Modeling control strategies for influenza A H1N1 epidemics: SIR models

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1 UPLEMENTO Revista Mexicana de Física JUNIO 2012 Modeling control strategies for influenza A H1N1 epidemics: IR models G. Gómez Alcaraz Depto. de Matemáticas, Fac. de Ciencias, Universidad Nacional Autónoma de México, México, D.F. gomal@servidor.unam.mx C. Vargas-De-León Unidad Académica de Matemáticas, Universidad Autónoma de Guerrero, México Facultad de Estudios uperiores Zaragoza, Universidad Nacional Autónoma de México, México. Recibido el 23 de Marzo de 2010; aceptado el 27 de Abril de 2011 We present a review of models of swine flu A H1N1. We discuss how control of epidemic critically depends on the value of the Basic Reproduction Number R 0. The R 0 for new influenza estimated 1.5 [5. By means of suitable Volterra-type Lyapunov functions, we establish the global stability of the steady state of an disease transmission models with control measures type IR. We conclude that R 0 plays an important role in the designing disease control strategies. Keywords: IR epidemic model; influenza; basic reproduction number; control measures; global stability; direct Lyapunov method. Presentamos una revisión de los modelos de la Influenza A H1N1. Discutimos que el control de la epidemia depende del valor del número reproductivo básico R 0. El R 0 de la influenza A H1N1 es estimado en 1,5 [5. Por medio de las funciones de Lyapunov tipo Volterra, demostramos la estabilidad global de los puntos de equilibrios de un modelo epidemiológico tipo IR que incorpora medidas de control. Concluimos que el R 0 de la enfermedad juega un rol importante en el diseño de las medidas de control de la epidemia. Descriptores: Modelo epidemiológico tipo IR; influenza; número reproductivo básico; medidas de control; estabilidad global; método directo de Lyapunov. PAC: Hq; a; Cc; xd. 1. Introduction On April 9, 2009 it became apparent to public health officials in Mexico City that an outbreak of influenza was in progress late in the influenza season. Virus samples were obtained and the virus determined to be a novel strain of influenza A of the H1N1 serotype. Preliminary tests conducted by the Centers for Disease Control and Prevention CDC indicated that the virus was a novel reassortant, containing genetic elements of influenza viruses found in swine, birds and human beings [1. Influenza virus, an enveloped virus of the Orthomyxoviridae family, has a unique capacity for genetic variation that is based in two molecular features of the virus family [2. The modeling of infectious diseases is a tool which has been used to study the mechanisms by which diseases spread, to prognostication of the development of an outbreak and to evaluate strategies to control an epidemic. For influenza outbreak historically we have: The basic model of disease influenza: model type IR compartmental model of disease transmission. The transfer diagram: leads to the following systems of differential equations for this IR model [3 are: d βi, di βi κi, 1 dr κi, 2 This first mathematical model is the basic epidemiological model for the description of an influenza epidemic, was developed in 1927 by Kermack and McKendrick [3. This model is known as the usceptible-infectious-recovered IR model, and is shown as a flow diagram in Fig. 1. To simulate an influenza epidemic the model is analyzed on a computer and one infected individual I is introduced into a closed population where everyone is susceptible. Each infected individual I transmits influenza, with rate β, to each susceptible individual they encounter. The number of susceptible individuals decreases as the incidence i.e., the number of individuals infected per unit time increases. At a certain point the epidemic curve peaks, and subsequently declines, because infected individuals recover and stop transmitting the virus. Only a single influenza epidemic can occur in a closed population because there is no inflow of susceptible individuals. The severity of the epidemic and the initial FIGURE 1. The transfer diagram for the IR model without vital dynamics

