Effects of anti-infection behavior on the equilibrium states of an infectious disease
EEffects of anti-infection behavior on the equilibrium states of an infectiousdisease
Andrés David Báez Sánchez a , Nara Bobko a, ∗ a Mathematics Department, Federal University of Technology, Av. Sete de Setembro, 3165, 80230-901, Curitiba,Paraná, Brazil.
Abstract
We propose a mathematical model to analyze the effects of anti-infection behavior on the equilibriumstates of an infectious disease. The anti-infection behavior is incorporated into a classical epidemio-logical SIR model, by considering the behavior adoption rate across the population as an additionalvariable. We consider also the effects on the adoption rate produced by the disease evolution, usinga dynamic payoff function and an additional differential equation. The equilibrium states of theproposed model have remarkable characteristics: possible coexistence of two locally stable endemicequilibria, the coexistence of locally stable endemic and disease-free equilibria, and even the possibilityof a stable continuum of endemic equilibrium points. We show how some of the results obtained maybe used to support strategic planning leading to effective control of the disease in the long-term.
Keywords:
SIR model, Stability, Behavioral Epidemiology, Game Theory.
1. Introduction
The propagation of an infectious disease can be affected by changes in the population behaviorand, at the same time, the population behavior concerning the disease can change due to changesin the perception of the epidemiological situation [1, 2, 3]. Most recently, in the context of theCOVID-19 pandemic, has been clear the relevant role played by human behavior on the diseasedynamic [4, 5, 6] and also has become evident the changes produced on the population behavior andpolicymakers due to the increase in the number of infected and death cases [7, 8, 9].Even before the COVID-19 emergency, there was a well-recognized demand for mathematicalmodels of infectious diseases considering aspects of the population behavior [10, 11]. ∗ Corresponding author
Email addresses: [email protected] (Andrés David Báez Sánchez), [email protected] (NaraBobko)
Preprint submitted to September 11, 2020 a r X i v : . [ q - b i o . P E ] S e p any mathematical and computational models for infectious diseases based on SIR models havealready considered some type of anti-infection strategies. Some works have incorporated implicitlythe possibility of a dynamic preventive behavior, by considering rates of infection or transmission daymay depend on some of the epidemiological variables S, I, or R [12, 13, 14, 15, 16, 17, 18, 19, 20].For other models considering behavioral features see [21].Vaccination, as a form of anti-infection behavior, has been considered assuming that part of thesusceptible population goes directly into the removed population or adding additional compartmentsfor partially immune population [22, 23, 24]. For other models considering vaccination see [25, 26, 27].In [28] a model for vaccination-related behavior is considered using an additional variablecorresponding with the rate of vaccination at birth. This new variable interacts with the infectiondynamics in the SIR model and is affected by a differential equation that depends on the infectedpopulation I . In the present work, we use a similar idea and introduce a behavioral variable related tothe adoption rate across the population of some anti-infection behavior. This variable is incorporatedinto a classical epidemiological SIR model. The dynamics effects on the adoption rate are introducedusing an additional differential equation and a dynamic linear payoff depending on the epidemiologicalvariables.We focus on the study of equilibrium states as an attempt to understand the long-term character-istic and consequences of the interplay between population behavior and disease dynamics.The equilibrium states of the proposed model have remarkable characteristics: possible coexistenceof two locally stable endemic equilibria, the coexistence of locally stable endemic and disease-freeequilibria, and even the possibility of a stable continuum of endemic equilibrium points. We willdescribe how some of the results obtained may be used to support strategic planning leading toeffective control of the infectious disease in the long-term.The paper is organized as follows. In Section 2 we develop the mathematical model and discusssome basic characteristics. In Section 3 we discuss the existence and stability of its equilibriumpoints, which is the main focus of the present work. We will show that the set of equilibrium pointsof the proposed model, have some remarkable characteristics in the context of epidemiological models:coexistence of two locally stable endemic equilibria, the coexistence of locally stable endemic anddisease-free equilibria, and the possibility of a stable continuum of endemic equilibrium points. InSection 4 we use some of the results to obtain thresholds for parameters leading to effective long-termcontrol of the epidemic disease. We conclude with some final remarks in Section 5 and an Appendixpresenting proofs of some of the results established in the paper.2 . A Mathematical Model for an Infectious Disease with an Anti-Infection behavior Compartment models, and particularly SIR models, have been extensively used for mathematicalmodeling of infectious diseases [11]. The main idea behind SIR models is to consider a populationdivided into three disjoint categories or compartments: susceptible individuals, infected individuals,and removed (recovered or deceased) individuals, denoted by S , I , and R respectively. If N denotesthe total population, then we have N = S + I + R .Depending on the modeling approach, the variables S , I , and R may be considered as theabsolute numbers of individuals in each group or as the proportion of individuals relative to thetotal population. In this work, we consider this latter approach. Therefore, considering the timedependency, we have that S ( t ) + I ( t ) + R ( t ) = 1 for all t .Within these considerations, an SIR model with vital dynamics and constant population can bestated as dSdt = µ − β S I − µ SdIdt = β S I − µ I − γIdRdt = γ I − µ R, (1)with S (0) + I (0) + R (0) = 1 . The positive real numbers µ , β , and γ can be interpreted as birth-mortality rate, infection rate, and recovery rate respectively. The constant population considerationis implicit into the system, since N ( t ) = 1 is the only solution of dNdt = dSdt + dIdt + dRdt = µ (1 − N ) satisfying N (0) = 1 . For more details about SIR-type models see [25, 26].Now, consider that there is some behavior or action that can be taken to avoid or reduce theimpact of the infection. This behavior can be interpreted as a vaccination initiative, a preventivehygienic measure, a quarantine restraint, or a combination of similar actions. Let x be the proportionof the population following this anti-infection behavior.When the population is considering this behavior or action, the perception of the benefit obtainedby following it, may not always be constant. In fact, depending on the epidemiological state, thebenefit may vary. For example, in a situation with a small proportion of infected, the benefit ofadopting the anti-infection behavior may be considered irrelevant for some part of the population.On the other hand, in a situation where the majority of the population has no immunity, thebenefits may be considered high. To analyze this kind of situation, we propose to consider that there3xists a perceived payoff or benefit obtained from the anti-infection behavior that depends on theepidemiological variables S , I , and R according to a function p given by p ( S, I, R ) = − a c + a I I + a S S + a R R, (2)where a c , a I , a S , and a R are positive constants. The constant a c can be interpreted as the fixedcost of adopting the anti-infection behavior, and the constants a I , a S , and a R can be interpreted asthe behavior-adoption benefit associated with the proportion of infected, susceptible, and removedmembers of the population, respectively. As we have considered that S + I + R = 1 , we have that − a c + a I I + a S S + a R R = − a c + a I I + a S S + a R (1 − S − I )= − ( a c − a R ) + ( a I − a R ) I + ( a S − a R ) S = − a + a I + a S. Therefore, the payoff functions can be simplified to obtain p ( S, I ) = − a + a I + a S. (3)Based on the SIR model (1) and the payoff function (3), we propose the following model consideringsimultaneously the epidemiological variables ( S, I, R ) and the behavioral state x : dSdt = µ − (1 − x ) β S I − µ SdIdt = (1 − x ) β S I − µ I − γIdRdt = γ I − µ Rdxdt = x (1 − x )( − a + a I + a S ) (4)with initial conditions in [0 , , and N (0) = S (0) + I (0) + R (0) = 1 . The three initials equations areessentially the SIR model (1) with a variable infection rate depending on the behavioral variable x . If x = 1 , there is no infection at all. If x = 0 , the diseases follow the classical SIR dynamics. The fourthequation may be seen as a logistic equation for x with a growth rate depending on the variables S and I and on the cost/payoff parameters a , a , a . Thus, depending on the interplay betweenthese values over time, the adoption rate x may increase or decrease, leading also to a dynamicallydecreasing or increasing infection rate. The differential equation for x can also be obtained from thereplicator equations in evolutionary game theory (see [29]), applied to a two-behavior game (followor not follow the anti-infection behavior) with a symmetric payoff given by − a + a I + a S .4he main goal of the present work is to study the long-term behavior of model (4) in terms of itsequilibrium points. To achieve this, we will consider a simplified model obtained by re-scaling someof the parameters. Considering τ = tµ ; (cid:101) β = βµ ; (cid:101) γ = γµ ; (cid:101) a = a µ ; (cid:101) a = a µ ; (cid:101) a = a µk = 1 + γµ = 1 + (cid:101) γ and R = βµ + γ = (cid:101) β (cid:101) γ = (cid:101) βk . (5)and replacing in (4), we obtain dSdτ = 1 − (1 − x ) kR S I − SdIdτ = (1 − x ) kR S I − kIdRdτ = ( k − I − Rdxdτ = x (1 − x )( − (cid:101) a + (cid:101) a I + (cid:101) a S ) , (6)with initial conditions in [0 , and N (0) = S (0) + I (0) + R (0) = 1 .Note that the parameter k > and the parameter R is also a positive real number. Theparameter R is called the basic reproduction number and has a fundamental role in the descriptionof the equilibria stability in the classical SIR model [25, 26]. The parameter R can be interpretedas the number of cases one case generates, on average, in an uninfected population. It represents ameasure of the effectiveness of the infection. We introduce below the term R p , that will be importantin the forthcoming analysis of equilibrium points R p = (cid:101) a − k (cid:101) a (cid:101) a − k (cid:101) a . Note that R p depends both on the payoffs associated with the anti-infection behavior and on thepopulation parameter k = 1 + γµ . We will see in Section 3 that under the effects of the anti-infectionbehavior, the constant R p plays a similar role to the one played by the basic reproduction number R in the classical SIR model.We end this section proving that the variables in (6) properly represent population proportions,in the sense that S, I, R and x belongs to the interval [0 , for all t (cid:62) , and that N ( τ ) = S ( τ ) + I ( τ ) + R ( τ ) = 1 . Lemma 1.
