earthquakes and tsunamis, or in areas with relatively poor sanitation. In recent years, cholera outbreaks have remained severe in less developed countries, including Haiti, Yemen, Zimbabwe, and the Congo. Researchers estimate that there are 1,300,000–4,000,000 cases of cholera worldwide each year, and 21,000–143,000 deaths due to infection [38].
As is well known, mathematical models play a critical role in comprehending the dynamics of disease transmission. Over the past decade or so, many scholars have studied the spread of cholera through the development of various mathematical models. Capasso and Paveri-Fontana [6] proposed the first model of indirect cholera transmission in response to the cholera epidemic that occurred in the Mediterranean region in the summer of 1973, and the model included only infected population and pathogen. Codeço [7] in 2001 extended this model by adding equations for susceptible populations and showed the aquatic hosts have an important influence on cholera transmission. Subsequently, many more researchers have extended the model in [7], such as, Tien and Earn [30] extended the SIR model by adding pathogen concentration compartment (
It is worth noting that strengthening personal hygiene, improving environmental sanitation, and using safe drinking water are the most important strategies for controlling and preventing cholera. Therefore, vaccination has become an effective measure against cholera that has received widespread interest and became the subject of many clinical and theoretical studies. Many researchers as well have taken the cholera vaccine into account and have come up with a wealth of theoretical results. In [2,16], it was shown that backward branching can exist in models of all susceptible individuals vaccinated when the basic reproduction number is less than one. Cai et al. [5] modeled SAIVB infectious diseases with vaccine age and showed that not considering vaccine age underestimates the risk of transmission of cholera outbreaks.
Note that most of the aforementioned models are either ordinary differential equations (ODEs) or hybrids with partial differential equations (PDEs), which do not introduce spatial heterogeneity, resulting in an inadequate comprehension of the spatial propagation of cholera epidemics. Due to the differences in temperature, climate and geographic characteristics among different countries and regions, which in turn may contribute to the spatial distribution of the disease. Therefore, it is particularly important to integrate geographic and pathogen characteristics to develop cholera models with spatial heterogeneity. Much work has been done on the spatial dynamics of cholera transmission. Zhang et al. [41] developed a reaction diffusion model incorporating both direct and indirect transmission routes, and investigated the effects of diffusion coefficients and multiple transmission routes on disease propagation. Wang and Feng [34] proposed a model with different diffusion rates and spatial heterogeneity to study threshold dynamics and asymptotic behavior. Wang and Wu [35] investigated a degenerate cholera diffusion model, focusing on the asymptotic behavior of the positive steady state at small and large diffusion rates in susceptible individuals. Other studies on asymptotic behavior are continuing and are detailed in [4,9,36,39].
As mentioned earlier, the existing studies have mainly focused on how spatial heterogeneity and individual mobility contribute to the cholera propagation, prevention, and control. Actually, vaccine effectiveness, the prevalence of asymptomatic individuals, and spatial heterogeneity are all essential features that must be taken into account in the propagation and sustenance of cholera. Throughout this article, we establish a model of cholera with general incidence, spatial transmission, and incomplete immunity, ignoring person-to-person transmission (direct transmission) and assuming no transmission of environmental viruses, concentrating on the effects of these features on the spatial and temporal distribution of the disease and its prevention and control.
The rest of the article is organized as follows. The model is presented in Section 2. In Section 3, the global existence, ultimate boundedness, and the existence of attractors of model solution are studied. In Section 4, some relations between the principal eigenvalues and the basic reproduction number
2 Model formulation
Let
where
and
Schematic of the propagation of model (1).
