Special Issue in honor of Alan C. Lazer Electronic Journal of Differential Equations, Special Issue 01 (2021), pp. 315–325. ISSN: 1072-6691. URL: https://ejde.math.txstate.edu or https://ejde.math.unt.edu SIR EPIDEMIC MODELS WITH SPATIAL SPREAD IN BOUNDED DOMAINS HELMUT KNOLLE, JAIRO SANTANILLA Abstract. The spread of an infectious disease which confers immunity after recovery from infection, can be described by a SIR model, i.e. a system of three differential equations for the dependent variables S, I, and R, which are the numbers (densities) of susceptible, infectious and recovered (immune) individuals, respectively. The equations for S and I are typically nonlinear. In this article, we consider two spatio-temporal SIR models. The first model is similar to reaction-diffusion systems in chemistry. A simple birth-and-death process is incorporated, and it is assumed, that a fraction f of the newborns is vaccinated and is then immune for life. We show how the smallest eigenvalue of the eigenvalue problem associated with the linearized equation for I is related to the basic reproduction number R0, a key concept in the mathematical theory of infectious diseases. Here it is defined by a variational principle. We show that the disease-free equilibrium is asymptotically stable if R0 < 1, or if R0 ≥ 1 and f > 1− 1/R0, and unstable if R0 > 1 and f < 1− 1/R0. In the other model we assume that the population consists of sedentary individuals who leave their home only temporarily. Both models suggest that restriction of mobility may be counterproductive for the control of an epidemic outbreak. 1. Introduction We consider an infectious disease, which is transmitted directly and confers life- long immunity after recovery from infection. For this type of disease, the local spread of infection in a homogeneously mixing population has been described by the equations, dS dt = −βSI, dI dt = βSI − γI, dR dt = γI, where t is time, S the number of susceptibles, I the number of infectives, R the number of immunes [2], β and γ are positive constants. The trivial solution S = N =constant, I = 0, R = 0 is called the disease-free equilibrium (DFE). It is asymptotically stable if Nβ < γ. Otherwise, if Nβ > γ, a small number of infectives can trigger a large epidemic, but in the long run I tends to zero anyway. Things are different, when fresh susceptibles are introduced by births. Therefore, when the long-term behavior of an infectious disease in a population is considered, a birth- and-death process must be incorporated into the model. If we assume constant 2020 Mathematics Subject Classification. 92D30. Key words and phrases. SIR epidemic model; spatial spread; basic reproduction number; mass action; disease-free equilibrium; stability; nonlinear parabolic differential equations. ©2021 This work is licensed under a CC BY 4.0 license. Published March 18, 2022. 315 316 H. KNOLLE, J. SANTANILLA EJDE/SI/01 population size and age-independent birth and death rate, this leads to the model dS dt = µ(S + I +R)− βSI − µS dI dt = βSI − γI − µI dR dt = γI − µR The basic reproduction number is defined as R0 = Nβ/(µ+γ). The DFE is stable if R0 < 1, and unstable if R0 > 1. An endemic equilibrium (EE), i.e., an equilibrium with I > 0 , exists if and only if R0 > 1. The question rises, how the theory must be adapted when we are dealing with a population spread over a large area. Then the variables S, I, and R are time and space-dependent, the parameters β and γ are space-dependent, and the assumption of homogeneous mixing is no longer justified, because infection will be more likely between close