The set
Ω = { x ∈ [0 , , S ≥ , I ≥ , R ≥ and S + I + R = 1 } is positively invariantunder (6) . roof. Since x ( τ ) = 1 and x ( τ ) = 0 are stationary solutions of dxdτ = x (1 − x )( − (cid:101) a + (cid:101) a I + (cid:101) a S ) , the uniqueness of solutions ensures that x ( τ ) ∈ [0 , for all τ (cid:62) , whenever x (0) ∈ (0 , .Furthermore, from (6) we have that dNdτ = µ (1 − N ) . Since N (0) = 1 , follows that S ( τ )+ I ( τ )+ R ( τ ) = N ( τ ) = 1 for all τ (cid:62) .To prove that S , I , and R are positives, we analyze the behavior of the solutions with initialconditions at the border of R (cid:62) .Case 1. If S (0) = 0 then dSdτ (0) = 1 > , therefore S grows locally.Case 2. If I (0) = 0 then dIdτ (0) = 0 , therefore I ( τ ) will remain non-negative.Case 3. If R (0) = 0 then dRdτ (0) = ( k − I (0) . In this case, if I (0) = 0 , then dRdτ (0) = 0 , whence R will remain non-negative. On the other hand, if I (0) > then dRdτ (0) > since k > . Thus R grows locally.
3. Equilibrium States
In this subsection, we determine all the possible equilibrium points of model (6) and its conditionsfor existence. The following lemma summarizes the results regarding the six different classes ofequilibrium points that can be obtained.
Lemma 2.
Any equilibrium point P = ( ¯ S, ¯ I, ¯ R, ¯ x ) of model (6) satisfies that ¯ I = k (cid:0) − ¯ S (cid:1) and ¯ R = (cid:0) − k (cid:1) (cid:0) − ¯ S (cid:1) . Thus all equilibrium points are determined by the values of ¯ S and ¯ x .Furthermore, all the equilibrium points of model (6) fall into one of the following categories: P : ¯ S = 1 and ¯ x = 0 ; P : ¯ S = 1 and ¯ x = 1 ; P : ¯ S = 1 and ¯ x ∈ [0 , , s.t. (cid:101) a = (cid:101) a ; P : ¯ S = R and ¯ x = 0 , s.t. R > ; P : ¯ S = R p and ¯ x = 1 − R p R , s.t. R > R p > and (cid:101) a (cid:54) = k (cid:101) a ; P : ¯ S = R (1 − ¯ x ) and ¯ x ∈ (cid:16) , R − R (cid:17) , s.t. R > and k (cid:101) a = (cid:101) a = k (cid:101) a . roof. The equilibrium points of (6) are the solutions in Ω of the non-linear system − (1 − ¯ x ) kR ¯ S ¯ I − ¯ S = 0(1 − ¯ x ) kR ¯ S ¯ I − k ¯ I = 0( k −
1) ¯ I − ¯ R = 0¯ x (1 − ¯ x )[ − (cid:101) a + (cid:101) a ¯ I + (cid:101) a ¯ S ] = 0 . (7)Note from the first equation that ¯ S can not be equal to zero. Now, adding the first two equationsin (7), we obtain that any equilibrium point must satisfy − ¯ S = k ¯ I . Therefore ¯ I = 1 k (cid:0) − ¯ S (cid:1) (8)and thus, from third equation in (7), follows that ¯ R = (cid:18) − k (cid:19) (cid:0) − ¯ S (cid:1) . (9)Thus, if ¯ S = 1 , then (8) and (9) implies that ¯ I = ¯ R = 0 and the expressions for equilibrium types P , P and P can be obtain from fourth equation in (7).If ¯ S (cid:54) = 1 , then (8) implies that ¯ I (cid:54) = 0 . Thus, from second equation in (7), we obtain (1 − ¯ x ) R ¯ S = 1 , which implies that in this case ¯ x (cid:54) = 1 and therefore ¯ S = 1 R (1 − ¯ x ) . (10)Equation (10) implies the expression for equilibrium P but additionally, can be used jointly withequation (8) and the fact that R p = (cid:101) a − k (cid:101) a (cid:101) a − k (cid:101) a , obtain by basic manipulations of the fourth equationin (7), the expressions and conditions defining P and P . Model (6) has more possible equilibrium points that the classic SIR model. Indeed, the classicalSIR model has only two equilibrium points: a disease-free equilibrium and an endemic equilibriumthat corresponds precisely to equilibria P and P . In addition, model (6) have other disease-freeequilibria ( P and P ) and other endemic equilibria ( P and P ).The equilibrium points P and P differs only in the last component: in P no one is adoptingthe anti-infection behavior and in P all population does. Although P seems an ideal scenario, itmay not be realistic even if the prevention policy has an insignificant cost.7quilibrium type P also differs from P only in the last component. However, note that P represents an infinite set of equilibrium, since for each ¯ x we obtain a different equilibrium point.In particular, P include P and P when ¯ x = 0 and ¯ x = 1 , respectively. In fact, P represents aconnected path between these two disease-free equilibria.Note that the family of equilibria P exists only if (cid:101) a = (cid:101) a . In terms of the original parameters,this is equivalent to a c = a S , that is, the fixed cost has to be exactly equal to the payoff associatedwith the proportion of susceptible members of the population. Such equality between parametersmay be unrealistic, thus we consider P of minor practical interest. This also applies to equilibriumfamily P which has also a condition for its existence involving equality between parameters.As mentioned before, P corresponds to the endemic equilibrium of the classical SIR model andhas the same existence condition ( R > ) in that context.In turn, the equilibrium point P does not coincide with any equilibrium of the classic SIR modeland can be considered as a more realistic scenario. In the P case, the infection is present ( ¯ I (cid:54) = 0 )and only a part of the population adopted the anti-infection behavior. Note also that the condition R p < R , implies that the proportion of the susceptible population in P is greater than in P .Consequently, the proportion of infected