Exact Solution to a Dynamic SIR Model
aa r X i v : . [ m a t h . C A ] D ec Exact Solution to a Dynamic SIR Model ∗ Martin Bohner † , Sabrina Streipert ‡ and Delfim F. M. Torres § Department of Mathematics & Statistics, Missouri University of Scienceand Technology, 65409 Rolla, MO, USA Centre for Applications in Natural Resource Mathematics,School of Mathematics and Physics, University of Queensland,4067 St Lucia, QLD, Australia R&D unit CIDMA, Department of Mathematics, University of Aveiro,3810-193 Aveiro, Portugal
Abstract
We investigate an epidemic model based on Bailey’s continuous differential system.In the continuous time domain, we extend the classical model to time-dependent co-efficients and present an alternative solution method to Gleissner’s approach. If thecoefficients are constant, both solution methods yield the same result. After a briefintroduction to time scales, we formulate the SIR (susceptible-infected-removed) modelin the general time domain and derive its solution. In the discrete case, this providesthe solution to a new discrete epidemic system, which exhibits the same behavior as thecontinuous model. The last part is dedicated to the analysis of the limiting behaviorof susceptible, infected, and removed, which contains biological relevance.
MSC 2010:
Keywords: dynamic equations on time scales; deterministic epidemic model; closed-form solution; time-varying coefficients; asymptotic behavior.
Modeling infectious diseases is as important as it has been in 1760, when Daniel Bernoullipresented a solution to his mathematical model on smallpox. It was however not until the ∗ This is a preprint of a paper whose final and definite form is with
Nonlinear Analysis: Hybrid Systems ,ISSN: 1751-570X, available at https://doi.org/10.1016/j.nahs.2018.12.005 . Submitted 16/May/2018;Revised 10/Oct/2018; Accepted for publication 18/Dec/2018. † Martin Bohner: [email protected] ‡ Sabrina Streipert: [email protected] § Delfim F. M. Torres: delfi[email protected] x + y ,where x is the number of susceptible and y the number of infected individuals. Let k bethe actual number of individuals a susceptible interacts with and p be the probability thata susceptible gets infected at contact with an infected individual. Then pk yx + y is the rate atwhich one susceptible enters the group of infected [1]. This leads to the rate of change forthe group of susceptible as d x d t = − pk yx + y x. Similarly, the infected increase by that rate, but some infected leave the class of infected,due to death for example, at a rate c , which yields the differential equation in y asd y d t = pk yx + y x − cy. To obtain a system with time independent sum, a third group is added, the group of removedindividuals for example, denoted by z , with the dynamics given byd z d t = cy. Many modifications of the classical model have been investigated such as models with vitaldynamics, see [2–4]. To model the spreading of diseases between different states, a spatialvariable was added, which led to a partial differential system, see [5, 6]. Already in 1975,Bailey discussed in [1] the relevance of stochastic terms in the mathematical model of epi-demics, which is still an attractive way of modeling the uncertainty of the transmission andvaccines, see [7–10]. Although these modifications exist, so far there has been no success ingeneralizing the epidemic models to a general time scale to allow modeling a noncontinu-ous disease dynamics. A disease, where the virus remains within the host for several yearsunnoticed before continuing to spread, is only one example that can be modeled by timescales. We trust that this work provides the foundation for further research on generalizingepidemic models to allow modeling of discontinuous epidemic behavior.