In this article, we formulate the following general hypotheses:
Functions
There exists a Hölder continuous function
3 Well-posedness of the problem
Throughout this section, we shall confirm that model (1) has a unique global nonnegative solution and admits a connected global attractor. Let
where
Furthermore, set
By the aforementioned setting, we can rewrite model (1) as follows:
where
The local solutions of the model (1) on
Lemma 1
For any
Proof
For
which means
It yields from the general results in [21, Corollary 4] that (1) admits positive solution
Lemma 2
For any
Proof
From Lemma 1, we learn that model (1) has a unique solution
Combining [17, Lemma 1] and the comparison theorem leads to
where
Thus, there exists positive constant
which implies that
where
Choosing sufficiently large
for
where
for
where
Then,
where
The arrangement (6) can be derived as follows:
Combining (7) and (8), we obtain
These, together with Lemma 1, we realize that model (1) has a unique nonnegative global solution, i.e.,
On the basis of the model (1), we have
Hence, we obtain
where
it follows that
In the next step, we show that
It is apparent that (11) holds when
Multiplying the third equation of (1) by
Since
Thus, (12) becomes
According to the boundedness of
By using Young’s inequality,
Hence, (12) can be estimated by
where
Similarly, we multiply the fourth equation of (1) by
Since
where
By again using Young’s inequality by setting
Similarly, we also let
Hence, we can obtain
where
Multiplying the fifth equation of (1) by
By again applying Young’s inequality by choosing
Thus, (15) becomes
By combining (13), (14), and (16), we have
where
By applying interpolation inequality
Let
Hence, we can obtain
where
where
Denote by
where
As the fifth equation of model (1) has no diffusion, the weak compactness of the solution semiflow
for any bounded set
Lemma 3
For any
Proof
Let
Let
is precompact. Hence, we rewrite
Obviously,
It then follows that for
Therefore,
4 Basic reproduction number and steady states
Model (1) admits a disease free steady-state
Define by
and
We denote
According to [32], we can define the spectral radius of
where
where
Hence, combining the expressions of
Therefore, we have
This indicates that
So there exists a strictly positive eigenfunction
which is equivalent to
Let’s further consider the following eigenvalue problem
where
Multiplying the first equation of the problem (20) by
This shows that
In addition, we consider the following eigenvalue problem
which is equivalent to
Similar to the method of (21), one derives
Next, consider the following eigenvalue problem
where
From (22)–(25) and multiplying the first equation of problem (18) by
Therefore, we finally obtain
Note that in case the diffusion coefficients are all 0, i.e.,
where
For the relation between
Remark 1
The following facts were established.
If
If
By substituting
According to [29] and [32, Lemma 2.2], it is straightforward to derive the following result.
Lemma 4
Lemma 5
If
Proof
From (17), one can derive
where
Let
According to [39, Lemma 2.5], one has
where
As a result,
5 Stability of steady states
This section focuses on the analysis of the threshold dynamics of model (1) on
Theorem 1
If
Proof
Similar to [32, Theorem 3.1], it is clear that
Choose a constant
Thus,
Next, we address the uniform persistence of the disease. Before the proof, we present a few lemmas that will be utilized afterward.
Lemma 6
Let
For any
For any
Proof
(
Hence,
By the the maximum principle,
Note that problem (29) and (30) possesses a unique steady state, defined by
Choose
(
By [33, Lemma 1.26], it follows that
Similarly, we can also derive
This is,
Lemma 7
If there exists
then there exists
Proof
In the case of
By the comparison theorem, we learn that (32) there exists positive functions
This means that there exists
According to the comparison theorem, we learn that (32) there exists positive function
In the other
As a result, the
By the comparison principle, we obtain
By using the comparison principle again, one has
In the following, we present the uniform persistence of model (1).
Theorem 2
If
In addition, the model possesses at least one positive steady state.
Proof
According to Lemmas 6 and 7 and the process of [28, Theorem 3], define the set
where
Claim I: For any
For any
Combining the conditions (
This proves Claim I.
Claim II:
By arguing by contradiction, we suppose that there exists
So there exists
Thus, one has
Assuming that the linear system
admits a solution
By
Next, we define the continuous function
Evidently,
Theorem 3
In case
Proof
As a start, let us study the local stability of
By
Denote
Then for some
where
where
for
Then we can derive
where
where
By combining (35) and (36), we can also derive
As a result, by (36),
Let
Let
Moreover,
Let
where
Hence,
where
Integrating (40) and (41) yields
Finally, combining (36)–(37), (42) and
This proves the local stability of
Further, let us prove the global attractiveness of
Claim I: For all
From (4) and (5), it follows that
Using the principle of comparison, it follows that
Inspired by [8], we define
Clearly,
where
Definition by
we immediately obtain that
Claim II.
