A Model Analysis for the Transmission Dynamics of Avian Influenza

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1 A Model Analysis for the Transmission Dynamics of Avian Influenza A. R. Kimbir 1, T. Aboiyar 1 P. N. Okolo * (corresponding author) 1 Department of Mathematics/Statistics/Computer Science, University of Agriculture, Makurdi, Nigeria Department of Natural Sciences, Nasarawa State Polytechnic, Lafia, Nasarawa State, Nigeria * of the corresponding author: patricknoahokolo@yahoo.com Abstract This paper examines the transmission dynamics of avian influenza. A nonlinear mathematical model for the problem is formulated analysed. For the prevalence of the disease the ease of analysis, we considered the model in proportions of susceptible, infectious, isolated recovered compartments. The basic reproduction number was computed used to prove the stability of the disease free equilibrium states. It is proved that the basic reproduction number is a decreasing function of the culling rate of infected birds. It is further proved that the disease free equilibrium state is locally asymptotically stable whenever the basic reproduction number is less than unity. Key words: Avian influenza, Mathematical model, Basic reproduction number. Disease free equilibrium 1.0: Introduction Avian influenza or bird-flu (also called influenza A virus) is a virus that infects wild birds (such as ducks, gulls shore birds) domestic poultry (such as chickens, turkey, ducks geese). In recent times the term bird-flu has been used to describe the H5N1 avian influenza virus that occurs mainly in birds, can be deadly to them (Alexer, 000; Arora Arora, 008) Infected birds shed influenza virus in their saliva, nasal secretions faeces. Susceptible birds become infected when they have contact with contaminated secretions or excretions with surfaces that are contaminated by infected birds (De Jong Hien, 006). Fecal to - oral transmission is the most common mode of spread between birds. Highly pathogenic avian influenza can be spread from birds to people as a result of extensive direct contact with nasal discharge or fecal droppings in infected birds (The Writing Committee of the World Health Org. (WHO) Consultation on Human Influenza A/H5, 006). Highly pathogenic avian influenza virus subtype H5N1 has severely affected poultry populations in Southeast Asia since 003. Initial outbreaks were confined to Indonesia, Vietnam, Thail, Cambodia China. However from late July 005, the highly pathogenic virus spread in a north east direction, causing out breaks in wild birds poultry in eastern, central western Russia, Mongolia Central Kazakhstan (The Writing Committee of the World Health Org. (WHO) Consultation on Human Influenza A/H5, 006). In February, 006, highly pathogenic avian influenza virus of the H5N1 subtype was detected in chickens in Kaduna State in Northern Nigeria, the first African country reporting a confirmed highly pathogenic avian influenza (H5N1) outbreak (Monne et al, 008; Fusaro et al, 009; WHO, 006). According to Monne et al (008), by the end of February, 006, local laboratory tests had detected the virus in seven contiguous states in the north central parts of the country (Kaduna, Kano, Plateau, Katsina, Bauchi, Yobe Nasarawa) the Federal Capital Territory of Abuja. Since avian influenza virus is highly contagious easily spread, the most common method of control is the culling of the infected flocks. Another method is the quarantine of affected areas until the disease is no longer present. While vaccination is possible has been tested on a small scale, it is not widely considered a viable control method. The virus can also be killed by common disinfectants or heat (WHO, 004; Le Menach et al, 006). Persons recovering from natural infection according to Todar (008) acquire some resistance to re infection with the particular antigenic strain. Bodewes et al (010) also asserted that the induction of antibodies 15

2 of proper specificity will afford strain specific protection this strain specific immunity can be very long lasting. A number of mathematical models both deterministic stochastic have been used to predict the world wide spread of pemic influenza for comparing interventions aimed at preventing controlling avian influenza. See for example, Ferguson et al, (005); Derouich Boutayeb (008) Srinivasa (008). Okosun Yusuf (007); Iwani et al(007); Derouich Bontayeb (008) presented various mathematical models for avian influenza (H5N1). These models does not explicitly take into account any control measures. Using the data from the avian influenza epidemic in the Netherls, LeMenach et al (006) analysed a spartial farm-based model, which treats poultry farms as units, found that an immediate depopulation of infected flock following an accurate quick diagnosis would have a greater impact than simply depopulating surrounding flocks. Ferguson et al (005) used a simulation model of influenza transmission in Southeast Asia to evaluate the potential effectiveness of targeted mass prophylactic use of antiviral drugs as a containment strategy. On the same note, Longini et al (005) used a stochastic influenza simulation model for rural Southeast Asia to investigate the effectiveness of targeted antiviral prophylaxis, quarantine pre vaccination in containing an emerging influenza strain at the source. Although many of mathematical modelling studies tend to emphasize the use of pharmaceutical interventions, it could be useful to carry out modelling studies that focus on non pharmaceutical intervention such as culling of infected birds isolation of humans with symptoms. The main aim of this study is to build on the model by Okosun Yusuf (007), by incorporating the dynamics of wild domestic birds, culling of infected birds the isolation of infected individuals with avian influenza strain. The paper is organized as follows, in Section, we derive a model consisting of ordinary differential equations (ODE) that describes the interaction between birds human population the underlying assumptions. In Section 3, we compute the basic reproduction number use it to establish the local stability of the disease free equilibrium states. Our conclusions are discussed in Section 4..0 Model Formulation In describing the new model we subdivide the total avian (birds) population at time t, denoted by N B (t) into susceptible wild birds, S W (t), susceptible domestic birds, S D (t), infected wild birds, I W (t), infected domestic birds, I D (t), so that N B (t) = S W (t) + S D (t) + I W (t) (t). In the human population, we assume that humans infected with avian influenza cannot infect susceptible humans. Thus the total human population at time t, denoted by N H (t) is sub-divided into susceptible humans, S H (t), infected humans, I H (t), isolated infected humans, Q H (t), recovered humans, R H (t), so that N H (t) = S H (t) + I H (t) + Q H (t) + R H (t) The variables parameters used in the model are defined in Table 1. 16