We investigate a susceptible-infected-removed (SIR) model proposed by Norman Bailey in [1]of the form x ′ = − bxyx + y ,y ′ = bxyx + y − cy,z ′ = cy, (1)2ith initial conditions x ( t ) = x > y ( t ) = y > z ( t ) = z ≥ x, y, z : R → R +0 , and b, c ∈ R +0 . The variable x represents the group of susceptible, y the infected population, and z the removed population. By adding the group of removed, the total population N = x + y + z remains constant. In [11], assuming x, y >
0, the model is solved by rewriting the first twoequations in (1) as ( x ′ x = − bx + y y, y ′ y = bxx + y − c. Subtracting these equations yields x ′ x − y ′ y = − b + c, i.e., y ′ y = x ′ x + b − c, which is equivalent to (ln y ) ′ = (ln x ) ′ + ( b − c ) . Integrating both sides and taking the exponential, one gets y = xκe ( b − c )( t − t ) , where κ = y x . (2)If b = c , then, plugging this into the first equation in (1) yields a first order linear homoge-neous differential equation with the solution given by x ( t ) = x (1 + κ ) bb − c (cid:0) κe ( b − c )( t − t ) (cid:1) − bb − c . (3)Replugging yields the solution of (1) as x ( t ) = x (1 + κ ) bb − c (1 + κe ( b − c )( t − t ) ) − bb − c ,y ( t ) = y (1 + κ ) bb − c (1 + κe ( b − c )( t − t ) ) − bb − c e ( b − c )( t − t ) ,z ( t ) = N − ( x + y ) bb − c ( x + y e ( b − c )( t − t ) ) − cb − c . (4)If b = c , then (2) gives y = xκ , and the solution (4) of (1) is x ( t ) = x e − bκ ( t − t κ ,y ( t ) = y e − bκ ( t − t κ ,z ( t ) = N − ( x + y ) e − bκ ( t − t κ . In this work, we present a different method to solve (1), considering not only constant b, c but b, c : R → R + . This will allow us to find the solution to the model on time scales.To this end, define w := xy > x, y > w ′ = x ′ y − y ′ xy = − b ( t ) xyx + y y − (cid:16) b ( t ) xyx + y − c ( t ) y (cid:17) xy = − b ( t ) xy + c ( t ) xyy = ( c − b )( t ) w, time t b(t)c(t) Figure 1: Time-varying parameters b and c of“von Bertalanffy” type: b ( t ) = t √ π e − ln2( t )2 and c ( t ) = 0 .
55 (1 − e − . t − . ).which is a first-order homogeneous differential equation with solution w ( t ) = w e R tt ( c − b )( s ) d s , i.e., y ( t ) = κe R tt ( b − c )( s ) d s x ( t ) , (5)which is the same as (2) for constant b, c . We plug (5) into (1) to get x ′ = − b ( t ) x κe R tt ( b − c )( s ) d s x + κxe R tt ( b − c )( s ) d s = − b ( t ) κe R tt ( b − c )( s ) d s κe R tt ( b − c )( s ) d s x, which has the solution x ( t ) = x exp (cid:26) − κ Z tt b ( s ) (cid:16) κ + e R st ( c − b )( τ ) d τ (cid:17) − d s (cid:27) . (6)Note that, for constant b, c with b = c , (6) simplifies to (3). Hence, the solution to (1) isgiven by x ( t ) = x exp (cid:26) − κ R tt b ( s ) (cid:16) κ + e R st ( c − b )( τ ) d τ (cid:17) − d s (cid:27) ,y ( t ) = y exp (cid:26)R tt (cid:20) b ( s ) (cid:16) κe R st ( b − c )( τ ) d τ (cid:17) − − c ( s ) (cid:21) d s (cid:27) ,z ( t ) = N − (cid:16) y e R tt ( b − c )( s ) d s + x (cid:17) exp (cid:26) − κ R tt b ( s ) (cid:16) κ + e R st ( c − b )( τ ) d τ (cid:17) − d s (cid:27) . (7)The time-varying parameters b and c allow us to investigate epidemic models, where thetransmission rate peaks in early years before reducing, for example due to initial ignorancebut increasing precaution of susceptibles. This behavior could be described by the probabilitydensity function of the log-normal distribution. A removal rate that increases rapidly to aconstant rate could be modeled by a “von Bertalanffy” type function, see Figure 1. Usingthese parameter functions with initial conditions x = 0 . y = 1 . c . Zooming into the last part of the time interval, we see that the number ofsusceptibles converges. 4 time susceptibleinfected time susceptibleinfected Figure 2: Dynamics of the susceptible x andthe infected y populations with initial condi-tions x = 0 . y = 1 . b and c of Figure 1. Example 1.