population in P is lower than in P . Therefore, P can beinterpreted as a desirable situation where anti-infection behavior reduces the impactof the disease in the long-term .Note also that in this P scenario, for a fixed value of R , the larger is R p , the smaller is theproportion of infected people. This relationship between R , a parameter related only to the disease,and R p , a parameter related to the cost of intervention, allows an analysis of the effects of behaviorand cost/payoff changes in the disease dynamic. The best-case scenario would be one with a minimalvalue for ¯ I , or equivalently, a maximal value for ¯ x . This will occur if R p tends to and in the limitthis will imply (cid:101) a = (cid:101) a (existence condition of P ).The worst-case scenario for P would be one where R p goes to R because in this case, ¯ x goes tozero and P goes to P .Equilibrium type P represents an infinite set of endemics equilibrium points, one for each ¯ x ∈ (cid:16) , R − R (cid:17) . Unlike disease-free equilibria P , in P the value of ¯ x will affect the value of ¯ S ( ¯ I and ¯ R too). Note that if ¯ x approach R − R , then ¯ S approach . This means that if the proportion of thepopulation adopting the prevention behavior increase, the proportion of susceptible population alsoincreases (and the proportion of infected population decrease).Note that, when ¯ x goes to , P goes to P , and when ¯ x = R − R , the equilibrium P goes to a8 equilibrium point. In fact, when k (cid:101) a = (cid:101) a = k (cid:101) a both sets of equilibria P and P coexist andhave a linking point at (cid:16) , , , R − R (cid:17) . Lastly, note that equilibrium points P cannot co-exist withequilibrium point P , since its existence conditions are incompatible. We are interested in study the stability of equilibrium points of (6). Then, it will be useful toconsider the associated Jacobian matrix given by: J ( S,I,R,x ) = − (1 − x ) IkR − − (1 − x ) kR S IkR S (1 − x ) IkR (1 − x ) kR S − k − IkR S k − − − x ) x (cid:101) a (1 − x ) x (cid:101) a − x )( − (cid:101) a + (cid:101) a I + (cid:101) a S ) . The characteristic polynomial of J ( S, I, R, x ) can be written as: p ( λ ) = | J ( S, I, R, x ) − λI | = ( − − λ ) q ( λ ) (11)where q ( λ ) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − (1 − x ) IkR − − λ − (1 − x ) kR S IkR S (1 − x ) IkR (1 − x ) kR S − k − λ − IkR S (1 − x ) x (cid:101) a (1 − x ) x (cid:101) a (1 − x )( − (cid:101) a + (cid:101) a I + (cid:101) a S ) − λ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . P , P , P , and P It is clear from (11) that for any equilibrium point, the Jacobian has at least one negativeeigenvalue λ = − and that additional eigenvalues can be studied analyzing the equation q ( λ ) = 0 .This can be used to establish the following subsection result about the stability of equilibrium points P , P , P , and P whose complete proof is presented in the Appendix. Theorem 1.
Consider system (6) . Assume that (cid:101) a (cid:54) = (cid:101) a , (cid:101) a (cid:54) = (cid:101) a /k , R (cid:54) = 1 , and R (cid:54) = R p . If R < then i. P is locally asymptotically stable if (cid:101) a > (cid:101) a ; ii. P is locally asymptotically stable if (cid:101) a < (cid:101) a , iii. P and P do not exist. If R > and R < R p , then i. P is locally asymptotically stable if (cid:101) a > (cid:101) a ; ii. P is locally asymptotically stable if (cid:101) a < (cid:101) a ; P is not stable; iv. P do not exist. If R > and R > R p , i. P is locally asymptotically stable if (cid:101) a > (cid:101) a , and (cid:101) a < (cid:101) a /k ; ii. P is locally asymptotically stable if (cid:101) a > (cid:101) a , and (cid:101) a > (cid:101) a /k ; iii. P is locally asymptotically stable if (cid:101) a < (cid:101) a , and (cid:101) a < (cid:101) a /k ; iv. P and P are locally asymptotically stable if (cid:101) a < (cid:101) a , and (cid:101) a > (cid:101) a /k ; v. P is not stable.3.5. Comments on Theorem 1 In the classic SIR model (1), when the basic replication rate is sufficiently low ( R < ), thedisease-free equilibrium point is stable, so the infection does not become an epidemic. As describedin Theorem 1, this phenomenon also occurs in system (6) but in this case, there are two possibledisease-free equilibrium: P (zero behavior adoption) and P (complete behavior adoption). Thevalues of (cid:101) a and (cid:101) a determine which one is stable.When the disease is more infectious ( R > ), the classic SIR model admits only one possibility:the endemic equilibrium is stable and the disease-free equilibrium is unstable. Cases (2i) and (3ii) ofthe Theorem 1 are equivalent to this situation, since P is equivalent to the endemic equilibrium ofthe classical SIR model. However, in model (6) some more realistic behaviors may occur. Note forexample that it is possible that a disease-free equilibrium P and the endemic equilibrium P coexistsimultaneously, both being locally stable (Theorem 1 (3iii)). Figure 1(a) illustrates this interestingcase. Note also that in this situation, equilibrium points P and P also exist but are not stable.From Theorem 1 (2ii) and (3ii) another remarkable behavior can be observed, even if R > , it ispossible that the system has a disease-free and unique