Abstract
A compartment epidemic model for infectious disease spreading is investigated, where movement of individuals is governed by spatial diffusion. The model includes infection age of the infected individuals and assumes a logistic growth of the susceptibles. Global well-posedness of the equations within the class of nonnegative smooth solutions is shown. Moreover, spectral properties of the linearization around a steady state are derived. This yields the notion of linear stability which is used to determine stability properties of the disease-free and the endemic steady state.
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1 Introduction
We consider a compartment epidemic model for infectious disease spreading. The total population is divided into susceptible and infected individuals which move in space by diffusion. Infectious individuals are structured by infective age keeping track of the time elapsed since an individual first acquires the disease. A logistic growth is assumed for the susceptibles.
Let S(t, x) and I(t, a, x) be the densities of susceptible and infected individuals, respectively, at time \(t\ge 0\), position \(x\in \Omega \), and infection age \(a\in (0,a_m)\), where \(\Omega \subset \mathbb {R}^n\) with \(n\le 3\) is a bounded, smooth domain, and \(a_m\in [0,\infty )\) is the maximal invective age. The population of susceptible individuals is assumed to obey a logistic growth with intrinsic growth rate \(\kappa _1>0\) and carrying capacity \(\kappa _2>0\). Susceptible individuals are infected at a rate b(a, x) by infected individuals of invective age a and position x. Infectious individuals die naturally and disease-induced at a combined rate \(\mu (a,x)\ge 0\). They may recover and enter directly the class of susceptibles at a rate \(r(a,x)\ge 0\). We shall thus focus on the equations
for \(t>0\), \(a\in (0,a_m)\), and \(x\in \Omega \). The differentiation operator D in (1.1b) is defined as
and is thus, if I is continuously differentiable with respect to t and a, given by
For notational simplicity we will take a susceptible diffusion rate \(d_1=1\) and consider a diffusion coefficient \(d=d(a)>0\) for infected individuals dependent only upon infection age (though a space dependence does not alter the subsequent results). The equations are supplemented by the initial conditions
and the boundary conditions
for \((t,a,x)\in (0,\infty )\times (0,a_m)\times \partial \Omega \). Here, \(\delta \in \{0,1\}\) is fixed so that \(\delta =0\) corresponds to Dirichlet boundary conditions, while \(\delta =1\) yields Neumann boundary conditions with \(\partial _\nu I=\nabla I\cdot \nu \) denoting the derivative in normal direction \(\nu \) on the boundary \(\partial \Omega \) (we treat the two cases simultaneously).
Age-structured compartment epidemic models and age-structured population models in general have been investigated since many years (Thieme (2003); Webb (1985, 2008)). The particular case of equations (1.1) without spatial diffusion (and \(r=0\)) was studied in Cao et al. (2021). Therein, criteria for stability and instability of the disease-free and the endemic steady states were obtained in dependence on the corresponding basic reproduction number. Moreover, conditions for the occurrence of Hopf bifurcation were presented.
The inclusion of spatial heterogeneity in age-structured populations leads to additional technical difficulties in the analysis. We refer to the monograph of Webb (2008) for a general treatment of and a comprehensive overview on such problems. Regarding SIS- and SIR-models there are various linear and nonlinear variants involving age and spatial structure with Laplace diffusion. The list includes the pioneering works of Webb (1980, 1981, 1982) on epidemic models including incubation periods, followed by Fitzgibbon et al. (1995, 1996), Kubo and Langlais (1994), Langlais and Busenberg (1997) and, more recently, Chekroun and Kuniya (2019, 2020a, 2020b); Di Blasio (2010); Ducrot and Magal (2009); Ducrot et al. (2010); Ducrot and Magal (2011); Kim (2006); Kuniya and Oizumi (2015) though these references are non-exhaustive. The cited papers address various questions under different modeling hypotheses, for instance related to well-posedness of the equations, existence and stability properties of disease-free and endemic steady states, disease persistence, traveling wave solutions, or numerical simulations. We also refer to Kang and Ruan (2021) and the references therein for age-structured epidemic models describing long-distance spreading of diseases by nonlocal diffusion.
In this research we shall focus on the particular model (1.1). We first prove the existence and uniqueness of positive, global, smooth solutions by using a semigroup approach relying on results outlined in Webb (2008). We then show that the linearization of these equations around a steady state yields a strongly continuous semigroup, and we use the spectrum of its generator for determining linear stability properties of steady states. Without further assumptions we prove stability or instability of the disease-free steady state in dependence on the reproduction number. In a particular case of (1.1) assuming spatially homogeneous rates and Neumann boundary conditions we improve the local stability of this steady state to global stability. Moreover, we investigate the stability of the endemic steady state.
In the following Sect. 2 we present our main results. Section 3 is dedicated to the proof of the well-posedness of (1.1). The details on the linearized problem in the general case are given in Sect. 4 and then applied in Sect. 5. The application to a simplified version of equations (1.1) with Neumann boundary conditions and spatially homogeneous rates are presented in Sect. 6. Some technical results are postponed to the Appendix 1.
2 Main results
We first state the result on the well-posedness of (1.1) and then investigate linearized stability of steady states.
2.1 Well-posedness
In order to present our existence result, we set \(J:=[0,a_m]\) and take without loss of generality \(d_1=1\). We assume that
and
for some \(\rho >0\). Moreover,
and
Here, \(C^\rho \) stands for \(\rho \)-Hölder continuous functions. The regularity assumptions on the data are mainly imposed in order to derive smooth solutions. Denote \(\mathbb {R}^+:=[0,\infty )\) and \({\dot{\mathbb {R}}}^+:=(0,\infty )\).
We shall prove the following result on the existence, uniqueness, and regularity of global, positive solutions to (1.1):
Theorem 2.1
Assume (2.1) and \(p\in \big (\max \{3n/4,2\},\infty \big )\). Then, given initial values \( S_0\in L_p^+(\Omega )\) and \(I_0\in L_1\big (J,L_p^+(\Omega )\big )\), there is a unique positive global solution (S, I) to (1.1) such that
is a strong solution to (1.1a), while
satisfies (1.1b) in the sense that
for \(t>0\) and a.e. \(a\in (0,a_m)\). In fact, the solution map \((t,(S_0,I_0))\mapsto (S,I)(t)\) defines a global semiflow on \(L_p^+(\Omega )\times L_1(J,L_p^+(\Omega ))\).
The proof of Theorem 2.1 is based on a semigroup representation of solutions in the spirit of Webb (2008) and on Banach’s fixed point theorem. It is performed in several steps in Sect. 3. In fact, it is not restricted to the particular nonlinearities in (1.1) and may rather be a template for similar problems. That the solution map defines a global semiflow paves the way to consider qualitative aspects of the model.
2.2 Linearization around steady states
Assume (2.1) and fix an arbitrary steady state \((S_*,I_*)\) to (1.1), i.e. a time-independent solution. The regularizing effects of the Laplacian implies that we may assume without loss of generality the regularity
for some \(p>n\). The linearization of (1.1) around the steady state \((S_*,I_*)\) is
for \((t,a, x)\in \mathbb {R}^+\times (0,a_m)\times \Omega \), and subject to the initial conditions
and boundary conditions
for \((t,a,x)\in (0,\infty )\times (0,a_m)\times \partial \Omega \). We shall show that the solutions (S, I) to the linearized problem (2.3) are given by a strongly continuous semigroup on \(L_p(\Omega )\times L_1(J,L_p(\Omega ))\) with compact resolvent:
Theorem 2.2
Suppose (2.1) and \(p>(2\vee n)\). Let \((S_*,I_*)\) be a steady state to (1.1) satisfying (2.2). Then, the solution (S, I) to the linearized equation (2.3) is given as
where \((\mathbb {S}_*(t))_{t\ge 0}\) is a strongly continuous semigroup on \(L_p(\Omega )\times L_1(J,L_p(\Omega ))\). Its generator has a compact resolvent and thus a pure point spectrum without finite accumulation point.
Theorem 2.2 is a consequence of Theorem 4.2 and Corollary 4.3 from Sect. 4. There, we also present more precise information on the semigroup and its generator. Note that the semigroup \((\mathbb {S}_*(t))_{t\ge 0}\) lacks positivity and thus less information on the spectrum of its generator is available in general. We refer to Remark 4.6 for further details.