Considering a simple decreasing transmission rate to account for the risingprecaution of susceptibles and a simple decreasing removal rate accounting for medical ad-vances, for example by choosing b ( t ) = t +1 and c ( t ) = t +1 , the solution with t = 0 is givenby (7) as x ( t ) = x exp nR t − κ ( s +1)( κ + s +1) d s o = x κ +1+ t ( κ +1)( t +1) ,y ( t ) = y exp (cid:8) κ +1+ s − s +1 (cid:9) = y κ +1+ t ( κ +1)( t +1) ,z ( t ) = N − κ + tt +1 n x κ +1 − y ( κ +1)( t +1) o , where N = x + y + z and κ = y x . In order to formulate the time scales analogue to the model proposed by Norman Bailey, wefirst introduce fundamentals of time scales that we will use. The following introduces themain definitions in the theory of time scales.
Definition 2 (See [12, Definition 1.1]) . For t ∈ T , the forward jump operator σ : T → T is σ ( t ) := inf { s ∈ T : s > t } . For any function f : T → R , we put f σ = f ◦ σ . If t ∈ T has a left-scattered maximum M ,then we define T κ = T \ { M } ; otherwise, T κ = T . Definition 3 (See [13, Definition 1.24]) . A function p : T → R is called rd-continuousprovided p is continuous at t for all right-dense points t and the left-sided limit exists forall left-dense points t . The set of rd-continuous functions is denoted by C rd = C rd ( T ) =C rd ( T , R ) . Definition 4 (See [12, Definition 2.25]) . A function p : T → R is called regressive provided µ ( t ) p ( t ) = 0 for all t ∈ T , where µ ( t ) = σ ( t ) − t. he set of regressive and rd-continuous functions is denoted by R = R ( T ) = R ( T , R ) .Moreover, p ∈ R is called positively regressive, denoted by R + , if µ ( t ) p ( t ) > for all t ∈ T . Definition 5 (See [12, Definition 1.10]) . Assume f : T → R and t ∈ T κ . Then the derivativeof f at t , denoted by f ∆ ( t ) , is the number such that for all ε > , there exists δ > , suchthat (cid:12)(cid:12) f ( σ ( t )) − f ( s ) − f ∆ ( t )( σ ( t ) − s ) (cid:12)(cid:12) ≤ ε | σ ( t ) − s | for all s ∈ ( t − δ, t + δ ) ∩ T . Theorem 6 (See [12, Theorem 2.33]) . Let p ∈ R and t ∈ T . Then y ∆ = p ( t ) y, y ( t ) = 1 possesses a unique solution, called the exponential function and denoted by e p ( · , t ) . Useful properties of the exponential function are the following.
Theorem 7 (See [12, Theorem 2.36]) . If p ∈ R , then1. e ( t, s ) = 1 , and e p ( t, t ) = 1 ,2. e p ( t, s ) = e p ( s,t ) ,3. the semigroup property holds: e p ( t, r ) e p ( r, s ) = e p ( t, s ) . Theorem 8 (See [12, Theorem 2.44]) . If p ∈ R + and t ∈ T , then e p ( t, t ) > for all t ∈ T . We define a “circle-plus” and “circle-minus” operation.
Definition 9 (See [13, p. 13]) . Define the “circle plus” addition on R as p ⊕ q = p + q + µpq and the “circle minus” subtraction as p ⊖ q = p − q µq . It is not hard to show the following identities.