2 38 G. GÓMEZ ALCARAZ AND C. VARGA-DE-LEÓN rate of increase depend upon the value of the Basic Reproduction Number R 0. R 0 is defined as the average number of new infections that one case generates, in an entirely susceptible population, during the time they are infectious. If R 0 > 1 an epidemic will occur and if R 0 < 1 the outbreak will die out. The value of R 0 for any specific epidemic can be estimated by fitting the IR model to incidence data collected during the initial exponential growth phase. The value of R 0 may also be calculated retroactively from the final size of the epidemic. If the IR model is used, R 0 for influenza is equal to the infectivity/transmissibility of the strain β multiplied by the duration of the infectious period, with value approximately equal [5. Therefore once the value of R 0 has been obtained, the value of β can be determined [4. The IR model is the basis for all subsequent influenza models. The extension to model with demographics; specifically, immigration and emigration of individuals into the population. Analysis of this demographic model shows that influenza epidemics can be expected to cycle, with damped oscillations. The IR model can extended to predict the spatial - temporal dynamics of an influenza epidemic. The first spatial - temporal model of influenza was developed in the late 60s by Rvachev [6. He connected a series of analogous IR models in order to construct a network model of linked epidemics. He then modeled the geographic spread of influenza in the former oviet Union by using travel data to estimate the degree of linkage between epidemics in major cities. In the 80s, he and his colleagues Baroyan and Longini extended his network model and evaluated the effect of air travel on influenza pandemics [7,8. ince then other modeling studies have quantified the importance of air travel on geographic spread [9,10. The IR compartmental model of disease transmission incorporating only treatment. The compartmental model of disease transmission incorporating vaccination and treatment. The IR model of disease transmission incorporating quarantine and isolation measures. In Ref. 11 the model is a EIRQ. Mathematical models have been used to understand the spatial-temporal transmission dynamics of influenza, and too used to predict future epidemics or pandemics. The potential epidemiological impact of both behavioral and biomedical interventions has been investigated. Here present the model of influenza studies how can provide insights into circulating FIGURE 2. The transfer diagram for the IR model with vaccination, treatment, quarantine and isolation measures. strain of novel influenza A H1N1. This strain was formerly known as swine flu. 2. The IR epidemic model with control measures The IR compartmental model of infection disease transmission incorporating vaccination, treatment, quarantine and isolation measures. The transfer diagram in Fig. 2. The transfer diagram leads to the following system of differential equations: d Λ βi µ + φ γ + δ Q, di βi κ + µ + α + σ + γ II, dq γ µ + δ Q, dr κi + δ IQ I µr, 3 dq I γ I I µ + δ I Q I, dv φ µv, dt σi µt. Where susceptible individuals can be quarantined Q and then returned to the pool of susceptible individuals once it is determined they are uninfected. usceptible individuals can be vaccinated V and infected individuals I can be treated with antiviral drugs T. If they develop symptoms and become infectious they can be isolated Q I, as can the infected individuals I, send to recovered individuals with permanent immunity R. The assumptions are the following. Λ is the recruitment rate of susceptibles, µ is the per capita natural mortality rate, β is the transmission probabilities of susceptible individuals, κ is the rate constant for recovery, and α is the disease-related death rate of infectious individuals. The susceptible population is vaccinated at a constant rate φ, and there is no loss of effectiveness of vaccine. The parameters γ and δ are the rate of isolation of susceptible individuals and the rate at which return to susceptible class from class Q, respectively. The parameter γ I is the rate constant for individuals leaving the infectious class I for the quarantine class Q I. The parameter δ I is the rate at which individuals recover and return to recovered class R from class Q I. The parameter σ represent the therapeutic treatment coverage of infected individuals. Heuristically, 1/κ can be regarded as the mean infectious period, and 1/δ the mean period of susceptible individuals isolated. In the Table I is show some IR models, that are simplifications of model 2. The total population size N+I+Q +R+Q I +V +T is variable with N Λ µn αi. In the absence of disease, the population size N approaches the carrying capacity Rev. Mex. Fis

3 MODELING CONTROL TRATEGIE FOR INFLUENZA A H1N1 EPIDEMIC: IR MODEL 39 TABLE I. The basic reproduction numbers of several IR models for influenza A H1N1. IR Models with vital dynamics Vaccination, treatment, quarantine and isolation measures R c Quarantine and isolation measures Whit φ σ 0 R q Vaccination and treatment Whit γ γ I δ δ I 0 R v Without control measures Whit φ σ γ γ I δ δ I 0 R 0 Basic reproduction number βµ+δ Λ µ+δ µ+φ+µγ κ+µ+α+σ+γ I βµ+δ Λ µµ+δ +γ κ+µ+α+γ I βλ µ+φκ+µ+α+σ βλ µκ+µ+α Λ/µ. The differential equation for N implies that solutions of 2 starting in the positive orthant R 7 + either approach, enter, or remain in the subset of R 7 + defined by {, I, Q, R, Q I, V, T R 7 + :, I, Q, R, Q I, V, T 0, N Λ/µ}. This set is a positive invariant set of 2. The system 2 has a disease-free steady state with coordinates Q E, 0, Q, 0, 0, 0, 0, µ + δ Λ µ + δ µ + φ + µγ, γ Λ µ + δ µ + φ + µγ, 4 and it exists for all positive values of the parameters. Let β R c κ + µ + α + σ + γ I βµ + δ Λ µ + δ µ + φ + µγ κ + µ + α + σ + γ I. 5 is the Reproduction Control Number of the system 2. R c is the product of the number of susceptibles at the diseasefree steady state, the transmission coefficient β, and the average residence time 1/µ + κ + α + σ + γ I in the infectious individuals class. By straightforward computation, if R c > 1, system 2 has a unique endemic steady state E, I, Q, R, Q I, V, T in the interior of R 7 +, with coordinates R c, I ΛR c β 1 1Rc, Q γ µ + δ, R 1 µ κi + δ I Q I, Q I γ I µ + δ I I, V φ µ, T σ µ I 6 3. Global tability Now, we study the global stability behavior of the equilibria for the IQ subsystem of model 3 d Λ βi µ + φ + γ + δ Q, di βi κ + µ + α + σ + γ II, dq γ µ + δ Q, 7 which is independent of the R, Q I, V and T variables. The following results shows that if R c 1 the disease-free steady state is stable globally, and that the unique endemic steady state is globally asymptotic stability in the interior of feasible region when R c > 1. It can be verified that the non-negative octant R 3 + is positive invariant region with respect to system 7. Note that from the following differential inequality + I + Q Λ µ + I + Q, it easily follows that model 7 can be studied in the positive invariant and attractive set Ω {, I, Q R 3 + :, I, Q 0, + I + Q Λ/µ }. The system 7 has two steady states: the disease-free steady state and endemic steady state E, 0, Q E, I, Q with coordinates given by 4 and 6, respectively. Motivated by the works in Ref. 12 to 14, in this paper, we construct a Lyapunov functions for each steady state, using suitable linear combinations of Volterra-type functions to Rev. Mex. Fis