3 Table 1: Variables Parameters used in the model their description Variable/Parameter Description (t) Total number of wild birds at time t (t) Total number of domestic birds at time t N H (t) Total number of humans at time t S W (t) Total number of Susceptible wild birds at time t I W (t) Total number of Infected wild birds at time t S D (t) Total number of Susceptible domestic birds at time t I D (t) Total number of Infected domestic birds at time t S H (t) Total number of Susceptible humans at time t I D (t) Total number of Infected humans at time t Q H (t) Total number of Isolated humans with avian strain at time t R H (t) Total number of Recovered humans at time t β W Average birth rate in wild birds β D Average birth rate in domestic birds α W,, α A Infection transmission rates for birds η Destruction (culling) rate for infected birds δ B Natural death rate in birds d w Flu induced death rate in wild birds d D Flu induced death rate in domestic birds β H Average birth rate in humans δ H Natural death rate in humans d H Flu induced death rate in humans ε H Isolation rate for humans with avian stain θ H Flu induced death rate in Isolated humans (θ H < d H ) v Recovery rate without immunity γ Recovery rate with substantial immunity σ Loose of immunity rate in recovered humans A schematic flow diagram of the extended model for the birds population human population is shown in Figure 1 below. Wild Birds β W S W (t) δ B S W δ B I W α W ( I W )S W I W (t) (d W + η)i W Domestic Birds β D S D (t) ( I W )S D I D (t) (d D + η)i D δ B S D δ B I D 17

4 Humans νi H δ H S H (d H + δ H )I H β H N H S H (t) α A ( I W )S H I H (t) ε H I H νq H Q H (t) (d H + θ H )Q H γq H γi H σr H R H (t) δ H R H Figure 1: Schematic diagram of the transmission dynamics of avian influenza (H5N1) in birds human population..1 Susceptible Infected Wild Birds The population of susceptible wild birds is generated by birth of wild birds (at the rate β W ). It is reduced by infection, following contact with infected wild birds infected domestic birds (at the rate α W ), where α W is the infection transmission rate for wild birds further reduced by natural death (at the rate δ B ). Hence ds W = β W α W I W S W α W I D S W δ B S W, = β W α W ( I W ) S W δ B S W, The population of infected wild birds is increased through the infection of susceptible wild birds following contact with infected wild birds infected domestic birds. It decreased either by natural death (at the rate δ B ) avian induced mortality (at the rate d W ) by culling of infected wild birds (at the rate η). So that di W = α W ( I W ) S W (d W + δ B + η)i W,. Susceptible Infected Domestic Birds The population of susceptible domestic birds is generated by birth of domestic birds (at the rate β D ). It is reduced by infection, following contact with infected wild birds infected domestic birds (at the rate ), where is the infection transmission rate for domestic birds further reduced by natural death (at the rate δ B ). Thus 18