stable equilibrium. Figure 1(b) illustrates thissituation. In this case, the equilibrium P exists and is unstable and equilibrium P does not exit.Finally, besides P , we have the possibility of another endemic stable equilibrium: the equilibriumpoint P . Assuming that conditions of Theorem 1 (3i) are satisfied, equilibria P , P , and P exists,but only the last one is stable. This equilibrium is particularly interesting because it represents amore favorable epidemiological situation than the equilibrium point P . Indeed, as R p < R , thevalue of ¯ I at P is smaller than the value of ¯ I at P .10 x I (a) Coexistence of Stable Equilibrium P (inblue) and P (in red). Ix (b) P (in blue) stable with R > . x I (c) Locally Stable Equilibrium P (in orange). Ix (d) Coexistence of the Families P (in blue) and P (in red).Figure 1: Solution curves I ( τ ) × x ( τ ) of system (6) . Numerical simulations of solutions with differentinitial conditions. Temporal evolution is represented using dark green for initial trajectory points and agradual variation to yellow as time increase. The equilibrium points P , P , P , and P are denoted by thedots in the color black, blue, red, and orange, respectively, while the families of equilibria P and P aredenoted by the lines in the color blue and red, respectively (when they exist). In all cases pictured k = 2 and R = 5 . In 1(a) (cid:101) a = (cid:101) a = 1 and (cid:101) a = 2 . Thus R p = 3 and R > max { , R p } , ensuring that P and P areboth locally stable, while P and P are unstable (the instability of P is highlighted in zoom). In 1(b) (cid:101) a = 1 , (cid:101) a = 7 , and (cid:101) a = 2 . Thus R p = 0 . and R > max { , R p } , ensuring that P are locally stable, while P and P are unstable. In 1(c) (cid:101) a = 3 , (cid:101) a = 7 , and (cid:101) a = 2 . Thus R p = 3 , R > max { , R p } , and (cid:101) a /k > (cid:101) a > (cid:101) a ,ensuring that P are locally stable, while P , P , and P are unstable. In 1(d) (cid:101) a = (cid:101) a = 1 and (cid:101) a = 2 . Thus,Theorem 1 ensures that whole family P is stable. .6. On the stability of equilibria family P As mentioned before, the existence conditions for equilibrium families P and P involve equalitybetween some parameters which can be unrealistic. The corresponding stability analysis can not bedone using the standard approach based on the Jacobian matrix as in Theorem 1, because in thiscases, the corresponding Jacobian matrix have a null eigenvalue. In fact, solving equation (11) forequilibrium P equilibrium lead us to λ = − , λ = − , λ = 0 and λ = k ( R (1 − ¯ x ) − , and in a similar fashion, the P equilibrium points also have a null eigenvalue .It can be noted however that some points in the equilibrium family P may be locally stable, asillustrated in the Figure 1(d). When ¯ x = R − R , then P becomes (1 , , , ¯ x ) , so this point is a linkingpoint between P and P . Note that this point acts as the threshold between stable and unstableequilibrium points in P .Nevertheless, a closer look at system (6) and to the conditions for the existence of P , allow us todetermine some stability conditions for equilibria P presented in the following theorem. Theorem 2.
Assume that in model (6) we have R > and k (cid:101) a = (cid:101) a = k (cid:101) a , so the family ofequilibria P exists. If − R k ( R − k − < (cid:101) a , then the family of equilibria P is stable.Proof. Note first that if − R k ( R − k − < (cid:101) a then for all ¯ x ∈ (cid:16) , R − R (cid:17) we have that − (1 − ¯ x ) R k ¯ x ( k − < (cid:101) a , (12)because < ¯ x < R − R implies that − − ¯ x ¯ x < − R − . Now, if k (cid:101) a = (cid:101) a = k (cid:101) a , we have from thirdand fourth equation of the system (6) that dxdτ = x (1 − x )[ − (cid:101) a + k (cid:101) a I + (cid:101) a S ]= (cid:101) a x (1 − x )[( k − I − R ]= (cid:101) a x (1 − x ) dRdτ . Thus, we have that dxdR = (cid:101) a x (1 − x ) , Null eigenvalues appears also for P if (cid:101) a = (cid:101) a or R = 1 , and for P when (cid:101) a = (cid:101) a x in terms of R as x ( R ) = e (cid:101) a R e (cid:101) a R + C . This consideration allows us to eliminate the differential equation for x in (6) and using that S = 1 − I − R , we can reduce model (6) to a simplified epidemic model with a recovered-dependantinfection described as dIdτ = I [ f ( R ) (1 − I − R ) − k ] dRdτ = ( k − I − R, (13)where f ( R ) = (1 − x ( R )) kR . Recovered-dependent epidemic models as (13) were considered by theauthors in [20]. In particular, Theorem 4.3 in [20] establish the following result: If f is positive function, differentiable on [0 , and ( I ∗ , R ∗ ) is an endemic equilibrium point of (13) such that dfdR ( R ∗ ) < k − f ( R ∗ ) then ( I ∗ , R ∗ ) is a locally stable equilibrium point. Note that if f ( R ) = (1 − x ( R )) kR , f is in fact a positive differentiable function on R . Additionally,the following inequalities equivalences holds: dfdR ( R ∗ ) < k − f ( R ∗ ) − kR dxdR < k − f ( R ∗ ) − kR (cid:101) a x (1 − x ) < (1 − x ) k R k − − (cid:101) a x < (1 − x ) kR k − − (1 − x ) R kx ( k − < (cid:101) a , which we already showed in (12) is valid when − R k ( R − k − < (cid:101) a . Therefore, we conclude that thewhole family of equilibria P is stable.