4 40 G. GÓMEZ ALCARAZ AND C. VARGA-DE-LEÓN proved the global stability. Then we have the following results. Theorem 3.1 i If R c 1, then the diseasefree steady state P is globally asymptotically stable in R 3 + {, I, Q : 0, I 0, Q 0}. ii If R c > 1, then the unique endemic steady state, P, is globally asymptotically stable in the interior of R 3 +. Proof: i tability of the disease-free steady state. Define the global Lyapunov function by U : {, I, Q R 3 + : > 0, Q > 0 } R U, I, Q ln + I + δ Q Q Q ln Q µ + δ Q. 8 Then U, I, Q is C 1 on the interior of R 3 +, P is the global minimum of U, I, Q on R 3 +, and U, 0, Q 0. The time derivative of 8 computed along solutions of 7 is Using the identities U, I, Q 1 d 1 Filling 10 into 9 yields Using 10 and 11, we obtain 1 Q dq Q, + di + δ µ + δ [Λ βi µ + φ + γ + δ Q + [βi κ + µ + α + σ + γ I I + δ µ + δ 1 Q [γ µ + δ Q. Q Λ µ + φ + γ δ Q, 9 γ µ + δ Q. 10 Λ µ + φ + µq. 11 [ U, I, Q 1 µ + φ + µq βi µ + φ + µ + δ Q + δ Q + [βi κ + µ + α + σ + γ I I + δ [ 1 Q µ + δ Q µ + δ [ 1 µ + φ + µq βi + [βi κ + µ + α + σ + γ I I + δ Q Q µ + φ + µ Q 1 Q Q [ 1 µ + φ + µ Q µ + φ + µ Q + [β I κ + µ + α + σ + γ I I + δ Q µ + φ + µ Q + δ Q 2 2 Q Q Q Q + [ Q Q + δ Q Q Q Q Q µ + δ Q β κ + µ + α + σ + γ I I. + Q [ δ Q + δ Q + 1 Q Q Q Rev. Mex. Fis

5 MODELING CONTROL TRATEGIE FOR INFLUENZA A H1N1 EPIDEMIC: IR MODEL 41 Rewritten U, I, Q in terms of basic reproduction number 5, we have 2 U, I, Q µ + φ + µ Q δ Q Q Q Q Q 2 κ + µ + α + σ + γ I 1 R c I. Thus, R c 1 implies that U, I, Q 0. It is easy to see that U, I, Q 0 is equivalent to the fact that, Q Q and I 0, or if R c 1, { and Q Q. And the largest invariant set in, I, Q R 3 + : U, I, Q 0 } is the singleton set {P }. Therefore, it follows from the Laalle s Invariance Principle [15 that P is globally stable for R 0 1. Proof. ii tability of the endemic steady state. We define a function L, I, R as follows: L, I, Q ln + I I I ln I I + δ Q Q Q ln Q µ + δ Q. 12 It is clear that the point P is the only critical point and minimum point of L, I, Q, and L, I, Q at the boundary of the interior of R 3 +. Consequently, P is the global minimum point, L, I, Q 0, and the function is bounded from below. Hence L, I, Q is a global Lyapunov function. At endemic steady state, we have Λ µ + φ + γ δ Q + β I, 13 κ + µ + α + σ + γ I β, 14 γ µ + δ Q. 15 Filling 15 into 13 yields Λ µ + φ + µq + β I. 16 The derivative of the function 12 along the solutions of system 7 satisfies L, I, Q 1 d 1 + δ µ + δ + I I di I + δ 1 Q dq µ + δ Q, [Λ βi µ + φ + γ + δ Q + I I [β κ + µ + α + σ + γ I 1 Q Q Using 14, 15 and 16, we obtain [γ µ + δ Q. [ L, I, Q 1 µ + φ + µq + β I βi µ + φ + µ + δ Q + δ Q + βi I + δ [ 1 Q µ + δ Q β µ + δ [ µ + φ + µq Q µ + φ + µ Q + [β I βi + βi I + δ Q [ µ + φ + µ Q I I I + I µ + φ + µ Q µ + δ Q 1 1 Q Q + δ Q [ δ Q [ + β I I I + I + δ Q + δ Q Q Q + Q + 1 Q Q Q Q Q Q Q + 1 Rev. Mex. Fis