Corollary 10 (See [12]) . If p, q ∈ R , thena) e p ⊕ q ( t, s ) = e p ( t, s ) e q ( t, s ) , b) e ⊖ p ( t, s ) = e p ( s, t ) = e p ( t,s ) . heorem 11 (Variation of Constants, see [12, Theorems 2.74 and 2.77]) . Suppose p ∈ R and f ∈ C rd . Let t ∈ T and y ∈ R . The unique solution of the IVP y ∆ = p ( t ) y + f ( t ) , y ( t ) = y is given by y ( t ) = e p ( t, t ) y + Z tt e p ( t, σ ( s )) f ( s ) ∆ s. The unique solution of the IVP y ∆ = − p ( t ) y σ + f ( t ) , y ( t ) = y is given by y ( t ) = e ⊖ p ( t, t ) y + Z tt e ⊖ p ( t, s ) f ( s ) ∆ s. Lemma 12 (See [12, Theorem 2.39]) . If p ∈ R and a, b, c ∈ T , then Z ba p ( t ) e p ( t, c )∆ t = e p ( b, c ) − e p ( a, c ) and Z ba p ( t ) e p ( c, σ ( t ))∆ t = e p ( c, a ) − e p ( c, b ) . In this section, we formulate a dynamic epidemic model based on Bailey’s classical differentialsystem (1) and derive its exact solution. In the special case of a discrete time domain,this provides a novel model as a discrete analogue of the continuous system. We end thediscussion by analyzing the stability of the solutions to the dynamic model in the case ofconstant coefficients.Consider the dynamic susceptible-infected-removed model of the form x ∆ = − b ( t ) xy σ x + y ,y ∆ = b ( t ) xy σ x + y − c ( t ) y σ ,z ∆ = c ( t ) y σ ,x, y > , (8)with given initial conditions x ( t ) = x > y ( t ) = y > z ( t ) = z ≥ x, y : T → R + , z : T → R +0 , and b, c : T → R +0 . Theorem 13. If c − b, g ∈ R , then the unique solution to the IVP (8) is given by x ( t ) = e ⊖ g ( t, t ) x ,y ( t ) = e ⊖ ( g ⊕ ( c − b )) ( t, t ) y ,z ( t ) = N − e ⊖ g ( t, t ) (cid:0) x + y e ⊖ ( c − b ) ( t, t ) (cid:1) , here N = x + y + z , κ = y x , and g ( t ) := b ( t ) κκ (1 + µ ( t )( c − b )( t )) + e c − b ( σ ( t ) , t ) . Proof.
Assume that x, y, z solve (8). Since ( x + y + z ) ∆ = 0, we get z = N − ( x + y ), where N = x + y + z . Defining w := xy , we have w ∆ = x ∆ y − y ∆ xyy σ = − b ( t ) xy σ x + y y − (cid:16) b ( t ) xy σ x + y − c ( t ) y σ (cid:17) xyy σ = − b ( t ) xy σ + c ( t ) xy σ yy σ = ( c − b )( t ) w, which is a first-order linear dynamic equation with solution w ( t ) = e c − b ( t, t ) w , i.e., y ( t ) = κe ⊖ ( c − b ) ( t, t ) x ( t ) . (9)We plug (9) into (8) to get x ∆ = − b ( t ) xx σ e ⊖ ( c − b ) ( σ ( t ) , t ) κx + xe ⊖ ( c − b ) ( t, t ) κ = − b ( t ) e ⊖ ( c − b ) ( σ ( t ) , t ) κ e ⊖ ( c − b ) ( t, t ) κ x σ = − g ( t ) x σ , which has the solution x ( t ) = e ⊖ g ( t, t ) x . By (9), we obtain y ( t ) = y e ⊖ ( g ⊕ ( c − b )) ( t, t ) , and thus, z ( t ) = N − x ( t ) − y ( t ) = N − e ⊖ g ( t, t ) (cid:0) x + y e ⊖ ( c − b ) ( t, t ) (cid:1) . This shows that x, y, z are as given in the statement. Conversely, it is easy to show that x, y, z as given in the statement solve (8). The proof is complete.
Remark 14. If c − b ∈ R + and x > , y , z ≥ , then x, y, z ≥ for all t ∈ T . For T = R ,this condition is satisfied, since µ ( t ) = 0 for all t ∈ R . Remark 15. If b ( t ) = c ( t ) for all t ∈ T , then c − b ∈ R , and, by Theorem 13, the solutionof (8) is x ( t ) = e ⊖ g ( t, t ) x ,y ( t ) = e ⊖ g ( t, t ) y ,z ( t ) = N − e ⊖ g ( t, t ) ( x + y ) , where g ( t ) = b ( t ) κ κ .