neighbors than between persons at greater distance. In most cases, a human population consists of sedentary individuals who move away from their place of residence only temporarily to work, study, or participate in social life. Therefore, we introduce a probability distribution h: R2 → R+ for the probability of an infectious contact in unit time between a person living at place (x, y) and a person living at place (x+ u, y + v), and we define Î(x, y, t) = ∫∫ I(x+ u, y + v, t)h(u, v) du dv . Then the equation for I becomes It = βSÎ − γI − µI . This integro-differential equation can be approximated by a partial differential equa- tion of parabolic type, if I(x + u, y + v, t) as a function of u and v is expanded as a Taylor series up to second degree. Kendall [8] used this approach in his study of a model with only one spatial variable, and he focused on traveling waves, i.e., solutions of the form u(x, t) = u(x− ct), where c is the speed of propagation of an epidemic in space. Diekmann [5] has studied traveling waves of an epidemic in a more general setting. In another class of models, it is assumed that the individuals move at random, like the molecules of a gas. This leads to equations similar to the reaction-diffusion equations in chemistry. Allen et al. [1] consider the following spatial model of a SIS disease, which spreads over an open domain Ω in Rm St = dS∆S − β SI S + I + γI x ∈ Ω, t > 0 It = dI∆I + β SI S + I − γI x ∈ Ω, t > 0 ∂S ∂n = ∂I ∂n = 0 x ∈ ∂Ω, t > 0 . The infection mechanism described by the term βSI/(S + I) means that the in- cidence of infection is independent of the density of the population. This may be the case in sexually transmitted diseases (see e.g. Bailey [3]), but not in airborne infections such as Covid 19. Since the nonlinear term is bounded, Allen et al. can EJDE-2021/SI/01 SIR EPIDEMIC MODELS WITH SPATIAL SPREAD 317 easily prove that for given non-negative initial conditions a unique classical solution exists for all time. Then they prove that a unique DFE exists, and they define R0 = sup { ∫ βφ2∫ [dI | gradφ|2 + γφ2] : φ ∈ H1(Ω), φ 6= 0 } . It turns out, that the DFE is asymptotically stable if R0 < 1, and unstable if R0 > 1. Furthermore, a unique EE exists if R0 > 1. For the same model, Peng [9] studied the properties of equilibria, if one of the diffusion constants or both tend to zero or to infinity. The results show that reducing the mobility of the susceptible individuals can help to eliminate the infection, while reducing the mobility of the infectious individuals cannot. Wu and Zou [10] considered a similar model, in which the infection mechanism is given by the law of mass action, i.e. the incidence of infection is proportional to SI. This model presents a new challenge due to the unboundedness of the nonlinear term. The results reveal some fundamental differences in the conclusions to be drawn from spatial models with different infection mechanisms. 2. New model Assumptions. The model in Wu and Zou [10] is extended by the inclusion of an immune state and a simple birth-and-death process. Each person is either susceptible (healthy but susceptible to infection), infectious or immune. Every newborn is susceptible, but a fraction f of newborns become immune through vaccination. The immunity that is developed after vaccination or after surviving the infection is lifelong. Birth and death rates are the same for all three groups, regardless of age. The benefit of the greater complexity of our model, compared with that in [1, 10], outweighs the restrictions we impose on the parameters and the initial values. We assume