6 42 G. GÓMEZ ALCARAZ AND C. VARGA-DE-LEÓN µ + φ + µ Q 2 + β I 2 + δ Q 2 2 µ + φ + µ Q + βi δ Q Q Q Q Q 2. Q Q Q Q Therefore, L, I, Q is negative for all, I, Q > 0. And we have L, I, Q 0 if and only if, I I and Q Q holds. The largest compact invariant set in {, I, Q R 3 + : L, I, Q 0 } is the singleton {P }, where P is the endemic steady state. By Laalle s invariance principle [15 then implies that P is globally asymptotically stable in the interior of R Conclusions In this paper, we studied a mathematical model that describes the control measures for influenza A H1N1 epidemics. Our investigate the qualitative behavior of the IR model with vaccination, treatment, quarantine and isolation measures, such as positive invariance, the boundedness, the stability of the steady states and qualitatively explore the mechanism of control of influenza outbreaks. Usually, it is difficult to obtain the global properties of population models in dimensions greater than or equal to three. We used the Lyapunov function method and analyzed the global stability of the disease-free steady states and the endemic steady states, constructed an Volterra-type Lyapunov functions for each steady state. Next we explore qualitatively the control measures of the influenza outbreak using the basic reproduction number of the IR models of the Table I. In the usual IR endemic model with vital dynamics ee Table, decreasing the average infectious period 1/κ decreases the endemic infectious fraction. If 1/κ is decreased enough, then the basic reproduction number R 0 is reduced below 1, so that the disease dies out. To the model with vital dynamics we add the assumption that in unit time a fraction φ of the susceptible class is vaccinated there is no loss of effectiveness of vaccine, and σ is the rate at which individuals is recovered by prophylactic treatment with antivirals, or therapeutic treatment. In the IR model with vaccination and treatment ee Table I, we can see that both the effective infectious period 1/κ+µ+α+σ and R v decrease as the treatment rate constant σ increases. Is important the role of the vaccination rate φ in controlling the spread of the disease, if φ is increased then the basic reproduction number R v is decreased. The quarantine process is another method for reducing the average infectious period by isolating some infectives, so that they do not transmit the infection. In the IR model with quarantine and isolation ee Table I, we can see that both the effective infectious period 1/κ + µ + α + γ I and R q decrease as the quarantine rate constant γ I increases. We observed that the quarantine reproduction number R q in their IR model with quarantine and isolation ee Table I was independent of the mean residence time in the quarantine class Q I. This independence also occurs in the models in this paper, since the mean time in Q I is 1/δ I, and the expressions for R q do not involve the parameter δ I. It is not surprising that the average length 1/δ I of the quarantine period does not affect the threshold quantity R q, since the models assume that people in the quarantine class Q I do not infect others and people are not infectious when they move out of the quarantine class. However, the quarantine reproduction number R q do depend on the parameter γ I, which governs the transfer rate out of the infectious class into the quarantine class. In practice, the parameters γ and δ are the most easily controlled, that are the rate of isolation of susceptible individuals and the rate at which return to susceptible class from class Q, respectively. Increasing the average period of susceptible individuals isolated 1/δ, decreases the endemic infectious fraction. If 1/δ is increased By example, prolonged school closures then the basic reproduction number R q is decreased. The IR model 3 is incorporated vaccination, treatment, quarantine and isolation measures, the strategies to reduce the basic reproduction number, R c, are similar to IR model with vaccination and treatment, and the IR model with quarantine and isolation, respectively. If φ, δ I, 1/δ, σ, γ I are increased then the basic reproduction number, R c, is decreased. We conclude that it is necessary to the development of new biologically complex models to understand the dynamics of transmission of diseases and their control strategies. Rev. Mex. Fis

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