Claim I guarantees that
With the aforementioned analysis, we can deduce that
6 Asymptotic profiles of the endemic steady state
In this section, we study the asymptotic behavior of the steady state to evaluate the impact of diffusion rate on the propagation of the infection. The efficacy of vaccines is increasing with the development of medical technology, we consider ignoring the immune loss rate, i.e., taking
It is easy to come up with
then
here,
Lemma 8
Let
The
The lower bounds of
Proof
Arranging the first four equations of (45) and integrating over
Therefore, one obtains
Similarly,
and
Thus, it follows that
(
Thus, we obtain
Therefore, it is feasible to launch
6.1 The case of
D
1
→
0
Throughout the present section, we discuss the asymptotic behavior of
Theorem 4
Suppose that
where
and
Proof
In this proof, we suppose that
Step 1: The
It follows from Lemma 8 and the ellipse
It clearly follows from the fact that
Evidently, if
It is clear that
Let
Then, we derive
Combining the Hölder inequality and (47), the following result can be obtained
where
then one can conclude that
Similarly, we can also derive
Combining Sobolev’s embedding theorem
By a similar argument as the
Step 2: Convergence of
Note that
Fix
Choosing
where
Rewrite the
By (50), there holds
According to the standard
For sufficiently large
Similarly, we can also derive
where
either
Assuming that
Bringing in the third equation of (45) yields
Combining this with the Lemma 8(
Then for any small
Along with the first equation of (45), this implies that
and
Taking an arbitrarily large
and
Clearly, (54) and (55) contain unique positive solutions, denoted by
Further by a similar proof as in [10, Lemma 2.4], it follows that
and
Thus, as
By the arbitrariness of
Similarly, one can conclude that
Consider the
Define
Starting from the standard compactness argument for elliptic equations, as
By the Harnack-type inequality, one has
Define
Moreover, due to the uniqueness of the principal eigenvalue, 0 is the principal eigenvalue of (19) and (23). Combined with (18) and Lemma 4, this contradicts
Step 3: Convergence of
From (45), we know that
By (52), for any small
Thus,
So, for sufficiently large
where
For adequately large
with
By
Furthermore, by (58), one has
where
Arguing similarly as before, this leads to
Combining the above results, it follows that
Therefore, it can be conveniently derived that
6.2 The case of
D
3
→
0
We investigate the asymptotic behavior of
Theorem 5
Suppose that
where
and
Proof
We will deal with proof by means of belowing three claims.
Step 1: The endemic steady state
Arguing in a similar way to Step 1 of the proof of Theorem 4, the elliptic
where the positive constant
Using the approach of Step 1 in Theorem 4 yields the
Step 2: Convergence of
By (63), the elliptic
Then
where
Step 3: Convergence of
It is noteworthy that
From (64) and (65), It is possible to employ the sub-supersolution comparison argument as in Claim III of Theorem 4 (see [19, Theorem 1.1]) to conclude that
6.3 The case of
D
1
→
∞
or
D
3
→
∞
In the following, we analyze the asymptotic behavior of
Theorem 6
Suppose that
where
Proof
By Theorem 4, the
Then, the elliptic
Therefore,
where
The proof that is similar to the Theorem 4 leads to
It is evident that the Harnack-type inequality also holds for
Therefore, 0 is the principal eigenvalue of (19) and (23). Combined with (18) and Lemma 4. This contradiction with
As far as Theorem 7 is concerned, the method of proof is very similar to Theorem 6, so that we simply revise it slightly and then omit the details. It turns out to be as follows.