2.3 Linear stability
Due to Theorem 2.2, one may characterize stability properties of steady states based on the linearization of (1.1) around these steady states. That linearized stability indeed determines the (asymptotic) stability of steady states in certain nonlinear population models including age- and spatial structure has recently been shown in Walker (2023) and Walker and Zehetbauer (2022). We refrain, however, to prove this for problem (1.1).
Herein, we shall just call a steady state \((S_*,I_*)\) linearly stable if the generator of the semigroup associated with the linearization (2.3) of (1.1) (given in Theorem 2.2) has a spectrum lying entirely in the half plane \(\textrm{Re}\, \lambda <0\) while we call the steady state linearly unstable if there is a spectral point in the half plane \(\textrm{Re}\, \lambda >0\) (see Definition 4.4 and Remark 4.6 below for more details in this regard).
We assume for simplicity that
We provide a stability analysis in \(L_p(\Omega )\times L_1(J,L_p(\Omega ))\) of the trivial and the disease-free steady states. To state precise results we introduce the principal eigenvalue \(\mu _0\) of the Laplacian \(-\Delta \) on \(\Omega \) subject to either Dirichlet boundary conditions (hence \(\mu _0>0\)) or Neumann boundary conditions (hence \(\mu _0=0\)).
Theorem 2.3
Suppose (2.1), (2.4), and let \(p>(2\vee n)\).
- \(\mathbf{(a)}\):
-
The trivial steady state \((S_*,I_*)=(0,0)\) to (1.1) is linearly unstable in the space \(L_p(\Omega )\times L_1(J,L_p(\Omega ))\) if \(\kappa _1>\mu _0\) and linearly stable if \(\kappa _1<\mu _0\).
- \(\mathbf{(b)}\):
-
There is a disease-free steady state \((S_*,I_*)=({\tilde{S}}_*,0)\) to (1.1) with a smooth function \({\tilde{S}}_*>0\) if and only if \(\kappa _1>\mu _0\). In this case, the disease-free steady state \((S_*,I_*)=({\tilde{S}}_*,0)\) is unique (and given by \({\tilde{S}}_*=\kappa _2\) for Neumann boundary conditions), and there is a number \({\textsf{R}}_0>0\) such that it is linearly stable in the space \(\hbox {L}{_{p}}(\Omega )\times {{ L}}_1({{ J,L}}_{{ p}}(\Omega ))\) if \({\textsf{R}}_0<1\) and linearly unstable if \({\textsf{R}}_0>1\).
- \(\mathbf{(c)}\):
-
Let \(\kappa _1>\mu _0\). There is no endemic state \((S_*,I_*)\) to (1.1) with \(S_*, I_*\ge 0\) and \(I_*\not \equiv 0\) if \({\textsf{R}}_0\le 1\).
The proof of Theorem 2.3 is presented in Sect. 5. The reproduction number \({\textsf{R}}_0>0\) is defined in (5.3a) and corresponds to the spectral radius of a compact irreducible operator (defined in (5.3b)) depending on \({\tilde{S}}_*\). It is open whether there is an endemic state \((S_*,I_*)\) with \(S_*, I_*\ge 0\) and \(I_*\not \equiv 0\) if \({\textsf{R}}_0>1\) (see Remark 5.5 in this regard). However, the existence of an endemic steady state in case that \({\textsf{R}}_0>1\) is easily obtained when assuming Neumann boundary conditions and spatially homogeneous rates m and b as shown in the next subsection.
2.4 Linear stability in a particular model with Neumann boundary conditions
We give a more detailed account of Theorem 2.3 in the particular case of Neumann boundary conditions
so that \(\mu _0=0\), and spatially homogeneous data
that is, data only depending on age a. In this particular situation, besides the trivial steady state \((S_*,I_*)=(0,0)\) and the disease-free steady state \((S_*,I_*)=(\kappa _2,0)\), there is also an endemic steady state \(({{\bar{S}}}_*,{{\bar{I}}}_*)\) provided that \({\textsf{R}}_0>1\), where
with
It is given as
Linear stability and instability of these steady states is determined by the basic reproduction number \({\textsf{R}}_0\):
Theorem 2.4
Assume (2.4), (2.5), let \(p>(2\vee n)\), and let \({\textsf{R}}_0>0\) be defined in (2.6).
- \(\mathbf{(a)}\):
-
The trivial steady state \((S_*,I_*)=(0,0)\) is linearly unstable in the space \(L_p(\Omega )\times L_1(J,L_p(\Omega ))\).
- \(\mathbf{(b)}\):
-
If \({\textsf{R}}_0 <1\), then the disease-free steady state \((S_*,I_*)=(\kappa _2,0)\) is globally linearly stable in the space \(L_p(\Omega )\times L_1(J,L_p(\Omega ))\); that is, it is linearly stable and attracts any solution starting from positive initial values. If \({\textsf{R}}_0 >1\), then \((S_*,I_*)=(\kappa _2,0)\) is linearly unstable in \(L_p(\Omega )\times L_1(J,L_p(\Omega ))\).
- \(\mathbf{(c)}\):
-
For \(1<{\textsf{R}}_0 <3\), the endemic steady state \(({{\bar{S}}}_*,{{\bar{I}}}_*)\) is linearly stable in the space \(L_p(\Omega )\times L_1(J,L_p(\Omega ))\).
Part (a) and the local stability statements of part (b) of Theorem 2.4 have been observed already in Theorem 2.3. The proofs of the remaining statements of Theorem 2.4 are given in Sect. 6. In fact, when \({\textsf{R}}_0 <1\) we prove that
for any solution (S, I) to (1.1) corresponding to positive nontrivial initial values \((S_0,I_0)\) so that there is no further steady state in this case (in accordance with Theorem 2.3 (c)). Clearly, one expects \(({{\bar{S}}}_*,{{\bar{I}}}_*)\) to be linearly stable whenever \({\textsf{R}}_0 >1\).
3 Well-posedness: Proof of Theorem 2.1
We prove Theorem 2.1 in several steps. After introducing some notation we derive the existence of a local solution and then establish further properties.
3.1 Preliminaries and notation
For two Banach spaces E and F we write \(\mathcal {L}(E,F)\) for the Banach space of bounded linear operators from E to F, and we set \(\mathcal {L}(E):=\mathcal {L}(E,E)\). Similarly, \(\mathcal {K}(E,F)\) and \(\mathcal {K}(E)\) stand for compact linear operators.
For fixed \(\delta \in \{0,1\}\) and \(p\in (1,\infty )\), we set
and introduce the scale of Banach spaces
By \(\Delta _\mathcal {B} \) we denote the Laplacian defined on \(W_{p,\mathcal {B}}^2(\Omega )\). Moreover, for fixed \(a\in J\), also the operator
has domain \(W_{p,\mathcal {B}}^2(\Omega )\). Then \(\Delta _\mathcal {B} \) and A(a) are generators of positive analytic contraction semigroups on \(L_p(\Omega )\) for each \(p\in (1,\infty )\), see Amann (1983); Rothe (1984). In fact, since
it follows from Amann (1995, II.Corollary 4.4.2) that A generates a positive parabolic evolution operator
on \(L_p(\Omega )\) in the sense of Amann (1995, II.Section 2.1). In particular,
is, for given \(\sigma \in [0,a_m)\) and \(v^0\in L_p(\Omega )\), the unique solution
to the Cauchy problem
The contraction properties
are valid, where
Recall the interpolation relations
with real interpolation functor \((\cdot ,\cdot )_{\theta ,p}\) and
with complex interpolation functor \([\cdot ,\cdot ]_{1/2}\) (see Triebel (1978, 4.4.3/Theorem)). From Amann (1995, II. Lemma 5.1.3) and the embedding \(W_{p,\mathcal {B}}^{2\theta }(\Omega )\hookrightarrow L_q(\Omega )\) for \(\theta =\frac{n}{2}(\frac{1}{p}-\frac{1}{q})\) we then infer parabolic regularizing properties in the sense that, given \(0\le \vartheta \le \theta \le 1\) with \(2\vartheta , 2\theta \notin \{\delta +\frac{1}{p}\}\) and \(1<p\le q\le \infty \), there are \(\varpi \in \mathbb {R}\) and \(M\ge 1\) such that
and
Let \(p\in \big (\max \{\frac{3n}{4},2\},\infty \big )\) and let \(S_0\in L_p(\Omega )\) and \(I_0\in L_1(J,L_p(\Omega ))\) be fixed in the following.
