= Λ μs βs I N, (1) (μ + d + r)i, (2)

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1 Advanced Studies in Biology, Vol., 29, no. 8, Mathematical Model of the Influenza A(HN) Infection K. Hattaf and N. Yousfi 2 Laboratory Analysis, Modeling and Simulation Department of Mathematics and Computer Science Faculty of Sciences Ben M'sik, University Hassan II Mohammedia, P.O Box 7955 Sidi Othman, Casablanca Morocco Abstract The aim of this work is to give a mathematical model describing and modeling the transmission of influenza A (HN) virus, and discuss how this model can provide insights into the future of the currently circulating novel strain of influenza A (HN). Weprove that the disease will die out if the basic reproductive number R < while the disease may become endemic if R >. Stability analysis of the both endemic and free steadystate are also studied. Finally, we give some numerical simulations to illustrate our results and predict the evolution of the disease in Morocco. Keywords: Influenza A(HN), basic reproductive number, stability. Introduction The 29 flu pandemic is a global outbreak of a new strain of HN influenza virus, often referred to as "swine flu" in the media. Although the virus, first detected in April 29, contains a combination of genes from pigs, birds, and humans, it cannot be spread by eating pork products or being around pigs. The outbreak began in Veracruz, Mexico, with evidence that there had been an ongoing epidemic for months before it was officially recognized as such. The Mexican government closed most of Mexico City's public and private facilities in an attempt to contain the spread of the virus. However the virus continued to spread globally, clinics were overwhelmed by people infected, and the World Health Organization (WHO) and US Centers for Disease Control Corresponding author. k.hattaf@yahoo.fr 2 Corresponding author. nourayousfi@gmail.com

2 384 K. Hattaf and N. Yousfi (CDC) stopped counting cases and in June declared the outbreak to be a pandemic. n 8 December 29, Ministry of Public Health of Morocco has reported 3 new cases of pandemic influenza A (HN) and 2 deaths. Today and since June 29 were reported the first case, there are 228 cases and 4 deaths. In addition, on 4 December 29, worldwide more than 27 countries and overseas territories or communities have reported laboratory confirmed cases of pandemic influenza HN 29, including at least 8768 deaths [4]. In the mathematical biology literature, several mathematical models have been proposed for modeling the spread of infectious diseases. In this paper we give a mathematical model describing and modeling the transmission of influenza A (HN) virus. The rest of the paper is organized as follows. Section 2, describes our model for the transmission of HN. The analysis of the model is presented in section 3. In section 4, we present some numerical simulations to illustrate our theoretical results. Finally, theconclusion are summarized in Section 5. 2 HN Model In this section, we introduce a simple model for the transmission of HN. The host population is divided into the three epidemiological classes: namely susceptible S, infective I and removed R. N denotes the total population. We assume that an individual can be infected only through contacts with infectious individuals. Our HN model is given by the following nonlinear system of differential equations ds dt di dt dr dt = Λ μs βs I N, () = βs I N (μ + d + r)i, (2) = ri μr, (3) where S() = S,I() = I,R() = R are given and the definitions of above model parameters are listed in Tab.. Parameter Λ μ d β r Definition Recruitment rate Natural mortality rate HN induced mortality rate Effective contact rate Recovery rate Table : Parameter definitions

3 HN model 385 We first show that system () is dissipative, that is all solutions are uniformly bounded inapropersubset Ω IR 3 +. Let (S, I, R) IR 3 + be any solution with non-negative initial conditions. Then Adding equations of () we get lim sup S(t) Λ t + μ. dn dt =Λ μn di < Λ μn, then N(t) Λ μ + N()e μt. Thus N Λ, as t +. Therefore all feasible solutions of system () μ enter the region Ω={(S, I, R) IR 3 + : N Λ μ } Thus, Ω is positively invariant and it is sufficient to consider solutions in Ω. It can be shown that all solutions of system () starting in Ω remain in Ω for all t. The continuity of right side of the system () and its derivatives implies that unique solutions exit on maximal interval. Since solutions approach, enter in Ω they are eventually bounded and hence exist for t. Therefore the model is mathematically and epidemiologically well posed. Notice that the system () has a basic reproductive number of R = β μ + r + d (4) Recall that R is defined as the average number of secondary infectious produced by a single infectious individual during the entire infectious period. We show that the disease will die out if R < while the disease may become endemic if R >. 3 Analysis of the model In this section, we show that there exists a disease free equilibrium point and one endemic equilibrium point. In addition we study the stability of these equilibrium points.