4. Controlling the infection through population behavior: Choosing the right payoffs
In this section, we use the results in Theorem 1 to find conditions on the behavioral payoffs, thatproduce a diminishing on the infected population at a stable equilibrium. This can be interpreted asspecific policy actions leading to reduce and control the infection in the long-term.According to system (6), an infectious disease with a small replication rate ( R < ), requiresno anti-infection behavior to be eradicated, since the possible stable points P and P are bothdisease-free. Nevertheless, the stability conditions in part 1. of Theorem 1 can be rewritten in terms13f the original parameters as follows: if a c > a s , then P is locally asymptotically stable; if a c < a s ,then P is locally asymptotically stable. This can be interpreted in terms of public policies, as aquantification of how much reduction on the fixed cost a c is necessary to achieve full adoption ofan anti-infection behavior; if a c is smaller than a s , then in the long-term everyone tends tofollow the prevention behavior, even if the disease is poorly infectious ( R < ) .We focus now on the situation when R > and therefore, the infectious disease may becameendemic. We aim to determine, in terms of R p , a c , a S , a I , and a R , successful intervention strategiesto control the disease. We consider two scenarios: Scenario 1:
Assume that a c < a S and therefore (cid:101) a < (cid:101) a . In this case, from parts 2. and 3.in Theorem 1 we have two possibilities: only the disease-free equilibrium P is stable (cases (2ii)and (3iv)), or P and the endemic equilibrium P are stables (case (3iii)).From the epidemiological point of view, we would like to avoid the case of stability of an endemicequilibrium. Therefore, to avoid the stability of P , we must ensure that (cid:101) a < (cid:101) a /k , thatis, besides a c < a S , we need that a c < a I k + a R (cid:0) − k (cid:1) .This is an ideal scenario that can be interpreted as disease eradication in the long-run. Scenario 2:
Consider now that a c > a S (so (cid:101) a > (cid:101) a ), and still R > .In this case, the locally stable points will always be endemic: P (cases (2i) and (3ii)) or P (case (3i)). Note however that, if P exists ( R p < R ), this equilibrium will represent a better situationthan P , since the proportion of infected in P will be lower than in P . Although R does not dependon the payoff parameters, R p does, therefore in order to obtain a lower proportion of infected,we must seek strategies such that the payoff parameters imply R p < R . Furthermore,it is not enough that P exists, we want P to be stable. Then, in addition to (cid:101) a > (cid:101) a and R p < R , we must also be sure that (cid:101) a < (cid:101) a /k .Note also that the components of P depend on the value of R p and if R p goes to 1, the proportionof infected persons predicted by this equilibrium decrease. Given an infectious disease with R > ,whereas it is not possible to change the inequality (cid:101) a > (cid:101) a , it is possible to decrease the number ofinfected people ensuring that (cid:101) a be less than (cid:101) a /k (so P is stable) and as close as possible to (cid:101) a .In this scenario, it is possible to quantify precisely the percentage of reduction on the infectedpopulation, produced by changes in the payoff parameters, as described in the following proposition. Proposition 1.