Theorem 7
Suppose that
where
7 Numerical simulations
With this section, we use numerical examples to validate our theoretical findings and explore the impacts of individuals movement and spatial heterogeneity on
First, we select
The spatio-temporal distribution of
If the parameters
The persistence of disease for model (1) with
Subsequently, we consider the effects of the spatial heterogeneity parameters on disease propagation. In Figure 4(a)--(c) describe the effect of the vaccination rate
Effect of spatial heterogeneity parameters on the disease distribution of model (1): (a)–(c) effect of
Next, let’s explore how diffusion coefficients affect disease propagation. Figure 5(a)--(c) displays the spatial distribution of
Distribution of different diffusion coefficients for
In addition, let’s be interested in the correlation between the primary parameters and the local reproduction number
The correlation between the local reproduction number
Finally, we analyze in depth the effect of the proportion of asymptomatic infected individuals as a percentage of the whole infected individuals
The effects of
8 Conclusion
To this end, we proposed a model of degenerate cholera with asymptomatic individuals in spatially heterogeneous environments that takes into account the rate of immune loss and general morbidity, and assumes that environmental viruses do not disperse. We studied the existence of global positive solutions to the model and proved that
In the numerical simulation section, except for simulating the extinction and persistence of disease (Figures 2 and 3), the impacts of some crucial parameters on the spatial and temporal distributions of the disease are also explored. To be more specific, the spatial heterogeneity of model will lead to a regional variability in the spatial distribution of disease (Figure 4). Furthermore, numerical simulations show that the spread of susceptible individuals does not modify the risk level of local outbreaks when the spread of asymptomatic and infected individuals is relatively small (Figure 5(a)–(c)). This is consistent with the conclusions of [37], which stated that cholera cannot be controlled by limiting the movement of susceptible individuals. We also found the spread of asymptomatic and symptomatic individuals suppresses spatial heterogeneity so that each individual is less differentiated across the region (Figure 5(d)–(i)). Therefore, it is not advisable to entirely limit the mobility of people during epidemic periods of the disease. Further, we also analyze how the proportion of asymptomatic infected individuals to the whole infected individuals
Unfortunately, in this article, we only considered the effects of environment spread on the disease and ignored host-to-host spread (direct transmission), which exists in real situations. However, we know that cholera outbreaks are also related to seasons and temperatures, therefore, a response diffusion cholera model with cyclical and climatic variations would be more meaningful and the results would be more effective in preventing and controlling the disease.
Acknowledgements
We would also gratefully like to thank the editor and anonymous reviewers for their careful review and helpful suggestions, which improved the research in this thesis.
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Funding information: This research is partially supported by the Tianshan Talent Training Program (Grant No. 2022TSYCCX0015), the National Natural Science Foundation of China (Grant No. 12361103).
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Author contributions: All authors have accepted responsibility for the entire content of this manuscript and consented to its submission to the journal, reviewed all the results and approved the final version of the manuscript. Shengfu Wang: formal analysis, conceptualization, writing-original draft, writing–review & editing. Linfei Nie: funding acquisition, supervision, and editing.
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Conflict of interest: The authors state no conflict of interest.
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Data availability statement: Data sharing is not applicable to this article as no data sets were generated or analyzed during the current study.
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- The properties of a new fractional g-Laplacian Monge-Ampère operator and its applications
- A fractional profile decomposition and its application to Kirchhoff-type fractional problems with prescribed mass
- Cauchy problem for a non-Newtonian filtration equation with slowly decaying volumetric moisture content
- Multiplicity of normalized semi-classical states for a class of nonlinear Choquard equations
- The second Yamabe invariant with boundary
- Peaked solitary waves and shock waves of the Degasperis-Procesi-Kadomtsev-Petviashvili equation
- Normalized solutions for the Choquard equations with critical nonlinearities
- Boundedness and long-time behavior in a parabolic-elliptic system arising from biological transport networks