3.2 Existence of a unique maximal solution
Given \(S\in L_p(\Omega )\) and \(I\in L_1(J,L_p(\Omega ))\) we use the abbreviations (dropping x-dependence for simplicity)
and
where as, for time-dependent functions
it is convenient to abbreviate
Then (1.1) can be written compactly as
subject to the initial conditions
Solutions S to (3.6a) are of the form
while integrating (3.6b) subject to (3.6c) formally along characteristics (and recalling the properties of the evolution operator \(U_A\)) yields a solution I in the form
Given \(T>0\) we introduce the Banach space
and define
Then, fixed points (S, I) of \(\mathcal {Y}\) correspond to solutions of (3.6). In order to prove that \(\mathcal {Y}\) has a fixed point we first note:
Lemma 3.1
For \(q=p/2\), the mappings
are uniformly Lipschitz continuous on bounded sets. Moreover, if \(2\theta >n/p\), then there is \(\alpha >0\) such that
are uniformly Lipschitz continuous on bounded sets.
Proof
The statements readily follow from the regularity assumptions (2.1) and the fact that pointwise multiplications
are continuous for some \(\alpha >0\), see Amann (1991, Theorem 4.1). \(\square \)
Proposition 3.2
Given \(R>0\) there is \(T=T(R)>0\) such that, if
then
has a unique fixed point (S, I).
Proof
Let \(T\in (0,1)\) and \(\Vert S_0\Vert _{L_p(\Omega )}+\Vert I_0\Vert _{L_1(J,L_p(\Omega ))}<R\). Considering \((S,I), ({\tilde{S}},{\tilde{I}})\in {\mathbb {X}_T}\) both with norm less than R, we have
by Lemma 3.1 so that we readily obtain that \(\mathcal {S}[S,I]\in C\big ([0,T],L_{p}(\Omega )\big )\) by (3.4) since \(t\mapsto t^{-n/2p} \) is integrable on (0, T) as \(2p>n\). Moreover,
and
From (3.3)–(3.5) and Lemma 3.1 we inferFootnote 1
for \(t\in [0,T]\), and similarly
To check continuity we use (3.3)–(3.5) together with Lemma 3.1 and write, for \(0\le t_2\le t_1\le T\),
Now, as \(\vert t_1-t_2\vert \rightarrow 0\), the first integral on the right-hand side goes to zero since the function \(B[S,I]\in C\big ([0,T],L_{p/2}(\Omega )\big )\) is uniformly continuous while the second and the third integral vanish since \(a\mapsto a^{-n/2p}\) respectively \(I_0\) are integrable. To see that the fourth integral vanishes in the limit one may use the strong continuity of the evolution operator \(U_A\) on \(L_p(\Omega )\) (Amann 1995, Equation II. (2.1.2)) and Lebesgue’s theorem. Finally, for the last integral one may use the strong continuity of the translations on \(L_1(J,L_p(\Omega ))\). Consequently, \(\mathcal {I}[S,I]\in C\big ([0,T],L_1(J,L_p(\Omega ))\big )\).
Summarizing, we have shown in (3.9)-(3.12) that, given
we can choose \(T=T(R)\in (0,1)\) such that
is a contraction, and the claim follows from Banach’s fixed point theorem. \(\square \)
Since \(T=T(R)\) in the proof of Proposition 3.2 depends only upon
it is standard to extend (S, I) to a maximal solution and to show that the solution map defines a semiflow:
Corollary 3.3
(S, I) can be extended to a maximal interval \([0,T_m)\) such that
satisfies
and
If \(T_m<\infty \), then
Moreover, the mapping \(\big (t,(S_0,I_0)\big )\mapsto (S,I)(t)\) defines a semiflow on the space \(L_p(\Omega )\times L_1(J,L_p(\Omega ))\).
Remark 3.4
It is worth noting that Corollary 3.3 remains valid for models that can be recast in the form (3.6) such that f and B satisfy (3.7) with \(\frac{n}{2}(\frac{1}{q}-\frac{1}{p})<1\) for some \(1<q\le p<\infty \).
3.3 Regularity
We derive further regularity properties of the solution (S, I) (it is for this step that we have imposed restrictive regularity assumptions on the data b and r).
Proposition 3.5
Let \(2\vartheta \in [0,2]\setminus \{\delta +\frac{1}{p}\}\). If \(S_0\in W_{p,\mathcal {B}}^{2\vartheta }(\Omega )\), then
is a strong solution to (1.1a) while, if \(I_0\in L_1(J,W_{p,\mathcal {B}}^{2\vartheta }(\Omega ))\), then
satisfies (1.1b) in the sense that
for \(t\in (0,T_m)\) and a.e. \(a\in (0,a_m)\).
Proof
Since
and \(f[S,I]\in C\big ([0,T_m),L_{p/2}(\Omega )\big )\), it readily follows from (3.13) that \(S\in C\big ((0,T_m),W_{p,\mathcal {B}}^{2\theta }(\Omega )\big )\) for \(2\theta <2-n/p\) with \(2\theta \notin \{\delta +1/p\}\). Similarly, as in the proof of Proposition 3.2 (see also the proof of Lemma 7.1 in the Appendix) one derives from
and \(B[S,I]\in C\big ([0,T_m),L_1(J,L_{p/2}(\Omega ))\big )\) that \(I\in C\big ((0,T_m),L_1(J,W_{p,\mathcal {B}}^{2\theta }(\Omega ))\big )\) for \(2\theta <2-n/p\) with \(2\theta \notin \{\delta +1/p\}\). Now, since \(n<4p/3\), we find some \(2\theta \in (n/2p,2-n/p)\setminus \{\delta +1/p\}\) so that, according to Lemma 3.1, there is \(\alpha >0\) such that
for each \(\varepsilon >0\) small. Thus, we infer from Amann (1995, II.Theorem 1.2.2) that
is a strong solution to
Letting then \(\varepsilon \) tend to zero we obtain that
is a strong solution to (1.1a). Moreover, if \(S_0\in W_{p,\mathcal {B}}^{2\vartheta }(\Omega )\) for some \(2\vartheta \in [0,2]{\setminus }\{\delta +1/p\}\), then
Similarly, setting
we deduce from (3.14) and the properties of evolution operators that, for \(t\in [0,T_m-\varepsilon )\) and \(a\in J\),
Now, since
for \(2\theta <2-n/p\), it follows from (3.5) (see Lemma 7.1 in the Appendix) that
In addition, if \(I_0\in L_1(J,W_{p,\mathcal {B}}^{2\vartheta }(\Omega ))\) for some \(2\vartheta \in [0,2]\setminus \{\delta +1/p\}\), then
Moreover, (3.14) and the differentiability properties of the evolution operator \(U_A\) stated in Amann (1995, II.Equation (2.1.6)) imply
for \(t\in (0,T_m)\) and a.e. \(a\in (0,a_m)\) (in fact, for every \(a\in (0,a_m)\) provided \(I_0\in C\big ((0,a_m), L_p(\Omega )\big )\) is continuous). \(\square \)
Note that taking \(2\vartheta =0\) in Proposition 3.5 we obtain the regularity of the solution (S, I) claimed in Theorem 2.1.
Remark 3.6
Assuming additionally \(b\in BC^1(J,C({{\bar{\Omega }}}))\) and \(I_0\in C^1(J,L_p(\Omega ))\), one can show analogously to Walker (2010, Proposition 1) that the partial derivatives \(\partial _t I(t,a)\) and \(\partial _a I(t,a)\) exist and
in \(L_p(\Omega )\) for \(t\in (0,T_m)\) and \(a\in (0,a_m)\).
3.4 Positivity
Since the semigroup \(\big (e^{t\Delta _\mathcal {B}}\big )_{t\ge 0}\) and the evolution operator \(\big (U_A(a,\sigma )\big )_{0\le \sigma \le a\le a_m}\) are positive operators on \(L_p(\Omega )\) (as well as on the spaces \(W_{p,\mathcal {B}}^{2\theta }(\Omega )\)) and since there is \(\omega (R)>0\) such that
provided that \(S,I\ge 0\) with \(\Vert (S,I)\Vert _{L_\infty (\Omega )\times L_1(J,L_\infty (\Omega ))}\le R\) (see Sect. 3.3 for such local bounds), it is a standard iteration argument to derive that the solution (S, I) from Corollary 3.3 corresponding to non-negative initial values \(S_0\in L_p^+(\Omega )\) and \(I_0\in L_1(J,L_p^+(\Omega ))\) satisfies \({S}(t)\in {L}_{p}^+(\Omega )\) and \({{I}}({{t}})\in {{L}}_1({{J,L}}_{{p}}^+(\Omega ))\) for \(t\in [0,T_m)\).