Consider system (6) and assume that R > and (cid:101) a < (cid:101) a < (cid:101) a /k . A reductionof p % in (cid:101) a produce a reduction of (cid:16) (cid:101) a (cid:101) a − k (cid:101) a (cid:17) p percentage points in the infected population on theendemic equilibrium state P and a relative reduction of (cid:101) a (cid:101) a − (cid:101) a p % . roof. We can compute the percentage point reduction by computing the difference between the oldvalue of the proportion of the infected population at the equilibrium point P (denoted by ¯ I ) andthe new value (denoted by (cid:98) I ) obtained after the reduction on a . Note that ¯ I = k (cid:16) − R p (cid:17) = k (cid:16) − (cid:101) a − k (cid:101) a (cid:101) a − k (cid:101) a (cid:17) = (cid:101) a − (cid:101) a (cid:101) a − k (cid:101) a , so we have that ¯ I − (cid:98) I = (cid:101) a − (cid:101) a (cid:101) a − k (cid:101) a − (1 − p ) (cid:101) a − (cid:101) a (cid:101) a − k (cid:101) a = (cid:101) a (cid:101) a − k (cid:101) a p . Therefore, the reduction of p % in (cid:101) a can be interpreted as a reduction, in the long-term, of (cid:16) (cid:101) a (cid:101) a − k (cid:101) a (cid:17) p percentage points in the proportion of infected population.The corresponding relative reduction can be obtained as ( ¯ I − (cid:98) I )100¯ I = (cid:101) a (cid:101) a − k (cid:101) a (cid:101) a − k (cid:101) a (cid:101) a − (cid:101) a p = (cid:101) a (cid:101) a − (cid:101) a p. So, a reduction of p % in (cid:101) a can be interpreted as a reduction, in the long-term, of (cid:101) a (cid:101) a − (cid:101) a p % in theproportion of infected population. Example
Recent measles outbreaks have been associated with a lack of effective vaccination,mainly due to misinformation on the inherent risks of vaccines [30]. In terms of the model proposedin this paper, erroneously high valuations on vaccination risk could be interpreted as a high value for a c or equivalently, a high value for (cid:101) a . In this context, it is relevant to ask how much (cid:101) a must bereduced to obtain, for example, a reduction of 1 percentage point on the infected population in thelong-term. Under conditions on Proposition 1, this desired one percentage point reduction can beobtained by a reduction of (cid:16) (cid:101) a − k (cid:101) a (cid:101) a (cid:17) % in (cid:101) a .To obtain useful insights from last expression, besides considering the limitations and partialvalidity of using the proposed model for this specific disease, one should also be able to have estimationof k , (cid:101) a , (cid:101) a , and (cid:101) a . These last parameters were just introduced in the present paper and as such,there are not estimations available yet.For illustration purposes, we present in Figure 4 a heat map for p , the percentage reduction on a , depending on the values of (cid:101) a and (cid:101) a , that would be necessary to obtain a 1 percentage point15eduction on the infected population in the long-term, considering the value of a as a normalizedquantity equals to 1 and an estimated value of k equals to 3.8.From this estimations, we have for example that, if in comparison with (cid:101) a , (cid:101) a is 10 times greaterand (cid:101) a is a half, then, to obtain a 1 percentage point reduction on the infected population in thelong-term it is necessary at least a reduction of 8.1% on (cid:101) a . Figure 2:
Heat map for p . Necessary percentage reduction on a ( p ), depending on the values of (cid:101) a and (cid:101) a ,to obtain a reduction of 1 percentage point on the measles infected population in the long-term
5. Final Comments
The main contribution of this paper is the introduction of a mathematical model to analyze theinterplay between infectious disease and anti-infection behavior adoption across the population. Wefocused on equilibrium states (Lemma 2) and showed the appearance of remarkable characteristicsin the context of epidemiological models (Theorem 1), such as the coexistence of two locally stableendemic equilibria, the coexistence of a locally stable endemic and a disease-free equilibrium, andeven the possibility of a stable continuum of endemic equilibrium points (Theorem 2). We determine An estimation of k could be obtained from the equality R = βµ + γ = βµk in (5) so k = βR µ . For measles, R is commonly considered between 12-18, and in this example we consider it equals to 18. As discussed in [31], thisestimation may not be adequate for all kinds of populations. The risk of transmission of an infectious disease is closelyrelated to the infection rate β and we consider the worst-case scenario where both parameters are equals. For measles,we consider this value equal to 90% [32] The constant µ can be estimated as the inverse of the mean life expectancyand we are considering µ = x , for the epidemiological variables, may not be linear. Also,it would be reasonable to consider that the payoff parameters are not necessarily constants andmay vary on time. Different ways to model the variation and the effects of the behavioral variable x can also be considered. Other models different from SIR can be suitable for specific situations,including models considering delay differential equations to incorporate delayed effects/variations onthe behavior adoption rate. We consider that the results obtained in the present work open valuablepaths of research. Appendix A. Proof of Theorem 1