a uniform diffusion constant for S, I and R, and a space-independent population density at t = 0. Notation. Ω a bounded, open connected domain in R2 ω area of Ω H1(Ω) Hilbert space of functions u defined in Ω which have square summable first derivatives and satisfy ∂u/∂n = 0, x ∈ ∂Ω S(x, t) density of susceptibles at point x at time t I(x, t) density of infectives at point x at time t R(x, t) density of immunes at point x at time t d constant of diffusion f fraction vaccinated (f < 1) µ birth and death rate N population size β(x) product of frequency of contact and infectivity at point x γ recovery rate. The model consists of the following differential equations: St = (1− f)µ(S + I +R) + d∆S − βSI − µS x ∈ Ω, t > 0 , (2.1) It = d∆I + βSI − γI − µI x ∈ Ω, t > 0 , (2.2) Rt = fµ(S + I +R) + d∆R+ γI − µR x ∈ Ω, t > 0, . (2.3) 318 H. KNOLLE, J. SANTANILLA EJDE/SI/01 Since S, I, and R are densities, the domain of definition of these equations is the closed non-negative orthant of R3. We assume throughout that β(x) ≥ 0, γ > 0, µ > 0, and d > 0. The term βSI in (2.1) and (2.2) describes transmission of the infection according to the law of mass action. The boundary and initial conditions are ∂S ∂n = ∂I ∂n = ∂R ∂n = 0 on ∂Ω, t > 0, (2.4) S(x, 0) = S0(x) ≥ 0, I(x, 0) = I0(x) ≥ 0, (2.5) R(x, 0) = R0(x) ≥ 0 x ∈ Ω . (2.6) Concerning the initial values we assume S0(x) + I0(x) +R0(x) = N/ω, (2.7)∫ Ω I0 > 0 . (2.8) In the following, ∫ means the integral over Ω. Since the nonlinearity in (2.1) and (2.2) is unbounded, the proof of global existence and positivity is more difficult than for the model of Allen et al. [1]. We will apply a theorem by Fellner, Morgan & Tang [6], in which the following notation is used. For ui : Ω×(0, T )→ R, i = 1, 2, . . . ,m, the i-th density and u = (u1, u2, . . . , um), we consider the system ∂tui − di∆ui = gi(x, u), (x, t) ∈ Ω× (0, T ), (2.9) ∇xui · ν = 0, (x, t) ∈ ∂Ω× (0, T ), (2.10) ui(x, 0) = ui0 x ∈ Ω (2.11) where di > 0. The domain Ω, the initial data, and the nonlinearities satisfy the following assumptions. (H1) (Smooth domain) Ω ⊂ Rn is a bounded domain with ∂Ω of Cn class such that Ω lies locally on one side of ∂Ω. (H2) (Bounded, nonnegative initial data) For all i = 1, 2, . . . ,m, ui0 ∈ L∞(Ω) and ui0(x) ≥ 0 for a.e. x ∈ Ω. (H3) (Mass control) There exist K0 ≥ 0 and K1 ∈ R such that m∑ i=0 gi(x, u) ≤ K0 +K1 m∑ i=0 ui for all (x, u) ∈ Ω× Rm+ . (H4) (Local Lipschitz and quasi-positivity) For all i = 1, 2, . . . ,m, gi(x, · ) is locally Lipschitz, gi(·, u) ∈ L∞(Ω) and gi(x, u) ≥ 0 for all (x, u) ∈ Ω× Rm+ satisfying ui = 0. (H5) (Super-quadratic growth) There exist ε > 0 sufficiently small and K > 0 such that |gi(x, u)| ≤ K(1 + |u|2+ε ) for all (x, u) ∈ Ω× Rm, i = 1, 2, . . . ,m. Proposition 2.1 ([6]). Under assumptions (H1)–(H5) there exists a unique non- negative global solution u to (2.9)–(2.11). Theorem 2.2. Let Ω ⊂ R2 be a bounded connected domain with ∂Ω of C2 class such that Ω lies locally on one side of ∂Ω. Assume that β, S0, I0, R0 ∈ L∞(Ω), and (2.8) are satisfied. Then there exists a unique global nonnegative-nontrivial solution of (2.1)–(2.5). EJDE-2021/SI/01 SIR EPIDEMIC MODELS WITH SPATIAL SPREAD 319 Proof. Corresponding to the right-hand side of (2.1)–(2.3), we let u = (u1, u2, u3) and g1(x, u1, u2, u3) = (1− f)µ(u1 + u2 + u3)− β(x)u1u2 − µu1 , g2(x, u1, u2, u3) = β(x)u1u2 − γu2 − µu2 , g3(x, u1, u2, u3) = fµ(u1 + u2 + u3) + γu2 − µu3 , for (u1, u2, u3) ∈ R3 and x ∈ Ω. Assumptions (H1) and (H2) are obviously satisfied. Assumption (H3) is satisfied with K0 ≥ 0 and K1 = 0, since 3∑ i=0 gi(x, u) = 0 for all (x, u) ∈ Ω× R3 + . Since β ∈ L∞(Ω), it follows that gi(· , u) ∈ L∞(Ω). It is also clear that each gi(x, ·) is locally Lipschitz. Furthermore, gi(x, u) ≥ 0 for all (x, u) ∈ Ω× R3 + satisfying ui = 0. Hence assumption (H4) is satisfied. Finally, we show that (H5) is satisfied. (A) We claim that there exists c1 > 0 such that |g1(x, u)| ≤ c1(1 + |u|2+ε ). |g1(x, u1, u2, u3)| = | − fµu1 + (1− f)µ(u2 + u3)− β(x)u1u2| ≤ fµ|u1|+ β(x)|u1||u2|+ (1− f)µ(|u2|+ |u3|) ≤ a1(|u1|+ |u1||u2|+ |u2|+ |u3|), where a1 = max{fµ, ‖β‖∞, (1− f)µ}. Let v = max{|u1|, |u2|, |u3|} and assume v > 1. Then |g1(x, u)| ≤ 3a1(v + v2) ≤ 6a1v 2 ≤ 6a1(1 + |u|2+ε )(ε > 0). Suppose |u1| ≤ 1, |u2| ≤ 1, |u3| ≤ 1, then |g1(x, u)| ≤ a1(|u1|+ |u1||u2|+ |u2|+ |u3|) ≤ 4a1 ≤ 6a1(1 + |u|2+ε ). Then item (A) follows with c1 = 6a1. (B) As in item (A) with a2 = max{‖β‖∞, γ + µ}, there exits c2 > 0 such that |g2(x, u)| ≤ c2(1 + |u|2+ε ). (C] As above with a3 = max{µ, fµ+ γ}, there exits c3 > 0 such that |g3(x, u)| ≤ c3(1 + |u|2+ε ). Finally (H5) is satisfied with K = max{c1, c2, c3}. The fact that the solution is nontrivial, follows from assumption (2.8). � It also follows from our assumptions and the above proposition that S, I, and R are C1 as functions of t, and C2 as functions of x. Theorem 2.3. Every solution of (2.1)–(2.7) satisfies S + I +R = N/ω, x ∈ Ω, t > 0 (2.12) Proof. We define u := S + I + R. Then u satisfies (2.4) and the initial condition u(x, 0) = N/ω, x ∈ Ω. By adding the three differential equations we obtain ut = St + It +Rt = d∆(S + I +R) = d∆u. 320 H. KNOLLE, J. SANTANILLA EJDE/SI/01 The constant N/ω satisfies this equation, as well as the boundary and initial con- ditions. Since the solution of this problem is unique, the solution (S, I,R) of (2.1)–(2.7) satisfies equation (2.12). � Theorem 2.4. There is a unique equilibrium without infection, (Seq, 0, Req), with Seq = (1− f)N/ω and Req = fN/ω. In the proof we will use the following result. Lemma 2.5. The boundary value problem d∆u− cu = 0 x ∈ Ω, (d > 0, c > 0) ∂u ∂n = 0 x ∈ ∂Ω has only the solution u ≡ 0. The proof follows immediately after multiplying the equation by u and integrat- ing over Ω. By Green’s formula and the boundary condition, we have 0 = − ∫ d| gradu|2 dx− ∫ cu2dx ≤ − ∫ cu2 dx which implies u ≡ 0. Proof of Theorem 2.4. With I ≡ 0 and St = Rt = 0 we have (1− f)µ(S +R) + d∆S − µS = 0 x ∈ Ω, (2.13) fµ(S +R) + d∆R− µR = 0 x ∈ Ω . (2.14) Obviously, (Seq, Req) is a solution of (2.13)–(2.14) and (2.4). Equations (2.13)– (2.14) can be written as d∆S − fµS = −(1− f)µR, d∆R− (1− f)µR = −fµS . These nonhomogeneous linear equations with the boundary conditions (2.4) have at most one solution, because the homogeneous equations d∆S − fµS = 0 and d∆R− (1− f)µR = 0 with boundary condition (2.4) have only the trivial solution. Hence (Seq, 0, Req), is the only equilibrium. � 3. Stability and instability of the disease-free equilibrium To study the stability of the equilibrium without infection, following Allen et al. [1], we define R0 = sup { N/ω ∫ βφ2∫ [d| gradφ|2 + (γ + µ)φ2] : φ ∈ H1(Ω), φ 6= 0 } . (3.1) It turns out that stability of the DFE depends on R0 and f . Furthermore, an endemic equilibrium (EE), i.e. an equilibrium, in which ∫ I2 > 0, cannot exist, if R0 < 1. Indeed, when R0 < 1 and ∫ I2 > 0 with I ∈ H1(Ω), applying Green’s formula and (1d), we have 0 = ∫ Id∆I − ∫ (γ + µ)I2 + ∫ βSI2 EJDE-2021/SI/01 SIR EPIDEMIC MODELS WITH SPATIAL SPREAD 321 ≤ − ∫ d grad I|2 − ∫ (γ + µ)I2 +N/ω ∫ βI2 < 0 , a contradiction. Hence there is no EE if R0 < 1. The following classical theorem from calculus of variations [4] is the cornerstone of our stability