- Initial boundary value problem and exponential stability for the planar magnetohydrodynamics equations with temperature-dependent viscosity
- Nonexistence of mean curvature flow solitons with polynomial volume growth immersed in certain semi-Riemannian warped products
- Analysis of a vector-borne disease model with vector-bias mechanism in advective heterogeneous environment
- Normalized solutions for the Kirchhoff equation with combined nonlinearities in ℝ4
- Nonexistence of the compressible Euler equations with space-dependent damping in high dimensions
- Multiplicity of solutions for a nonhomogeneous quasilinear elliptic equation with concave-convex nonlinearities
- On semilinear inequalities involving the Dunkl Laplacian and an inverse-square potential outside a ball
- Besov regularity for the elliptic p-harmonic equations in the non-quadratic case
-
Uniqueness and nondegeneracy of ground states for
- Existence of solutions for a class of quasilinear Schrödinger equations with Choquard-type nonlinearity
- Weighted Hardy-Adams inequality on unit ball of any even dimension
- Existence and regularity for a p-Laplacian problem in ℝN with singular, convective, and critical reaction
- Temporal periodic solutions of non-isentropic compressible Euler equations with geometric effects
- Nodal solutions for a zero-mass Chern-Simons-Schrödinger equation
- The modified quasi-boundary-value method for an ill-posed generalized elliptic problem
- Nonlocal heat equations with generalized fractional Laplacian
- Choquard equations with recurrent potentials
- Local and global well-posedness of the Maxwell-Bloch system of equations with inhomogeneous broadening
- The Riemann problem for two-layer shallow water equations with bottom topography
- Global stability and asymptotic profiles of a partially degenerate reaction diffusion Cholera model with asymptomatic individuals
- On Kirchhoff-Schrödinger-Poisson-type systems with singular and critical nonlinearity
- Energy-variational solutions for viscoelastic fluid models
- Stability on 3D Boussinesq system with mixed partial dissipation
- Existence and multiplicity results for non-autonomous second-order Hamiltonian systems
- Erratum
- Erratum to “Normalized solutions for the Kirchhoff equation with combined nonlinearities in ℝ4”
Articles in the same Issue
- Regular Articles
- Existence of normalized peak solutions for a coupled nonlinear Schrödinger system
- Orbital stability of the trains of peaked solitary waves for the modified Camassa-Holm-Novikov equation
- Global boundedness in a two-dimensional chemotaxis system with nonlinear diffusion and singular sensitivity
- Existence and uniqueness of solution for a singular elliptic differential equation
- The bounded variation capacity and Sobolev-type inequalities on Dirichlet spaces
- Vanishing and blow-up solutions to a class of nonlinear complex differential equations near the singular point
- Existence, uniqueness, localization and minimization property of positive solutions for non-local problems involving discontinuous Kirchhoff functions
- Concentration phenomena for a fractional relativistic Schrödinger equation with critical growth
- Beyond the classical strong maximum principle: Sign-changing forcing term and flat solutions
- Monotonicity of solutions for parabolic equations involving nonlocal Monge-Ampère operator
- On a nonlinear Robin problem with an absorption term on the boundary and L1 data
- Multiple solutions for the quasilinear Choquard equation with Berestycki-Lions-type nonlinearities
- Gradient estimates for a class of higher-order elliptic equations of p-growth over a nonsmooth domain
- Global existence and decay estimates of the classical solution to the compressible Navier-Stokes-Smoluchowski equations in ℝ3
- Global existence and finite-time blowup for a mixed pseudo-parabolic r(x)-Laplacian equation
- Multiple positive solutions for a class of concave-convex Schrödinger-Poisson-Slater equations with critical exponent
- A minimization problem with free boundary for p-Laplacian weakly coupled system
- Multiplicity of semiclassical solutions for a class of nonlinear Hamiltonian elliptic system
- k-convex solutions for multiparameter Dirichlet systems with k-Hessian operator and Lane-Emden type nonlinearities
- Infinitely many solutions for Hamiltonian system with critical growth
- On optimal control in a nonlinear interface problem described by hemivariational inequalities
- Variational–hemivariational system for contaminant convection–reaction–diffusion model of recovered fracturing fluid
- Potential and monotone homeomorphisms in Banach spaces
- Critical fractional Schrödinger-Poisson systems with lower perturbations: the existence and concentration behavior of ground state solutions
- Supercritical Hénon-type equation with a forcing term
- Concentration of blow-up solutions for the Gross-Pitaveskii equation
- Mountain-pass-type solutions for Schrödinger equations in R2 with unbounded or vanishing potentials and critical exponential growth nonlinearities
- Regularity for critical fractional Choquard equation with singular potential and its applications