In this Appendix, we present the proof of Theorem 1 based on the Jacobian matrix and charac-teristic polynomial (11).As mentioned before, any equilibrium point has at least one eigenvalue λ = − , and the othereigenvalues can be studied by analyzing the equation q ( λ ) = 0 for P , P , P , and P . This isdescribed as follows. Case: P = (1 , , , In this case, we have q ( λ ) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − − λ − kR kR − k − λ
00 0 − (cid:101) a + (cid:101) a − λ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . Thence, the additional eigenvalues are λ = − , λ = k ( R − and λ = (cid:101) a − (cid:101) a . R < and (cid:101) a < (cid:101) a , then all eigenvalues will be negative and, consequently, P is locallyasymptotically stable. If R > or (cid:101) a > (cid:101) a , then P is not stable. Case: P = (1 , , , In this case, we obtain q ( λ ) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − − λ − k − λ
00 0 (cid:101) a − (cid:101) a − λ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . Thence, the additional eigenvalues are λ = − , λ = − k, and λ = (cid:101) a − (cid:101) a . Therefore, it is sufficient that (cid:101) a < (cid:101) a for P to be locally asymptotically stable. If (cid:101) a > (cid:101) a , then P is not stable. Case: P = (cid:16) R , k (cid:16) − R (cid:17) , (cid:0) − k (cid:1) (cid:16) − R (cid:17) , (cid:17) In this case, we have that q ( λ ) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − (cid:16) − R (cid:17) R − − λ − k − R (cid:16) − R (cid:17) R − λ R −
10 0 − (cid:101) a + (cid:101) a (cid:16) − R (cid:17) k + (cid:101) a R − λ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = q ( λ ) q ( λ ) , where q ( λ ) = R ( (cid:101) a − (cid:101) a k ) − (cid:101) a + (cid:101) a kkR − λ and q ( λ ) = λ + λ (cid:18)(cid:18) − R (cid:19) R + 1 (cid:19) + k (cid:18) − R (cid:19) R . Thence, the additional eigenvalues are λ = R ( (cid:101) a − (cid:101) a k ) − (cid:101) a + (cid:101) a kkR (the root of q ) and the roots of thequadratic polynomial q . If R > , then the coefficients of q are all positives and therefore from theRouth–Hurwitz criterion, we conclude that eigenvalues associated with this polynomial must havenegative real part. Note also that in this case λ < , if and only if R ( (cid:101) a − (cid:101) a k ) − (cid:101) a + (cid:101) a k < or,equivalently, R ( (cid:101) a − (cid:101) a k ) < (cid:101) a − (cid:101) a k .Therefore, λ < if and only if • (cid:101) a − (cid:101) a k > and R < (cid:101) a − (cid:101) a k (cid:101) a − (cid:101) a k = R p , or 18 (cid:101) a − (cid:101) a k < and R > (cid:101) a − (cid:101) a k (cid:101) a − (cid:101) a k = R p , or • (cid:101) a − (cid:101) a k = 0 and (cid:101) a − (cid:101) a k > .If any of these conditions are satisfied, then P is locally asymptotically stable. Case: P = (cid:16) R p , k (cid:16) − R p (cid:17) , (cid:0) − k (cid:1) (cid:16) − R p (cid:17) , − R p R (cid:17) In this case we obtain q ( λ ) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − R p − λ − k R R p ( R p − R p − − λ − R R p ( R p − (cid:101) a ( R − R p ) R p R (cid:101) a ( R − R p ) R p R − λ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = − λ − C λ − C λ − C , where C = R p C = ( R p − R R p (( R − R p )( (cid:101) a − (cid:101) a ) + kR R p ) , and C = ( (cid:101) a − (cid:101) a k )( R − R p )( R p − R R p . According to the Routh-Hurwitz criterion, the roots of − q (also roots of q ) will have the negativereal part if, and only if, C > , C > and C C − C > . If < R p < R , then we have immediately that C = R p > . Furthermore, in this case for C > it is necessary and sufficient that (cid:101) a − (cid:101) a k > . (A.1)Additionally, note that C C − C = ( R p − R − R p ) R (cid:20) ( (cid:101) a k − (cid:101) a ) + kR R p R − R p (cid:21) since ( (cid:101) a − (cid:101) a k ) R p = ( (cid:101) a − (cid:101) a k ) (cid:101) a − (cid:101) a k (cid:101) a − (cid:101) a k = (cid:101) a − (cid:101) a k .Considering that R > , R p > and R > R p , then C C − C > ⇔ ( (cid:101) a k − (cid:101) a ) + kR R p R − R p > ⇔ (cid:101) a < k (cid:18)(cid:101) a + R R p R − R p (cid:19) . (A.2)19hat is, P is locally asymptotically stable, if and only if, (A.1) and (A.2) are satisfied (s.t. theconditions of P existence).Remember that the existence conditions for P are (cid:101) a (cid:54) = k (cid:101) a (A.3) < R p < R . (A.4)Since R p = (cid:101) a − k (cid:101) a (cid:101) a − k (cid:101) a , to analyze inequality (A.4) we separate (A.3) in two cases:Case 1: (cid:101) a − k (cid:101) a > .Multiplying (A.4) by (cid:101) a − k (cid:101) a we have (cid:101) a − k (cid:101) a < (cid:101) a − k (cid:101) a < R ( (cid:101) a − k (cid:101) a ) ⇒ − k (cid:101) a < − k (cid:101) a < R ( (cid:101) a − k (cid:101) a ) − (cid:101) a ⇒ k (cid:101) a > k (cid:101) a > − R ( (cid:101) a − k (cid:101) a ) + (cid:101) a . Joining the last inequality with the hypothesis considered in this case we have (cid:101) a > k (cid:101) a > k (cid:101) a > − R ( (cid:101) a − k (cid:101) a ) + (cid:101) a . (A.5)Case 2: (cid:101) a − k (cid:101) a < .Analogously to the previous case, we will have (cid:101) a < k (cid:101) a < k (cid:101) a < − R ( (cid:101) a − k (cid:101) a ) + (cid:101) a . Note that, in order to ensure P stability, is necessary that (cid:101) a > (cid:101) a k (condition (A.1)), whichonly occurs in (A.5). 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