analysis. Theorem 3.1. The smallest eigenvalue of the problem d∆u− qu+ λpu = 0, in Ω, ∂u ∂n = 0 on ∂Ω, (3.2) where d > 0 is a constant, q, p ∈ C(Ω), and q, p > 0, is λmin = inf {∫ [d| gradu|2 + qu2]∫ pu2 : u ∈ H1(Ω), u 6= 0 } (3.3) and the associated eigenfunction can be chosen to be positive. Corollary 3.2. There exists a u ∈ H1(Ω), u(x) > 0 for x ∈ Ω such that d∆u− (γ + µ)u+ 1 R0 (βN/ω)u = 0, x ∈ Ω ∂u ∂n = 0 on ∂Ω . (3.4) Proof. In (3.2) we set q = γ+µ, p(x) = β(x)N/ω. Then the fraction in (3.3) is the reciprocal of the fraction in the definition of R0 (see (3.1)). Since the supremum of a fraction is equal to the infimum of its reciprocal, we have λmin = 1/R0. Since this is the smallest eigenvalue of (3.2), there is a positive function (an eigenfunction for 1/R0), that satisfies (3.4). � Theorem 3.3. Let λ∗ be the smallest eigenvalue of the problem d∆u− (γ + µ)u+ (1− f)(βN/ω)u+ λu = 0, x ∈ Ω ∂u ∂n = 0 on ∂Ω (3.5) Then λ∗ < 0 if (1− f)R0 > 1, and λ∗ > 0 if (1− f)R0 < 1. Proof. Let w be a positive eigenfunction to the eigenvalue λ∗ of (3.5). Then d∆w − (γ + µ)w + (1− f)(βN/ω)w + λ∗w = 0, x ∈ Ω ∂w ∂n = 0 on ∂Ω (3.6) Multiplying (3.4) by w and (3.6) by u, integrating over Ω, and subtracting, the boundary conditions and Green’s formula lead to ∫ d(w∆u− u∆w) = 0. It follows that (1− f − 1/R0) ∫ (βN/ω)uw + λ∗ ∫ uw = 0. Since both integrals are positive, (1 − f)R0 − 1 and λ∗ must have opposite signs. (The idea of this proof is due to Allen et al. [1]). � In the following theorem we assume that at t = 0 the density of immunes is equal to the density in equilibrium. This is realistic, because in the disease-free equilibrium all immune individuals are immune by vaccination, and R can become greater than Req only after infection and recovery of unvaccinated individuals. 322 H. KNOLLE, J. SANTANILLA EJDE/SI/01 Theorem 3.4. Assume that R(x, 0) = Req in Ω, f > 1 − 1/R0 if R0 ≥ 1, and f = 0 if R0 < 1. Then the DFE is asymptotically stable. Proof. With σ := S − Seq, ρ := R−Req, equations (2.1)–(2.3) take the form σt = d∆σ − fµσ + (1− f)µ(I + ρ)− βSeqI − βσI x ∈ Ω, t > 0 (3.7) It = d∆I − (γ + µ)I + βSeqI + βσI x ∈ Ω, t > 0 (3.8) ρt = d∆ρ− (1− f)µρ+ fµ(σ + I) + γI x ∈ Ω, t > 0 (3.9) Since S + I +R = N/ω, we have σ+ I + ρ = 0, and thus σ can be eliminated from (3.8) and (3.9), and ρ from (3.7). This yields σt = d∆σ − µσ − βSeqI − βσI x ∈ Ω, t > 0 (3.10) It = d∆I − (γ + µ)I + βSeqI − β(ρ+ I)I x ∈ Ω, t > 0 (3.11) ρt = d∆ρ− µρ+ γI x ∈ Ω, t > 0 (3.12) Stability of the disease-free equilibrium depends essentially on equation (3.11). The linear part of (3.11) can be written as It = (d∆−B)I where B = (γ+µ)−βSeq. The nonlinear part is −β(ρ+ I)I. We will show that β(ρ+ I)I is o(I). Let g(t) = ∫ Ω ρ(x, t)dx After integrating equation (3.12) over Ω, using Green’s formula, and the boundary condition, we have g′(t) = −µg(t) + γ ∫ Ω I(x, t)dx . From ρ(x, 0) = 0 for x ∈ Ω, it follows that g(0) = 0. Since I is bounded, g′(t) + µg(t) ≤ γ sup t>0 ∫ Ω I(x, t)dx. Then g(t) ≤ γ ( sup t>0 ∫ Ω I(x, t) dx ) e−µt ∫ t 0 eµs ds = γ ( sup t>0 ∫ Ω I(x, t) dx )∫ t 0 eµ(s−t) ds ≤ γ supt>0 ∫ Ω I(x, t)dx µ . Therefore, ρI is o(I). Now we consider the eigenvalues of the linear problem d∆I −BI + λI = 0 with boundary condition (2.4). By our assumption and Theorem 3.3, the smallest eigen- value is positive. Therefore, the equilibrium I = 0 of the linear approximation is uniformly asymptotically stable, and now Henry’s result [7, Theorem 5.1.1] implies that the equilibrium of the nonlinear equation is uniformly asymptotically stable. Finally, we conclude that σ also approaches zero as t→∞, since ρ+ I+σ = 