4 386 K. Hattaf and N. Yousfi 3. Free equilibria The system () always has disease free equilibria of the form E f =( Λ,, ). μ Let E(S, I, R) bet any arbitrary equilibrium. Then the characteristic equation about E is given by μ βi ( S ) βs ( I ) N N N N βi N ( S ) βs ( I ) (μ + r + d) βs N N N r μ Proposition 3.. βs I N 2 I N 2 =. (5). If R <, then the disease free equilibrium, E f, is locally asymptotically stable. 2. If R >, then E f is unstable. Proof. At E f, (5) reduces to where the eigenvalues are (λ + μ) 2 (β μ r d λ) =, (6) λ = μ, λ 2 = μ, λ 3 = β μ r d. Observe that λ =if β μ r d =. Rewriting this as β μ + r + d ==R, confirms our computation of R. It is clear that λ and λ 2 are negative. Moreover, λ 3 is negative when R <, thus E f is locally asymptotically stable. 3.2 Endemic equilibria Here we study the existence and stability of the endemic equilibrium points. It is easily verified that the system () has one endemic equilibrium point E (S,I,R ), where S = R N, I = μ(r ) R (μ + r) N, R = r(r ) R (μ + r) N, N = ΛR (μ + r) μ[(r )d + R (μ + r)],

5 HN model 387 Proposition If R <, then the point E does not exists and E = E f when R =. 2. If R >, then E is locally asymptotically stable. Proof. If R <, it easy to show that E does not exists and E = E f when R =. We assume that R >, at E, (5) reduces to (μ + λ)(λ 2 + a λ + a 2 )=, (7) where the roots are λ = μ, λ 2 and λ 3 are given by the solution of λ 2 + a λ + a 2 =. (8) where a = μ + μβ(r ) R (μ + r), a 2 = μβ(r ) R 2 (μ + r) [μ + β(r )]. Clearly when R >, both a and a 2 >, then all roots of (8) have negative real parts. Consequently, E is locally asymptotically stable whenever R >. 4 Numerical simulation of the model In this section, we give the numerical simulation of the model which illustrates the theoretical results and predict the evolution of the disease in Morocco. The parameters ( time unit= day) are set up as follows: Natural mortality rate of individuals, μ, is assumed to be inversely related to life expectancy at birth which is approximately 7 years in Morocco. Then μ = per day. The recruitment rate, Λ, controls the total population sizes because the asymptotic carrying capacity of the population is Λ. We assume that μ our population has reached its limiting values, thus Λ=μN. The disease-induced mortality rate, d varies from country to country. The case fatality rate (CFR), that is, the number of reported deaths per number of reported cases as of 6 July 29. This rate varied from.%

6 388 K. Hattaf and N. Yousfi to 5.% [3]. n 8 December 29, Ministry of Public Health of Morocco has reported 3 new cases of pandemic influenza A (HN) and 2 deaths. Today and since June 29 were reported the first case, there are 228 cases and 4 deaths. Thus the CFR was.63%. Taking the treatment period to be 5 days, we obtain the recovery rate as r = =.2 per individual per day. 5 According to (4), we get β =(μ + r + d)r. Therefore once the value of R has been obtained, the value of β can be determined. R has been estimated to range from.4.6 in Mexico [] and in Japan [2]. Initial values are: S = N =3 6, I =3and R =28. These initial data corresponding the state of Morocco on 8 December 29. Susceptible S x Infective I 2 x Removed R 2 x HN Population 3 x S I R Figure : shows the population of HN. We observe that the population of susceptible individuals, infected individuals and removed individuals converge asymptotically to endemic equilibrium state as time increases. Figure 2 and Figure 3, show the impact of varying the basic reproductive number R and initial values of reported cases of pandemic influenza A (HN). Figure 4, predicts the evolution of the disease in Morocco. The number of the infected individuals begins to increase from 8 December 29 where there are 3 cases, it will reach its maximum on 8 April 2 which is about one million and 9682 cases. Then this number of infected individuals decreases asymptotically to endemic equilibrium state which is 937 cases, it will arrive at this state on 22 July 2.

7 HN model x 5 I=3 I=3 I=3 5 Infective class I Figure 2: The infective class I is plotted for the three different values of I, 3, 3 and 3. 7 x 6 6 R=.2 R=.5 R=.6 R=2.3 5 Infective class I Figure 3: The infective class I is plotted for the 4 different values of R,.2,.5,.6 and x 6.8 Number of infected per day Figure 4: Modeling the evolution of influenza A (HN) in Morocco from 8 December 29 to 6 August 2.

8 39 K. Hattaf and N. Yousfi 5 Conclusion In this work, we give a mathematical model describing and modeling the transmission of influenza A (HN) virus. Analysis of the model shows that the disease free equilibrium is locally asymptotically stable if the basic reproductive number satisfies R < and the endemic equilibrium point is locally asymptotically stable if R <. In addition, the simulation of this model provides that the number of the infected individuals in Morocco begins to increase from 8 December 29, it will reach its maximum on 8 April 2 which is about one million and 9682 cases and it decreases asymptotically to endemic equilibrium state which is 937 cases, it will arrive at this state on 22 July 2. Therefore, the Ministry of Health of Morocco starts by vaccinating people with low immunity from 9 December 29, in order to reduce the number of infected cases. References [] C. Fraser, CA. Donnelly, S. Cauchemez, et al., Pandemic potential of a strain of influenza A (HN): early findings, Science, Vol. 324, (29), [2] H. Nishiura, C. Castillo-Chavez, M. Safan, G. Chowell, Transmission potential of the new influenza A(HN) virus and its age-specificity in Japan, Euro Surveill, (29), 4(22). [3] L. Vaillant, G. La Ruche, A. Tarantola, P. Barboza, Epidemiology of fatal cases associated with pandemic HN influenza 29, Euro Surveill. (29), 4(33). [4] World Health Organization: Received: December, 29