3.5 Global existence
Integrating (1.1) yields for \(t\in (0,T_m)\) the inequality (in fact, equality for Neumann boundary conditions, see Sect. 1 in the Appendix for a rigorous proof)
Since
we thus deduce from the positivity of (S, I) the \(L_1\)-estimate
for \(t\in (0,T_m)\). We shall then proceed with the following auxiliary result:
Lemma 3.7
(i) Let \(1\le q\le r\le \infty \) with \(\frac{n}{2}(\frac{1}{q}-\frac{1}{r})<1\). If
then
(ii) Let \(1\le r\le \infty \) with \(\frac{n}{2r}<1\). If
then
Proof
(i) By (3.13) we have
and therefore, using (3.17) and
we deduce from \(\Vert I(t)\Vert _{L_1(J,L_q(\Omega ))}\le c_0(T)\) for \(t\in [0,T]\) that
for \(t\in [0,T]\). This proves (i).
(ii) Set \(\frac{1}{q}:=\frac{1}{r}+\frac{1}{p}\) so that
Then we infer for \(t\in [0,T]\) from (3.14), (3.3), and (3.5) that
with \(\frac{n}{2r}<1\) whenever \(\Vert S(t)\Vert _{L_r(\Omega )}\le c_0(T)\) for \( t\in [0,T]\). Hence, Gronwall’s inequality implies
as claimed. \(\square \)
Now, since
by (3.18), we deduce from Lemma 3.7 (i) that
for \(n/2<r<n/(n-2)\) and hence
due to Lemma 3.7 (ii). Taking \(r=q=p\) in Lemma 3.7 (i) yields now
Consequently, \(T_m=\infty \) according to (3.15). This completes the proof of Theorem 2.1.
4 Linearized stability of steady states
We linearize (1.1) around a steady state and then derive properties of the associated linear semigroup. This allows us to introduce the notion of linear stability.
Throughout this chapter we assume (2.1) and fix an arbitrary steady state \((S_*,I_*)\) to (1.1) with regularity
for some \(p>( 2\vee n)\).
4.1 Linearization around steady states
Linearizing (1.1) around the steady state \((S_*,I_*)\) yields the problem
for \((t,a,x)\in \mathbb {R}^+\times [0,a_m]\times \Omega \), and subject to the initial conditions
and boundary conditions
Introducing
and setting
it follows
and \(A_1^*\) with domain \(W_{p,\mathcal {B}}^2(\Omega )\) generates a positive, compact, analytic semigroup \((e^{tA_1^*})_{t\ge 0}\) on \(L_p(\Omega )\) while the operator family A(a) with domain \(W_{p,\mathcal {B}}^2(\Omega )\) generates a positive parabolic evolution operator \((U_{A}(a,\sigma ))_{0\le \sigma \le a\le a_m}\) on \(L_p(\Omega )\). With this notation we can recast the linearization (4.2) as an equation in \(L_p(\Omega )\times L_1(J,L_p(\Omega ))\) of the form
subject to
Following Walker (2021) we next show that the solutions to (4.4) are given by a strongly continuous semigroup on the phase space \(L_p(\Omega )\times L_1(J,L_p(\Omega ))\).
4.2 The semigroup associated with the linearization (4.4)
In order to investigate the properties of the semigroup generated by the linearization (4.4) we write
and use a perturbation argument, first focusing on the diagonal part. In the following, we will require information on the operator family \(S_*Q^\lambda \) with
where
It follows from (3.5) and (2.1) that
where the compact embedding of \(W_{p,\mathcal {B}}^{1}(\Omega )\) into \(L_p(\Omega )\) ensures the compactness of the operator \(S_*Q^\lambda \) on \(L_p(\Omega )\). Moreover, for \(\lambda \in \mathbb {R}\), the operator \(S_*Q^\lambda \in \mathcal {L}(L_p(\Omega ))\) is an irreducible operator for \(p>n\) (Daners and Koch Medina (1992, Corollary 13.6)). Its spectral radius is thus characterized by the Krein-Rutman Theorem. We cite the following result in this context:
Lemma 4.1
(Walker (2013, Lemma 2.4, Lemma 2.5)) For \(\lambda \in \mathbb {R}\), the spectral radius \(r(S_*Q^\lambda )\) is positive and a simple eigenvalue of \(\hbox {S}_*{{Q}}^\lambda \in \mathcal {K}({{L}}_{{p}}(\Omega ))\) with an eigenvector \(\zeta _\lambda \in W_{p,\mathcal {B}}^1(\Omega )\) that is quasi-interior in \(L_p^+(\Omega )\). It is the only eigenvalue of \(S_*Q^\lambda \) with a positive eigenvector. The mapping
is continuous and strictly decreasing with
Now, in order to introduce the semigroup associated with (4.4), we recall from Walker (2021, Lemma 5.1) that there exists a mapping
such that \(B=B_{(S_0,I_0)}\) is for given \((S_0,I_0)\in L_p(\Omega )\times L_1(J,L_p(\Omega ))\) the unique solution to the Volterra equationFootnote 2
for \(t\ge 0\). If \((S_0,I_0)\in L_p^+(\Omega )\times L_1^+(J,L_p(\Omega ))\), then \(B_{(S_0,I_0)}(t)\in L_p^+(\Omega )\) for \(t\ge 0\). Now, given \((S_0,I_0)\in L_p(\Omega )\times L_1(J,L_p(\Omega ))\), define
and
Then \((\mathbb {T}_*(t))_{t\ge 0}\) defines a strongly continuous semigroup on \(L_p(\Omega )\times L_1(J,L_p(\Omega ))\):
Theorem 4.2
Suppose (2.1) and let \((S_*,I_*)\) be a steady state to (1.1) satisfying (4.1). Define \((\mathbb {T}_*(t))_{t\ge 0}\) on \(L_p(\Omega )\times L_1(J,L_p(\Omega ))\) with \(p>n\) according to (4.7).
(a) \((\mathbb {T}_*(t))_{t\ge 0}\) is a strongly continuous, eventually compact, positive semigroup on the space \( L_p(\Omega )\times L_1(J,L_p(\Omega ))\).
(b) Let \(\mathbb {A}_*\) be the infinitesimal generator of the semigroup \((\mathbb {T}_*(t))_{t\ge 0}\). Then \((\phi ,\psi )\in \textrm{dom}(\mathbb {A}_*)\) if and only if \((\phi ,\psi )\in W_{p,\mathcal {B}}^{2}(\Omega )\times C(J,L_p(\Omega ))\) and there exists \(\zeta \in L_1(J,L_p(\Omega ))\) such that \(\psi \) is the mild solution to
In this case,
(c) \(\mathbb {A}_*\) has compact resolvent.
(d) The spectral bound \(s(\mathbb {A}_*)\) and the growth bound \(\omega _0(\mathbb {T}_*)\) are equal.
Proof
(a) Semigroup: One may follow the lines of the proof of Webb (2008, Theorem 4) to show that \((\mathbb {T}_*(t))_{t\ge 0}\) defines a strongly continuous positive semigroup on \(L_p(\Omega )\times L_1(J,L_p(\Omega ))\) (see also Walker 2013, 2021). That this semigroup is eventually compact follows as in Walker (2021, Theorem 1.2 (a)) invoking Kolmogorov’s compactness criterion and using the compact embedding of \(W_{p,\mathcal {B}}^2(\Omega )\) into \(L_p(\Omega )\).
(b) Generator: The proof of the characterization of the generator \(\mathbb {A}_*\) is mostly along the lines of the proof of Walker (2021, Theorem 1.4 (a)), though a bit tricky. We provide some details here. The key is to derive a description of the resolvent \((\lambda -\mathbb {A}_*)^{-1}\).