- Double phase anisotropic variational problems involving critical growth
- Normalized solutions of NLS equations with mixed nonlocal nonlinearities
- Nontrivial solutions for resonance quasilinear elliptic systems
- Standing waves for Choquard equation with noncritical rotation
- Low regularity conservation laws for Fokas-Lenells equation and Camassa-Holm equation
- Modified quasilinear equations with strongly singular and critical exponential nonlinearity
- Limit profiles and the existence of bound-states in exterior domains for fractional Choquard equations with critical exponent
- Nests of limit cycles in quadratic systems
- Regularity of minimizers for double phase functionals of borderline case with variable exponents
- Boundedness, existence and uniqueness results for coupled gradient dependent elliptic systems with nonlinear boundary condition
- Ground state solutions for magnetic Schrödinger equations with polynomial growth
- Nonoccurrence of Lavrentiev gap for a class of functionals with nonstandard growth
- Dynamics for wave equations connected in parallel with nonlinear localized damping
- Boussinesq's equation for water waves: Asymptotics in Sector I
- Optimal global second-order regularity and improved integrability for parabolic equations with variable growth
- Normalized solutions of Schrödinger equations involving Moser-Trudinger critical growth
- Normalized solutions for the double-phase problem with nonlocal reaction
- Normalized solutions for Sobolev critical fractional Schrödinger equation
- Well-posedness of Cauchy problem of fractional drift diffusion system in non-critical spaces with power-law nonlinearity
- Improved results on planar Klein-Gordon-Maxwell system with critical exponential growth
- Existence and multiplicity of solutions for a new p(x)-Kirchhoff equation
- Quasiconvex bulk and surface energies: C1,α regularity
- Time decay estimates of solutions to a two-phase flow model in the whole space
- Bounds for the sum of the first k-eigenvalues of Dirichlet problem with logarithmic order of Klein-Gordon operators
- The properties of a new fractional g-Laplacian Monge-Ampère operator and its applications
- A fractional profile decomposition and its application to Kirchhoff-type fractional problems with prescribed mass
- Cauchy problem for a non-Newtonian filtration equation with slowly decaying volumetric moisture content
- Multiplicity of normalized semi-classical states for a class of nonlinear Choquard equations
- The second Yamabe invariant with boundary
- Peaked solitary waves and shock waves of the Degasperis-Procesi-Kadomtsev-Petviashvili equation
- Normalized solutions for the Choquard equations with critical nonlinearities
- Boundedness and long-time behavior in a parabolic-elliptic system arising from biological transport networks
- Initial boundary value problem and exponential stability for the planar magnetohydrodynamics equations with temperature-dependent viscosity
- Nonexistence of mean curvature flow solitons with polynomial volume growth immersed in certain semi-Riemannian warped products
- Analysis of a vector-borne disease model with vector-bias mechanism in advective heterogeneous environment
- Normalized solutions for the Kirchhoff equation with combined nonlinearities in ℝ4
- Nonexistence of the compressible Euler equations with space-dependent damping in high dimensions
- Multiplicity of solutions for a nonhomogeneous quasilinear elliptic equation with concave-convex nonlinearities
- On semilinear inequalities involving the Dunkl Laplacian and an inverse-square potential outside a ball
- Besov regularity for the elliptic p-harmonic equations in the non-quadratic case
-
Uniqueness and nondegeneracy of ground states for
- Existence of solutions for a class of quasilinear Schrödinger equations with Choquard-type nonlinearity
- Weighted Hardy-Adams inequality on unit ball of any even dimension
- Existence and regularity for a p-Laplacian problem in ℝN with singular, convective, and critical reaction
- Temporal periodic solutions of non-isentropic compressible Euler equations with geometric effects
- Nodal solutions for a zero-mass Chern-Simons-Schrödinger equation
- The modified quasi-boundary-value method for an ill-posed generalized elliptic problem
- Nonlocal heat equations with generalized fractional Laplacian
- Choquard equations with recurrent potentials
- Local and global well-posedness of the Maxwell-Bloch system of equations with inhomogeneous broadening
- The Riemann problem for two-layer shallow water equations with bottom topography
- Global stability and asymptotic profiles of a partially degenerate reaction diffusion Cholera model with asymptomatic individuals
- On Kirchhoff-Schrödinger-Poisson-type systems with singular and critical nonlinearity
- Energy-variational solutions for viscoelastic fluid models
- Stability on 3D Boussinesq system with mixed partial dissipation
- Existence and multiplicity results for non-autonomous second-order Hamiltonian systems
- Erratum
- Erratum to “Normalized solutions for the Kirchhoff equation with combined nonlinearities in ℝ4”