0. So, we have S → Seq, I → 0, and R→ Req; i.e., the DFE is asymptotically stable. � Theorem 3.5. Assume that R0 > 1 and f < 1 − (1/R0). Then the equilibrium (Seq, 0, Req) is unstable. EJDE-2021/SI/01 SIR EPIDEMIC MODELS WITH SPATIAL SPREAD 323 Proof. The assumptions and Theorem 3.3 imply that the smallest eigenvalue of the problem d∆u− (γ + µ)u+ (1− f)(N/ω)βu+ λu = 0 x ∈ Ω ∂u ∂n = 0 on ∂Ω is negative. Therefore, the equilibrium I = 0 of the linear approximation of equation (3.11) is unstable. Furthermore, this equation is quadratic in I. It follows from Henry’s result [7, Corollary 5.1.6], that the equilibrium I = 0 of the nonlinear equation (3.11) is unstable. This implies that the equilibrium (Seq, 0, Req) of system (2.1)–(2.4) is unstable. � 4. Kendall’s model extended Now we return to Kendall’s approach as sketched in the Introduction, but we avoid the assumption that β is constant, and allow β to depend on the place where a susceptible and an infective meet. Again Ω is a bounded domain in the plain with smooth boundary, but now its points are denoted by (x, y). For the probability distribution h we assume that it is zero outside a circle with radius r and center (0, 0). This means that movements of individuals are restricted to a circle C(r) with radius r. For our analysis it is necessary to assume certain symmetries of h, namely h(u, v) = h(v, u) = h(−u, v) = h(u,−v). This poses no problem at points with a distance greater than r from the boundary of Ω. But near the boundary movements of persons are usually one-sided. This inconsistency can be eliminated when we assume that population density tends to zero near the boundary. The infection parameter for a pair (S, I), where S is living at (x, y) and I is living at (x + u, y + v), has a value somewhere between β(x, y) and β(x + u, y + v), say (1−λ)β(x, y) +λβ(x+u, y+u) with 0 ≤ λ ≤ 1. This includes different patterns of mobility. If λ = 0, then the infection occurs always at (x, y), i.e. the susceptibles are immobile. If λ = 1, the infectives are immobile, but a susceptible can get the infection when the susceptible meets an infective at (x+u, y+v). For λ = 1/2 there is no difference in mobility. With this weighted mean of the infection parameter, the SIR model without birth-and-death process takes the form St = −S(x, y, t) ∫∫ C(r) I(x+ u, y + v, t) [ (1− λ)β(x, y) + λβ(x+ u, y + v) ] h(u, v) du dv It = −St − γI Rt = γI . (4.1) Expanding the Taylor series at (x, y) up to terms of second degree and using the symmetries, we obtain the approximation St = −1 2 S[σ(β∆I + 2λ(grad I)(gradβ)) + (λσ∆β + 2β)I] (4.2) It = −St − γI (4.3) Rt = γI, (4.4) 324 H. KNOLLE, J. SANTANILLA EJDE/SI/01 where σ = ∫∫ C(r) u2h(u, v) du dv which is the basic measure of mobility. The pair of equations (4.2)–(4.3) is strongly nonlinear. Therefore we cannot apply the same methods as in the preceding sec- tions, and we assume without proof that solutions exist and are nonnegative if initial conditions are so. Since (S + I +R)t = 0, the population density at each location is constant in time but may be varying in space. Let à = max(x,y)∈Ω[S(x, y, 0) + I(x, y, 0) + R(x, y, 0)]. The following theorem is the analogue of Theorem 3.4. Theorem 4.1. If 0 < [(λσ∆β)/2 +β]à < γ in Ω, then the disease-free equilibrium is stable. Proof. For any ε > 0, t1 > 0 and all (x, y) in Ω it is impossible that I(x, y, t) ≥ ε for all t > t1, because R(x, y, t) is bounded. Therefore limt→∞ I(x, y, t) = 0 in Ω, and I(x, y, t) is monotone decreasing for all (x, y) in Ω or it attains a positive maximum at some t > 0 and (x, y) in Ω. This would imply It = 0, grad