(i) To this end, we fix \(\lambda >0\) large enough (in particular in the resolvent set of \(\mathbb {A}_*\)) and write
for \((S_0,I_0)\in L_p(\Omega )\times L_1(J,L_p(\Omega ))\). Then, using the Laplace transform formula
and recalling (4.7c), we readily obtain
and using (4.7b), for \(a\in J\),
Invoking (4.9), (4.7a), and (4.5) we derive in particular that
Summarizing, we have shown that if \((S_0,I_0)\in L_p(\Omega )\times L_1(J,L_p(\Omega ))\) and
for \(\lambda >0\) large enough, then
and \(\psi \in C(J,L_p(\Omega ))\) is a mild solution to
subject to
(ii) Consider now an arbitrary \((\phi ,\psi )\in \textrm{dom}(\mathbb {A}_*)\subset L_p(\Omega )\times L_1(J,L_p(\Omega ))\). Defining (for \(\lambda >0\) large enough)
it readily follows from (4.10) that
while \(\psi \in C(J,L_p(\Omega ))\) is a mild solution to
with \(\zeta :=I_0-\lambda \psi \in L_1(J,L_p(\Omega ))\) and subject to
This is (4.8a), while (4.8b) follows from
(iii) Conversely, consider \((\phi ,\psi )\in W_{p,\mathcal {B}}^{2}(\Omega )\times C(J,L_p(\Omega ))\) with the property that there exists \(\zeta \in L_1(J,L_p(\Omega ))\) such that \(\psi \) is the mild solution to
Thus, for \(\lambda >0\) large enough and
we see that \(\psi \in C(J,L_p(\Omega ))\) is the mild solution to
and thus satisfies
Define now
and
Then, according to (4.10),
while \({{\bar{\psi }}}\in C(J,L_p(\Omega ))\) is the mild solution to
subject to
Since \({{\bar{\phi }}}=\phi \), it follows from (4.12) and (4.13) that
and hence \({{\bar{\psi }}}(0)=\psi (0)\) according to Lemma 4.1 for \(\lambda >0\) large enough (so that \(r(S_*Q^\lambda )<1\)). Consequently, \({{\bar{\psi }}}=\psi \) and therefore
This proves part (b).
(c) Compact Resolvent: In order to show that \(\mathbb {A}_*\) has compact resolvent, let \(\lambda >0\) again be sufficiently large (i.e. \(\lambda \) in the resolvent set of \(\mathbb {A}_*\) and \(r(S_*Q^\lambda )<1\)). Let \((S_{0,j},I_{0,j})_{j\in \mathbb {N}}\) be a bounded sequence in \(L_p(\Omega )\times L_1(J,L_p(\Omega ))\) and set
Then (4.10a) yields \(\phi _j=(\lambda -A_1^*)^{-1}S_{0,j}\) so that \((\phi _j)_{j\in \mathbb {N}}\) is a bounded sequence in \(W_{p,\mathcal {B}}^2(\Omega )\), the latter being compactly embedded in \(L_p(\Omega )\). It remains to show that \((\psi _j)_{j\in \mathbb {N}}\) is relatively compact in \(L_1(J,L_p(\Omega ))\) for which we first note from (4.10b)-(4.10c) that
and
In particular, since by (2.1d) and (3.5)
the sequence
is bounded in \(W_{p,\mathcal {B}}^{1}(\Omega )\), and we thus deduce from (4.15), (4.6), and \(r(S_*Q^\lambda )<1\) that
Setting
we next show that \(\{u_j;\, j\in \mathbb {N}\}\) is relatively compact in \(L_1(J,L_p(\Omega ))\) adopting the arguments from Baras et al. (1977) (there the case of semigroups was considered):
(i) We first fix \(\mu >0\) and define
Since
and
we see that, for every \(a\in J\), the sequence \((v_j^\mu (a))_{j\in \mathbb {N}}\) is relatively compact in \(L_p(\Omega )\). Next, in order to check equi-integrability we recall from Amann (1995, II. Equation (2.1.2)) that
and note for \(\xi >0\) that the set
is compact in \(\Delta _J^*\). We thus find for every \(\varepsilon >0\) and \(\xi >0\) some \(\eta >0\) such that
Taking \(\varepsilon >0\) arbitrary and \(\xi =\mu >0\), we use (3.3) to derive, for \(h\in (0,\eta )\) with \(0<a\le a+h\le a_m\),
and therefore
where the tilde refers to the trivial extension. Hence,
so that \(\{v_j^\mu ;\, j\in \mathbb {N}\}\) is equi-integrable and thus
(ii) We next consider the limit \(\mu \rightarrow 0\). Given \(\varepsilon >0\), \(\xi >0\), and using the notation from the previous part we have, for \(a\in J\) with \(a\ge \xi \) and \(0<\mu <\eta \),
while for \(0\le a\le \xi \) we have
Therefore,
so that
Together with (4.17) we conclude that \(\{u_j;\, j\in \mathbb {N}\}\) is relatively compact in \(L_1(J,L_p(\Omega ))\).
(iii) Finally, since
according to Amann (1995, II. Equation (5.3.8)), it readily follows from (4.16) and the Arzelà - Ascoli Theorem that \((U_A^\lambda (\cdot ,0)\psi _j(0))_{j\in \mathbb {N}}\) is relatively compact in \(C(J,L_p(\Omega ))\). Consequently, we deduce from the previous step (ii) and (4.14) that \((\psi _j)_{j\in \mathbb {N}}\) is relatively compact in \(L_1(J,L_p(\Omega ))\). Therefore, \(\mathbb {A}_*\) has indeed a compact resolvent.
(d) Spectral Bound: Since \(\mathbb {T}_*\) is eventually compact, it follows from Engel and Nagel (2000, IV. Corollary 3.12) that \(s(\mathbb {A}_*)=\omega _0(\mathbb {T}_*)\).
\(\square \)
Introducing now the perturbation
with \(P_*\) and N defined in (4.3b) and observing from (4.8) that \(\mathbb {A}_*+\mathbb {B}_*\) is exactly the linearized operator appearing in (4.4a) subject to (4.4b), we obtain the desired generation result for the linearization:
Corollary 4.3
Let the assumptions of Theorem 4.2 be satisfied. Then, the generator \(\mathbb {A}_*+\mathbb {B}_*\) of the strongly continuous semigroup \((\mathbb {S}_*(t))_{t\ge 0}\) on \(L_p(\Omega )\times L_1(J,L_p(\Omega ))\), associated with the linearized problem (4.4), has compact resolvent. In particular, the spectrum \(\sigma (\mathbb {A}_*+\mathbb {B}_*)\) is a pure point spectrum without finite accumulation point.
Proof
This follows from Theorem 4.2 and Engel and Nagel (2000, III. Proposition 1.12). \(\square \)
Definition 4.4
We call the steady state \((S_*,I_*)\) linearly stable in the space \(L_p(\Omega )\times L_1(J,L_p(\Omega ))\), if the generator \(\mathbb {A}_*+\mathbb {B}_*\) of the linearized problem satisfies
while we call \((S_*,I_*)\) linearly unstable if
Remark 4.5
Corollary 4.3 implies that the linear stability of the steady state \((S_*,I_*)\) is determined from (the real parts of) those \(\lambda \in \mathbb {C}\) for which there is a nontrivial \(\displaystyle ({{ S,I}})\in {{ W}}_{p,\mathcal {B}}^{2}(\Omega )\times {{ C(J,L}}_p(\Omega ))\) satisfying
(equality for the second component in the sense of Theorem 4.2) with notation introduced in (4.3).
Remark 4.6
Since the semigroup \((e^{t\mathbb {A}_*})_{t\ge 0}\) is eventually compact and due to the particular (nonlocal) form of the perturbation \(\mathbb {B}_*\), one can in fact show that the perturbation semigroup \(\mathbb {S}_*=(e^{t(\mathbb {A}_*+\mathbb {B}_*)})_{t\ge 0}\) is also eventually compact. Consequently,
that is, the growth bound of the semigroup \(\mathbb {S}_*\) and the spectral bound of its generator \(\mathbb {A}_*+\mathbb {B}_*\) coincide. A steady state \((S_*,I_*)\) is thus linearly stable in the sense of Definition 4.4 if and only if the semigroup \(\mathbb {S}_*\) associated with the linearization around this steady state has an exponential decay. One can further prove that this indeed implies the asymptotic stability of the steady state. The technical details of the proof follow along the lines of Walker (2023) (see also Walker and Zehetbauer (2022)).