8s an application of Theorem 13, we introduce the discrete epidemic model x ( t + 1) = x ( t ) − b ( t ) x ( t ) y ( t +1) x ( t )+ y ( t ) ,y ( t + 1) = y ( t ) + b ( t ) x ( t ) y ( t +1) x ( t )+ y ( t ) − c ( t ) y ( t + 1) ,z ( t + 1) = z ( t ) + c ( t ) y ( t + 1) , (10) t ∈ Z , with initial conditions x ( t ) = x > y ( t ) = y > z ( t ) = z ≥
0. Note that thesecond equation of (10) can be represented as y ( t + 1) = 11 + δ ( t ) y ( t ) , which implies that a fraction, namely δ , of the infected individuals remain infected. If therate with which susceptibles are getting infected is higher than the rate with which infectedare removed, i.e., ϕ = b xx + y > c , then the multiplicative factor δ = c − ϕ is greater than one,else less than one. Slightly rewriting the first equation into the form x ( t + 1) + ϕ ( t ) y ( t + 1) = x ( t )provides the interpretation that some susceptible individuals stay in the group of susceptibles,others become infected and contribute the fraction ϕ to the group of infected. A similarinference can be drawn from z ( t + 1) = z ( t ) + c ( t ) y ( t + 1) . The number of removed individuals is the sum of the already removed individuals and aproportion of infected individuals that are removed at the end of the time step.The following theorem is a direct consequence of Theorem 13.
Theorem 16. If c ( t ) − b ( t ) , g ( t ) = 0 for all t ∈ Z , where g ( t ) = b ( t ) κ (cid:2)Q ti = t (1 + ( c − b )( i )) (cid:3) + κ (1 + ( c − b )( t )) , then the unique solution to (10) is given by x ( t ) = x (cid:2)Q t − i = t (1 + g ( i )) (cid:3) − ,y ( t ) = y (cid:2)Q t − i = t (1 + ( c − b )( i ))(1 + g ( i )) (cid:3) − ,z ( t ) = N − (cid:16) x + y (cid:2)Q t − i = t (1 + ( c − b )( i )) (cid:3) − (cid:17) (cid:2)Q t − i = t (1 + g ( i )) (cid:3) − , where N = x + y + z and κ = y x . Example 17.
Consider a disease with periodic transmission rate, for example due to sen-sitivity of bacteria to temperature or hormonal cycles. In this case, we might choose b ( t ) = + sin( mt ) with m ∈ R \{ } . To account for medical advances, we let c ( t ) = t +1 . Note time SIR (a) T = Z time SIR (b) T = [0 , ∪{ , , , . . . , } Figure 3: the x =Susceptible ( S ), y =Infected ( I ), and z =Removed ( R )dynamics of Example 19. that c ( t ) = b ( t ) because ≤ b ( t ) ≤ < and g ( t ) = 0 for all t ∈ Z . The solution isthen given by Theorem 16 with g ( t ) = 2 + sin( mt ) h κ Q ti = t i i ) − sin( mi ) i + t )1+ t − sin( mt ) . Remark 18. If T = R , b, c ∈ R , b = c , and t = 0 , then, by Theorem 13, the solution to (8) is x ( t ) = x e − R t g ( s ) d s = x e − b R t κe − ( c − b ) s κe − ( c − b ) s d s = x e − bb − c ( ln(1+ κe − ( c − b ) t ) − ln(1+ κ ) )= x (cid:0) κe − ( c − b ) t (cid:1) − bb − c (1 + κ ) bb − c , which is consistent with (3) . If c = b , then Theorem 13 provides the solution as x ( t ) = x e − R t g ( s ) ds = x e − b R t κ κ ds = x e − bκ κ t , which is consistent with the continuous results. Example 19.
Let us consider the SIR model (8) with b = 0 . , c = 0 . , x = 0 . , y = 0 . , z = 0 . In Figure 3a, we show the solution in the discrete-time case T = Z determined by (10) ; inFigure 3b, we plot the solution to (8) for the partial continuous, partial discrete time scale T = [0 , ∪ { , . . . , } . Long Term Behavior
We start this section by recalling the following results.