I = 0, ∆I ≤ 0, hence 0 ≤ [(λσ∆β)/2 + β]SI − γI < [((λσ∆β)/2 + β)Ã− γ]I < 0, a contradiction. It follows that I(x, y, t) is monotone decreasing in t for all (x, y) in Ω. � Discussion. The relevance of our first model to practice is limited because the way infectious diseases spread has been very much modified by modern life. The analogy with the diffusion of a gas is unrealistic when, as in many countries today, there are different types of mobility with very different speeds and ranges. The assumptions of the model are most likely to be fulfilled in small towns and rural areas where there is no mass transport and in holiday resorts where people only move on foot. It seems paradoxical that, when γ is not constant, the new definition of R0 implies that it is a decreasing function of the diffusion constant d. But it is plausible if there is a subarea, in which βN/ω < γ + µ, but (N/ω) ∫ Ω β > ∫ Ω (γ + µ). The infectious individuals who drift into this subarea generate fewer infections there than in the rest of the area where βN/ω > γ + µ. Therefore R0 is reduced by high mobility of the population. Of course, this explanation of the paradox is vague. More information is given by Theorem 4.1, which is based on Kendall‘s approach. It shows that stability of the disease-free equilibrium depends also on the shape of the function β, in the following sense: • If β is constant, mobility has no influence on stability of the DFE. • If β is concave (∆β > 0) and λ > 0, high mobility increases the risk of an epidemic outbreak. • If β is convex (∆β < 0) and λ > 0, mobility can help to prevent an epidemic. EJDE-2021/SI/01 SIR EPIDEMIC MODELS WITH SPATIAL SPREAD 325 References [1] L. Allen, B. Bolker, Y. Lou, A. Nevai; Asymptotic profiles of the steady states for an SIS epidemic reaction-diffusion model, Discrete Contin. Dyn. Syst. (2008), 21, 1–20. [2] N. T. Bailey; The Mathematical Theory of Infectious Diseases. London, Griffin, 1975. [3] N. T. Bailey; An improved hybrid HIV/AIDS model geared to specific public health data and decision making, Mathematical Biosciences (1993), 117, 221–237. [4] Courant & Hilbert; Methoden der Mathematischen Physik I (3rd German edition), Springer, Berlin Heidelberg New York, 1967. [5] O. Diekmann; Thresholds and travelling waves for the geographical spread of infection, J. Math. Biology (1978) 6, 109–130. [6] K. Fellner, J. Morgan, B. Tang; Global classical solutions to quadratic systems with mass control in arbitrary dimensions, Annales de l’Institut Henri Poincaré C, Analyse non linéaire (2020), 37, 281–307 [7] D. Henry; Geometric Theory of Semilinear Parabolic Equations, Springer-Verlag, Berlin Hei- delberg New York, 1981. [8] D. Kendall; Mathematical models of the spread of infection, Mathematics and Computer Science in Biology and Medicine (1965), 213–225. [9] R. Peng; Asymptotic profiles of the positive steady state for an SIS epidemic reaction- diffusion model, J. Differ. Equations (2009), 247, 1096–111. [10] Y. Wu, X. Zou; Asymptotic profiles of steady states for a diffusive SIS epidemic model with mass-action infection mechanism, J. Differential Equations (2016), 261, 4424–4447. Helmut Knolle Scheyenholzstrasse 9, 3075 Rüfenacht, Switzerland. Institute Of Social And Preventive Medicine, University Of Berne Email address: helmut.knolle@bluewin.ch Jairo Santanilla Department of Mathematics, University of New Orleans, 2000 Lakeshore Dr., New Orleans, LA 70148, USA Email address: jsantani@uno.edu 1. Introduction 2. New model 3. Stability and instability of the disease-free equilibrium 4. Kendall's model extended Discussion References