5 Linearized stability: Proof of Theorem 2.3
We shall apply the results from the previous section. In the following, we are still imposing (2.1) with \(p>(2\vee n)\) and assume for simplicity (2.4). Recall that \(\kappa _1, \kappa _2>0\). We consider only steady states \((S_*,I_*)\) to (1.1) with regularity as in (4.1).
For the investigation of stability of steady states recall that the spectrum of the Laplacian is (counted according to multiplicity)
with \(0\le \mu _i\le \mu _{i+1}\) for \(i\ge 0\). In fact, \(\mu _0>0\) in the Dirichlet case \(\delta =0\) and \(\mu _0=0\) in the Neumann case \(\delta =1\).
5.1 The trivial steady state
The linear stability of the trivial steady state \((S_*,I_*)=(0,0)\) depends on the sign of \(\kappa _1-\mu _0\) (which is always positive for the Neumann case \(\delta =1\) but may be negative in the Dirichlet case \(\delta =0\)):
Proposition 5.1
The trivial steady state \((S_*,I_*)=(0,0)\) is linearly unstable if \(\kappa _1>\mu _0\) and linearly stable if \(\kappa _1<\mu _0\).
Proof
Using the notation introduced in (4.3) for \((S_*,I_*)=(0,0)\), we have
so that Remark 4.5 leads to investigating the eigenvalue problem
for a nontrivial \((S,I)\in W_{p,\mathcal {B}}^{2}(\Omega )\times C(J,L_p(\Omega ))\); that is,
with mild solution I. Hence \(I=0\) so that \(S\not =0\) and thus \(\kappa _1-\lambda \in \sigma (-\Delta _\mathcal {B})\). Consequently, \(s(\mathbb {A}_*)=\kappa _1-\mu _0\) is an eigenvalue. \(\square \)
5.2 The disease-free steady state
The existence of a disease-free steady state \((S_*,I_*)=( \tilde{S}_*,0)\) reduces to finding a positive non-trivial solution \( \tilde{S}_*\in W_{p,\mathcal {B}}^2(\Omega )\) to the semilinear equation
Clearly, in the Neumann case \(\delta =1\), the positive (constant) solution to (5.1) is \({\tilde{S}}_*=\kappa _2\).
To continue let us recall the following result from Amann (2005, Theorem 12, Theorem 16):
Lemma 5.2
Let \(q\in L_\infty (\Omega )\). Then the eigenvalue problem
has a smallest eigenvalue \(\lambda =\lambda _0(q)\in \mathbb {R}\) (in the sense that \(\textrm{Re}\, \lambda >\lambda _0(q)\) for every other eigenvalue \(\lambda \)). This principal eigenvalue is simple and the only eigenvalue with a positive eigenfunction \(u_0\), i.e. \(u_0\in W_{p,\mathcal {B}}^2(\Omega )\) for every \(p\in (1,\infty )\) and \(u_0>0\) in \(\Omega \). Moreover, if \(q_1,q_2\in L_\infty (\Omega )\) with \(q_1\le q_2\) and \(q_1\not \equiv q_2\), then \(\lambda _0(q_1)<\lambda _0( q_2)\). In fact, \(\lambda _0(0)=\mu _0\).
Now, if \({\tilde{S}}_*\in W_{p,\mathcal {B}}^2(\Omega ) \) is a positive non-trivial solution to (5.1), then necessarily
Conversely, it follows from Blat and Brown (1986) that if \(\kappa _1>\lambda _0(0)=\mu _0\), then (5.1) admits a positive solution \({\tilde{S}}_*\in W_{p,\mathcal {B}}^2(\Omega )\) (see also Walker (2011)). This solution is unique. Indeed, if there was another positive solution \({\hat{S}}_*\in W_{p,\mathcal {B}}^2(\Omega )\) to (5.1), then \(z:={\tilde{S}}_*-{\hat{S}}_*\) solves the eigenvalue equation
so that
According to (5.2) and the monotonicity of \(\lambda _0(q)\) with respect to q this yields the contradiction
The linear stability of the disease-free steady state \((S_*,I_*)=({\tilde{S}}_*,0)\) is then determined from the value of
where the family of compact operators
was introduced in (4.5) and properties of the spectral radius \(r({\tilde{S}}_*Q^\lambda )\) are stated in Lemma 4.1 for \(\lambda \in \mathbb {R}\).
Proposition 5.3
There is a disease-free steady state \((S_*,I_*)=({\tilde{S}}_*,0)\) with a smooth function \({\tilde{S}}_*>0\) if and only if \(\kappa _1>\mu _0\). In this case, \({\tilde{S}}_*\) is unique, (5.2) holds, and \({\textsf{R}}_0>0\) in (5.3a) is well-defined. For Neumann boundary conditions (i.e. \(\delta =1\)), we have \({\tilde{S}}_*=\kappa _2\).
Moreover, \((S_*,I_*)=({\tilde{S}}_*,0)\) is linearly stable in \({{L}}_{{p}}(\Omega )\times {{L}}_1({{J,L}}_{{p}}(\Omega ))\) if \({\textsf{R}}_0<1\) and linearly unstable if \({\textsf{R}}_0>1\).
Proof
We have already shown that there is a (unique) disease-free steady state \((S_*,I_*)=({\tilde{S}}_*,0)\) with \({\tilde{S}}_*>0\) (satisfying (5.2)) if and only if \(\kappa _1>\mu _0\). Thus, let \(\kappa _1>\mu _0\). According to Remark 4.5 we have to check the real parts of solutions \(\lambda \) to the eigenvalue problem
with a nontrivial \((S,I)\in W_{p,\mathcal {B}}^{2}(\Omega )\times C(J,L_p(\Omega ))\), where, due to (4.3),
That is, we have to investigate
where (5.5) entails
Assume first that \({\textsf{R}}_0<1\). Then either \(I(0)=0\) so that (5.4), the monotonicity of the principal eigenvalue, and (5.2) imply
hence \(\textrm{Re}\,\lambda <0\). Or \(I(0)\not =0\) so that (5.6) together with Walker (2021, Theorem 2.3 (b)) imply that \(\lambda \in \sigma (\mathbb {A})\), where \(\mathbb {A}\) is the generator of a strongly continuous, positive, eventually compact semigroup on \(L_1(J,L_p(\Omega ))\). Its spectral bound \(s:=s(\mathbb {A})\) is the unique real number with \(r({\tilde{S}}_*Q^s)=1\) according to Walker (2021, Proposition 5.2). Since \({\textsf{R}}_0=r(\tilde{S}_*Q^0)<1\), it follows from Lemma 4.1 that \(s=s(\mathbb {A})<0\) and hence again \(\textrm{Re}\,\lambda <0\). Consequently, if \({\textsf{R}}_0<1\), then \((S_*,I_*)=({\tilde{S}}_*,0)\) is linearly stable.
Conversely, if \({\textsf{R}}_0=r({\tilde{S}}_*Q^0) >1\), then there is \(\lambda >0\) such that \(r({\tilde{S}}_*Q^\lambda )=1\) by Lemma 4.1 and there is a nontrivial \(I(0)\in W_{p,\mathcal {B}}^1(\Omega )\) with \((1-{\tilde{S}}_*Q^\lambda )I(0)=0\). Then
satisfies (5.5). Moreover, owing to (5.2), we have
and thus \(\kappa _1-\lambda \) belongs to the resolvent set of \(-\Delta _\mathcal {B}+2\kappa _1 {\tilde{S}}_*/\kappa _2\). Hence,
is a nontrivial solution to (5.4). That is, \(\lambda >0\) is an eigenvalue and the disease-free steady state \((S_*,I_*)=(\tilde{S}_*,0)\) is thus linearly unstable. \(\square \)
5.3 Non-existence of endemic steady states for \({\textsf{R}}_0\le 1\)
An endemic steady state \((S_*,I_*)\) is a steady state to (1.1) with \(S_*, I_*\ge 0\) and \(I_*\not \equiv 0\). Note that, setting \(I_0:=I_*(0)\), this is equivalent to finding a positive element \((S_*,I_0)\in W_{p,\mathcal {B}}^2(\Omega )\times W_{p,\mathcal {B}}^1(\Omega )\) with \(I_0\not =0\) satisfying
As shown next, \({\textsf{R}}_0> 1\) is a necessary condition for the existence of an endemic state.