Lemma 20 (See [14, Lemma 3.2]) . If p ∈ R + , then < e p ( t, t ) ≤ exp (cid:26)Z tt p ( τ )∆ τ (cid:27) for all t ≥ t . Lemma 21 (See [15, Remark 2]) . If p ∈ C rd and p ( t ) ≥ for all t ∈ T , then Z tt p ( τ )∆ τ ≤ e p ( t, t ) ≤ exp (cid:26)Z tt p ( τ )∆ τ (cid:27) for all t ≥ t . The equilibriums of (8) are given as follows.
Lemma 22.
Suppose c ( t ) > for some t ∈ T . The equilibriums of (8) are given by theplane ( α, , N − α ) , where α ∈ [0 , N ] and N = x + y + z .Proof. Assume x, y, z are constant solutions of (8). Then, 0 = z ∆ ( t ) = c ( t ) y ( t ), so y ( t ) = 0for all t ∈ T . Therefore, − b ( t ) xyx + y = x ∆ = 0 = y ∆ = b ( t ) xyx + y − c ( t ) y for any 0 ≤ x ≤ N and the proof is complete. Theorem 23.
Consider (8) and assume T is unbounded from above. Assume b, c : T → R +0 , c − b ∈ R + , x , y > , and z ≥ . Moreover, assume ∃ L > Z tt ( c − b )( τ ) ∆ τ ≤ L for all t ≥ t (11) and Z ∞ t b ( τ )1 + µ ( τ )( c − b )( τ ) ∆ τ = ∞ . (12) Then all solutions of (8) converge to the equilibrium (0 , , N ) , where N = x + y + z .Proof. By Lemma 20, 0 < e c − b ( t, t ) ≤ e R tt ( c − b )( τ ) ∆ τ (11) ≤ e L , t ≥ t . By Lemma 21, since g ≥
0, we get e g ( t, t ) ≥ Z tt g ( τ ) ∆ τ = 1 + Z tt b ( τ )1 + µ ( t )( c − b )( τ ) κκ + e c − b ( τ, t ) ∆ τ ≥ κκ + e L Z tt b ( τ )1 + µ ( t )( c − b )( τ ) ∆ τ (12) → ∞ , t → ∞ . Then e ⊖ g ( t, t ) → t → ∞ , so thatlim t →∞ x ( t ) = 0 , lim t →∞ y ( t ) = 0 , lim t →∞ z ( t ) = N due to Theorem 13. 11 orollary 24. If b ( t ) = c ( t ) for all t ∈ T , then the conclusion of Theorem 23 holds provided Z ∞ t b ( τ ) ∆ τ = ∞ . Corollary 25. If b and c are constants, then the conclusion of Theorem 23 holds provided c − b ∈ R + and b ≥ c. Theorem 26.