Lemma 5.4
Let \(\kappa _1>\mu _0\) and let \({\tilde{S}}_*\) be as in Proposition 5.3. If \({\textsf{R}}_0=r({\tilde{S}}_*Q^0)\le 1\), then there is no positive solution \((S_*,I_0)\in W_{p,\mathcal {B}}^2(\Omega )\times W_{p,\mathcal {B}}^1(\Omega )\) to (5.7) with \(I_0\not =0\).
Proof
Let \({\textsf{R}}_0=r({\tilde{S}}_*Q^0)\le 1\) and assume for contradiction that there was a positive solution \((S_*,I_0)\in W_{p,\mathcal {B}}^2(\Omega )\times W_{p,\mathcal {B}}^1(\Omega )\) to (5.7) with \(I_0\not =0\). It follows from (5.1) and (5.7a) that \(z:=S_*-{\tilde{S}}_*\) solves
Moreover, we infer from (5.7a) and Lemma 5.2 that
Hence, \(\kappa _1\) belongs to the resolvent set of the operator
and consequently, since \(I_0>0\),
where we used the maximum principle from Amann (2005, Theorem 13). That is, \(S_*\le {\tilde{S}}_*\) in \(\Omega \) and thus \(S_*Q^0\le {\tilde{S}}_*Q^0\). The Krein-Rutman Theorem now yields for the spectral radii
and therefore that the eigenvector equation \(p=S_*Q^0p\) has no positive nontrivial solution in contradiction to \(I_0>0\) solving (5.7b). \(\square \)
Remarks 5.5
(a) As noted in the proof of Lemma 5.4, necessary conditions for the existence of an endemic steady state \((S_*,I_*)\) with \(S_*, I_*\ge 0\) and \(I_*\not \equiv 0\) are (see (5.7))
where \(({\tilde{S}}_*,0)\) is the disease-free steady state.
(b) Whether the condition \({\textsf{R}}_0>1\) is sufficient for the existence of an endemic steady state is, however, left open. In fact, one can show (see Blat and Brown 1986) that for every \(I_0\in L_p^+(\Omega )\) with \(\lambda _0(Q^0I_0)<\kappa _1\) there is a unique solution \(S_*=S_*(I_0)\in W_{p,\mathcal {B}}^2(\Omega )\) to (5.7a) with \(S_*(I_0)>0\) depending compactly and smoothly on \(I_0\) (by the implicit function theorem), where \( \tilde{S}_*=S_*(0)\). This then reduces problem (5.7) to finding a nontrivial positive fixed point of the smooth, compact operator F defined by \(F(I_0):=S_*(I_0) Q^0 I_0\). Noticing that \(DF(0)=\tilde{S}_* Q^0\) has a positive eigenvector associated with the eigenvalue \({\textsf{R}}_0>1\) by the Krein-Rutman Theorem, it would remain to find \(\rho >0\) such that \(\lambda _0(Q^0I_0)<\kappa _1\) for \(\Vert I_0\Vert _{L_p}\le \rho \) and \(r(S_*(I_0)Q^0)<1\) when \(\Vert I_0\Vert _{L_p}= \rho \). This then would allow one to apply the fixed point theorem of Amann (1976, Theorem 13.2) to derive the existence of a (unique) positive nontrivial solution \(I_0=F(I_0)\) and thus an endemic steady state \((S_*(I_0),I_*)\) with \(I_*(a):=U_A(a,0)I_0\). However, it is open whether this is indeed possible.
Nevertheless, when considering spatially homogeneous rates and Neumann boundary conditions, there exists a linearly stable endemic state if \({\textsf{R}}_0>1\) as stated in Theorem 2.4.
6 Linearized stability in a particular model: Proof of Theorem 2.4
For Neumann boundary conditions \(\delta =1\) and spatially homogeneous rates \(m=m(a)\) and \(b=b(a)\) we can improve the results from the previous section. Thus assume (2.4), (2.5), and \(p>(2\vee n)\). Recall that the principal eigenvalue of the Laplacian \(-\Delta _N\) subject to Neumann boundary conditions is \(\mu _0=0\). We write \(W_{p,N}^2(\Omega )\) in the following for the domain of \(-\Delta _N\). Since
and m is spatially homogeneous, the corresponding evolution operator is given by
In the previous section we showed that the trivial steady state \((S_*,I_*)=(0,0)\) is linearly unstable, and we have discussed the local linear stability of the disease-free steady state \((\tilde{S}_*,0)=(\kappa _2,0)\). We next investigate the latter’s global stability.
6.1 Global stability of the disease-free steady state \((S_*,I_*)=(\kappa _2,0)\)
We consider \(\hbox {S}_{*}={\tilde{S}}_*=\kappa _{2}\). Using (6.1), the operators \(Q^\lambda \) from (5.3b) become
where
Noticing
it readily follows from Lemma 4.1 for the spectral radius (see (2.6)) that
We have seen in Proposition 5.3 that \((S_*,I_*)=(\kappa _2,0)\) is linearly stable when \({\textsf{R}}_0 <1\) and linearly unstable when \({\textsf{R}}_0 >1\). In the former case, we can prove now its global stability. That is, if \({\textsf{R}}_0 <1\), then any solution to (1.1) subject to positive initial values \((S_0,I_0)\) converges to \((S_*,I_*)=(\kappa _2,0)\).
Proposition 6.1
Assume (2.4) and (2.5). Let \({\textsf{R}}_0 <1\). Consider any non-trivial \((S_0,I_0)\in L_p^+(\Omega )\times L_1^+(J, L_p\Omega ))\) and let (S, I) be the corresponding positive global solution to (1.1) provided by Theorem 2.1. Then
Proof
Since solutions become immediately smooth according to Theorem 2.1, we may restrict without loss of generality to initial values
(i) We first derive an upper bound on S. From (1.1a) we have
so that
by the comparison principle, where
is the solution to
Due to \({\textsf{R}}_0 <1\) and (6.2) we may choose \(\varepsilon _0>0\) such that
Since \(\lim _{t\rightarrow \infty }z(t)=\kappa _2\), we find for fixed \(\varepsilon \in (0,\varepsilon _0)\) some \(t_0>0\) with
(ii) Next, we derive an upper bound for I. Using (6.5) and (1.1b)-(1.1c) we obtain
subject to
Let G solve (see Walker (2021))
We claim that
Indeed, setting \(w(t,a):=G(t,a)-I(t+t_0,a)\), we have, for \(t\ge 0\) and \(a\in J\),
and therefore
with
Introducing for \(t\in [0,T]\) and \(B\in C([0,T],L_p(\Omega ))\)
it follows that \(\mathcal {K}\in \mathcal {L}\big (C([0,T],L_p(\Omega ))\big )\) is a positive compact (Volterra) operator with spectral radius zero (see the proof of Walker (2021, Lemma 5.1)). Therefore,
and consequently
for
Hence \(w\ge 0\) according to (6.8) so that (6.7) is true.
(iii) Next, we claim that
which, according to (6.7), is ensured by showing that
As for (6.10) we fix \(\alpha \in (n/2p,1)\) and note that the linear problem (6.6) with birth rate \((\kappa _2+\varepsilon )b(a)\) fits exactly into the setting of problems investigated in Walker (2021). In fact, since \(G(0)=I(t_0)\in L_1(J,W_{p,N}^{2\alpha }(\Omega ))\), it follows from Walker (2021, Corollary 1.3) that \(G(t)=e^{{t{\hat{\mathbb {A}}}}_{\varepsilon }}G(0)\), \(t\ge 0\), where \((e^{{t{\hat{\mathbb {A}}}}_{\varepsilon }})_{t\ge 0}\) is an eventually compact, positive semigroup on \(L_1(J,W_{p,N}^{2\alpha }(\Omega ))\). Therefore, Engel and Nagel (2000,V. Corollary 3.2) ensures that the spectrum of the generator \({\hat{\mathbb {A}}}_{{\varepsilon }}\) consists of eigenvalues only, while Engel and Nagel (2000, IV. Corollary 3.12) yields that the spectral bound of \({\hat{\mathbb {A}}}_{{\varepsilon }}\) coincides with the type of the semigroup. In fact, the spectral bound \(s_0:=s({\hat{\mathbb {A}}}_{{\varepsilon }})\) is the unique real number with \(r({\hat{Q}}_\varepsilon ^{s_0})=1\) according to Walker (2021, Proposition 5.2), where
Since as in (6.2)
it follows from (6.4) that \(s_0=s({\hat{\mathbb {A}}}_{{\varepsilon }})<0\), and since the spectral bound and the type of the semigroup coincide, we conclude that
as \(t\rightarrow \infty \). Consequently, since \(W_{p,N}^{2\alpha }(\Omega )\hookrightarrow C({{\bar{\Omega }}})\), we deduce (6.10) and therefore, owing to (6.7), also (6.9).