Consider (8) and assume T is unbounded from above. Assume b, c : T → R +0 , c − b ∈ R + , x , y > , and z ≥ . Moreover, assume ∃ M > b ( t ) ≤ M ( c − b )( t ) for all t ∈ T (13) and Z ∞ t ( c − b )( τ ) ∆ τ = ∞ . (14) Then all solutions of (8) converge to the equilibrium ( α, , N − α ) for some α ∈ (0 , N ) .Proof. Note first that (13) implies c ( t ) ≥ b ( t ) for all t ∈ T . By Lemma 21, we have e c − b ( t, t ) ≥ Z tt ( c − b )( τ ) ∆ τ (14) → ∞ , t → ∞ , so lim t →∞ e ⊖ ( c − b ) ( t, t ) = 0 . (15)Next, g ( t ) = b ( t )1 + µ ( t )( c − b )( t ) κκ + e c − b ( t, t ) ≤ b ( t )1 + µ ( t )( c − b )( t ) κe c − b ( t, t ) = κb ( t ) e c − b ( σ ( t ) , t ) ≤ κM ( c − b )( t ) e c − b ( σ ( t ) , t ) , and thus, using [12, Theorem 2.39], we get Z tt g ( τ ) ∆ τ ≤ M κ Z tt ( c − b )( τ ) e c − b ( σ ( τ ) , t ) ∆ τ = M κ (cid:20) − e c − b ( t, t ) (cid:21) < M κ. By Lemma 21, since g ≥ t ∈ T , we get1 ≤ Z tt g ( τ ) ∆ τ ≤ e g ( t, t ) ≤ exp (cid:26)Z tt g ( τ ) ∆ τ (cid:27) < e κM for all t ≥ t , so lim t →∞ e g ( t, t ) exists and is bounded from below by 1 and bounded from above by e κM .We therefore get that lim t →∞ e ⊖ g ( t, t ) exists and is greater than or equal to e − κM > α := lim t →∞ x ( t ) > , lim t →∞ y ( t ) = 0 , lim t →∞ z ( t ) = N − α due to (15) and Theorem 13. 12 orollary 27. If b and c are constants, then the conclusion of Theorem 26 holds provided b < c. Finally, we give a result that describes the monotone behavior of the solution y . Theorem 28. If c ( t ) ≥ b ( t ) for all t ∈ T or x x + y b ( t ) ≤ c ( t ) ≤ b ( t ) for all t ∈ T , then y isdecreasing. If x x + y b ( t ) ≥ c ( t ) , then y ∆ ( t ) ≥ .Proof. If x x + y b ( t ) > c ( t ), then y ∆ ( t ) = b ( t ) x ( t ) y ( σ ( t )) x ( t ) + y ( t ) − c ( t ) y ( σ ( t )) = (cid:20) b ( t ) x x + y − c ( t ) (cid:21) y ( σ ( t )) ≥ . If c ( t ) ≥ b ( t ) for all t ∈ T , then y ∆ ( t ) = b ( t ) x ( t ) y ( σ ( t )) x ( t ) + y ( t ) − c ( t ) y ( σ ( t )) ≤ b ( t ) x ( t ) y ( σ ( t )) x ( t ) + y ( t ) − b ( t ) y ( σ ( t ))= b ( t ) y ( σ ( t )) (cid:20) x ( t ) x ( t ) + y ( t ) − (cid:21) = − b ( t ) y ( t ) y ( σ ( t )) x ( t ) + y ( t ) ≤ t ∈ T . Next, we calculate (cid:18) xx + y (cid:19) ∆ = x ∆ y − y ∆ x ( x + y )( x σ + y σ ) = − b ( t ) xy σ x + y y − (cid:16) b ( t ) xy σ x + y − c ( t ) y σ (cid:17) x ( x + y )( x σ + y σ ) = ( c − b )( t ) xy σ ( x + y )( x σ + y σ ) . If x x + y b ( t ) ≤ c ( t ) ≤ b ( t ) for all t ∈ T , then y ∆ ( t ) = b ( t ) x ( t ) y ( σ ( t )) x ( t ) + y ( t ) − c ( t ) y ( σ ( t )) ≤ b ( t ) y ( σ ( t )) x x + y − c ( t ) y ( σ ( t ))= (cid:20) b ( t ) x x + y − c ( t ) (cid:21) y ( σ ( t )) ≤ t ∈ T . This completes the proof.
Example 29.
For T = Z , S (0) = 0 . , I (0) = 0 . , R (0) = 0 . , and b = 0 . , we get for c = 0 . the limit behavior for the solutions as shown in Figure 4a. Changing c to . such that b > c ,we get the behavior demonstrated in Figure 4b. Acknowledgement
Torres has been partially supported by FCT within CIDMA project UID/MAT/04106/2019,and by TOCCATA FCT project PTDC/EEI-AUT/2933/2014. The authors are very gratefulto three anonymous reviewers for several constructive comments, questions and suggestions,which helped them to improve the paper. 13 time
SIR (a) b = 0 . < . c time SIR (b) b = 0 . > . c Figure 4: the x =Susceptible ( S ), y =Infected ( I ), and z =Removed ( R )long term behavior of Example 29. References [1] N. T. J. Bailey,
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