(iv) Finally, we prove that
Given \(\varepsilon \in (0,\kappa _1)\) we infer from (6.9) that there is \(t_1\ge t_0\) with
and hence, due to (1.1a),
Since the strong maximum principle yields for every \(x_0\in \Omega \) some \(\varrho >0\) such that
we obtain \(S(t,x)\ge \xi (t)\) for \(t\ge t_1\) and \(x\in {{\bar{\mathbb {B}}}}(x_0,\varrho )\), where \(\xi \) solves
Therefore,
Letting \(\varepsilon \rightarrow 0\) and invoking (6.5), we derive
Finally, using again the \(L_\infty \)-bound from (6.5) and Lebesgue’s theorem we conclude that \(S(t)\rightarrow \kappa _2\) in \(L_p(\Omega )\) as \(t\rightarrow \infty \). Together with (6.9), this proves Proposition 6.1. \(\square \)
6.2 The endemic steady state \(({{\bar{S}}}_*,{{\bar{I}}}_*)\)
In case that the basic reproduction number satisfies
there is an endemic steady state \(({{\bar{S}}}_*,{{\bar{I}}}_*)\) given by
It is convenient to set \({\textsf{r}}_0:=1/{\textsf{R}}_0 \in (0,1)\). Then, the endemic steady state can be written as
In (4.3) we have
so that
and
In the following we still assume \(p>(2\vee n)\).
Proposition 6.2
Assume (2.4) and (2.5). For \(1<{\textsf{R}}_0 <3\), the endemic steady state \(({{\bar{S}}}_*,{{\bar{I}}}_*)\) to (1.1) is linearly stable in \(L_p(\Omega )\times L_1(J,L_p(\Omega ))\).
Proof
Let \(\lambda \) be a spectral point of the linearization, so that, according to Remark 4.5, there is a nontrivial \((S,I)\in W_{p,N}^{2}(\Omega )\times C(J,L_p(\Omega ))\) with
That is,
From (6.11b) we get
and plugged into (6.11c) this yields
with
In order to verify that \(\textrm{Re}\,\lambda <0\), we assume for contradiction that \(\textrm{Re}\,\lambda \ge 0\). Then \(\lambda +\kappa _1{\textsf{r}}_0 -\Delta _N\) is invertible and we infer from (6.11a) and (6.12) that
Recall that the eigenfunctions \((\phi _j)_{j\in \mathbb {N}}\) of the Neumann-Laplacian, corresponding to the eigenvalues counted according to multiplicity, build an orthonormal basis in \(W_{2,N}^1(\Omega )\). Then \(-\Delta _N\phi _j=\mu _j \phi \) entails \(e^{t\Delta _N}\phi _j=e^{-t\mu _j}\phi _j\) for \(t\ge 0\), and the operator \({{\bar{S}}}_*Q^\lambda \) leaves the eigenfunctions invariant. More precisely, from (6.1) we deduce
for every \(j\in \mathbb {N}\). Note that \(I(0)\in W_{p,N}^1(\Omega )\hookrightarrow W_{2,N}^1(\Omega )\) is nonzero as otherwise also \(S=0\). Hence, writing \(I(0)=\sum _j \xi _j\phi _j\), we derive from the identity (6.14) that
Taking any \(j\in \mathbb {N}\) with \(\xi _j\not =0\), the previous identity leads to the characteristic equation
Owing to \(\mu _j\ge 0\) we have
Clearly, (6.15) has no real solution \(\lambda \ge 0\) since in this case
For an arbitrary \(\lambda \in \mathbb {C}\) with \(\textrm{Re}\,\lambda \ge 0\) we write
and obtain from (6.15) and (6.16) the contradiction
since \({\textsf{r}}_0 -1<0\) and \(2\alpha +3{\textsf{r}}_0 -1\ge 3{\textsf{r}}_0 -1>0\) by our assumption that \(1/3<{\textsf{r}}_0 <1\). Therefore, we conclude that indeed \(\textrm{Re}\,\lambda <0\). Consequently, the endemic steady state \(({{\bar{S}}}_*,{{\bar{I}}}_*)\) is linearly stable when \(1<{\textsf{R}}_0 <3\). \(\square \)
Clearly, one expects \(({{\bar{S}}}_*,{{\bar{I}}}_*)\) to be linearly stable for all \({\textsf{R}}_0 >1\).
Notes
Here and in the following, if \(t>a_m\), then integrals \(\int _0^t\textrm{d}a\) equal \(\int _0^{a_m}\textrm{d}a\) and integrals \(\int _0^{a_m-t}\textrm{d}a\) vanish.
We recall again that for \(t>a_m\), integral \(\int _0^t\textrm{d}a\) equal \(\int _0^{a_m}\textrm{d}a\) and integrals \(\int _0^{a_m-t}\textrm{d}a\) vanish.
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I thank Lina Sophie Schmitz for helpful discussions on the topic.
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Appendix
Appendix
1.1 Regularity of I
We provide the missing step from the proof of Proposition 3.5.
Lemma 7.1
For \(\varepsilon >0\) small and some \(n/2p<2\theta <2-n/p\), let
be as in the proof of Proposition 3.5. Then
Proof
We proceed analogously to the proof of Proposition 3.2. According to Lemma 3.1 there is \(\alpha >0\) such that
and we obtain from (3.5), for \(0<t_2\le t_1\le T<T_m-\varepsilon \),
Now, as \(\vert t_1-t_2\vert \rightarrow 0\), the first integral on the right-hand side goes to zero since the function \(B[S_\varepsilon ,I_\varepsilon ]\in C([0,T],W_{p,\mathcal {B}}^{2\alpha }(\Omega ))\) is uniformly continuous while the second and the third integral vanish since \(a\mapsto a^{\alpha -1}\) respectively \(I_{0,\varepsilon }\) are integrable. To see that the fourth integral vanishes in the limit one may use the strong continuity (Amann 1995, Equation II. (2.1.2)) of the evolution operator \(U_A\) in \(\mathcal {L}\big (W_{p,\mathcal {B}}^{2\theta }(\Omega ),W_{p,\mathcal {B}}^{2}(\Omega )\big )\) and Lebesgue’s theorem. For the last integral one may use the strong continuity of the translations on \(L_1\big (J,W_{p,\mathcal {B}}^{2\theta }(\Omega )\big )\). Consequently, \(I_\varepsilon \in C\big ((0,T_m-\varepsilon ),L_1(J,W_{p,\mathcal {B}}^{2}(\Omega ))\big )\). Letting \(\varepsilon \rightarrow 0\) yields \(I\in C\big ((0,T_m),L_1(J,W_{p,\mathcal {B}}^{2}(\Omega ))\big )\). \(\square \)
1.2 Proof of the \(L_1\)-inequality (3.16)
We derive inequality (3.16). To this end, note from Gauss’ theorem that
since \(\partial _\nu u=0\) on \(\partial \Omega \) if \(\delta =1\) and \(\partial _\nu u\le 0\) on \(\partial \Omega \) if \(\delta =0\). Now, for \(a\in J\) fixed set
Integrating then
with A given in (2.1), we get from (7.1)
and therefore
Similarly, one derives for \(t>a\) that
We then recall (3.14) and use the previous two identities to obtain
Using (3.14) again we may rewrite the previous inequality as
Integrating (1.1c) and using (7.1) yields
Adding (7.2) and (7.3) yields (3.16).
Remark 7.2
When considering Neumann boundary conditions (\(\delta =1\)), then (7.2) and (7.3) are equalities and thus also (3.16).
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Walker, C. Well-posedness and stability analysis of an epidemic model with infection age and spatial diffusion. J. Math. Biol. 87, 52 (2023). https://doi.org/10.1007/s00285-023-01980-y
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DOI: https://doi.org/10.1007/s00285-023-01980-y