5 ds D = β D I W S D I D S D δ B S D, = β D ( I W ) S D δ B S D, The population of infected domestic birds is increased through the infection of susceptible domestic birds following contact with infected wild birds infected domestic birds. It decreased either by natural death (at the rate δ B ) avian induced mortality (at the rate d D ) by culling of infected wild birds (at the rate η). This yield di D = ( I W ) S D (d D + δ D + η)i D,.3 Susceptible, Infected, Isolated Recovered Humans The population of susceptible humans are increased by birth (at the rate β H ), recovered humans who lost immunity to return to susceptible humans (at the rate σ), recovered infected isolated humans without immunity (at the rate ν). It decreased by infection of susceptible humans following contact with infected wild birds infected domestic birds (at the rate α B ), where α B is the infection transmission rate for humans further reduced by natural death (at the rate δ H ). This gives ds H = β H N H α B ( I W ) S H δ H S H + νi H + νq H + σr H, Infected humans are generated through infection of susceptible humans following contact with infected wild birds infected domestic birds (at the rate α B ) reduced by natural death (at the rate δ H ) avian induced mortality (at the rate d D ). It is further reduced by isolation of infected humans (at the rate ε H ) recovered infected humans without immunity with substantial immunity (at the rate ν γ respectively). Thus, di H = α B ( I W ) S H (ε H + d H + δ H + ν + γ)i H, Isolated humans are generated by isolation of infected humans (at the rate ε H ) reduced by natural death (at the rate δ H ) avian induced mortality (at the rate θ H where, θ H < d H ; it is assumed that isolated individuals are given some treatment such as Tamiflu). It is further reduced by recovered isolated humans without immunity those with substantial immunity (at the rate ν γ respectively). Hence, dq H = ε H I H (ν + θ H + γ + δ H )Q H, The recovered humans are generated by the recovery of infected humans isolated humans (at the rate γ). Decreased by natural death (at the rate δ H ) losing immunity (at the rate σ). So that dr H = γi H + γq H (σ + δ H )R H, The above assumptions derivations leads to the following system of ordinary differential equations ds W = β W α W ( I W ) S W δ B S W, (1) di W = α W ( I W ) S W (d W + δ B + η)i W, () ds D = β D ( I W ) S D δ B S D, (3) 19

6 di D = ( I W ) S D (d D + δ D + η)i D, (4) ds H = β H N H α B ( I W ) S H δ H S H + νi H + νq H + σr H, (5) di H = α B ( I W ) S H (ε H + d H + δ H + ν + γ)i H, (6) dq H = ε H I H (ν + θ H + γ + δ H )Q H, (7) dr H = γi H + γq H (σ + δ H )R H, (8) For prevalence of the disease, it is necessary to consider the model in proportions of susceptible, infectious, isolated recovered compartments. Adding equations (1) - () equations (3) (4) gives d = β W δ B (d w + η)i W (9) d = β D δ B (d D + η)i D (10) Similarly, adding equations (5) (8) gives the rate of change of the total human population: dn H = β H N H δ H N H d H I H θ H Q H (11) We now define the proportion for each class as follows: So that s w = S W, i W = I W, s D = S D, i D = I D, s H = S H N H, i H = I H N H, q H = Q H N H, r H = R H N H, s W + i W = 1 s W = 1 i W, s D + i D = 1 s D = 1 i D s H + i H + q H + r H = 1 s H = 1 i H q H r H Thus, the system (1) (8) expressed in proportion is given below: ds W = β W α W (i W + i D )s W β W s W + (d w + η)i W s W (1) di W = α W (i W + i D )s w (d W + β W + η)i W + (d w + η)i W (13) ds D = β D (i W + i D )s D β D s D + (d D + η)i D s D (13) di D = (i W + i D )s D (d D + β D + η)i D + (d D + η)i D (14) ds H = β H α B (i W + i D )s H + v(i H + q H ) + σr H β H s H + d H s H i H + θ H s H q H (15) di H = α B(i W + i D )s H (ε + d H + β H + v + γ)i H +θ H i H q H + d H i H (16) 0

7 dq H = ε H i H (v + θ H + γ + β H )q H + d H i H q H + θ H q H (17) dr H = γi H + γq H (σ + β H )r H + d H i H r H + θ H q H r H (18)` The system (1) (18) can be reduced further by setting s W = 1 i W, s D = 1 i D s H = 1 i H q H r H di W = α W (i W + i D )(1 i W ) (d W + β W + η)i W + (d w + η)i W (19) di D = (i W + i D )(1 i D ) (d D + β D + η)i D + (d D + η)i D (0) di H = α B(i W + i D )(1 i H q H r H ) (ε + d H + β H + v + γ)i H +θ H i H q H + d H i H (1) dq H = ε H i H (v + θ H + γ + β H )q H + d H i H q H + θ H q H () dr H = γi H + γq H (σ + β H )r H + d H i H r H + θ H q H r H (3)` These are the governing equations of the model. 3.0 Model Analysis The nonlinrar system (1) (18) will be analysed so as to find the conditions for the existence stability of the disease free equilibrium states (DFEs). To achieve this, we will compute the Basic Reproduction number use it to determine if the disease can be eliminated from the population or not. 3.1 Invariant Region The avian influenza model (1) (18) in proportions will be analyzed to establish the biological feasible region as follows. The system (1) (18) is split into two parts, namely the avian population where n B (t) = s w (t) + i W (t) + s D (t) + i D (t) the human population where n H (t) = S H (t) + i H (t) + q H (t) + r H (t). Consider the feasible region Ω = Ω B Ω H R + 4 R + 4 with Ω B = {(s w, i W, s D, i D ) R + 4 : s w + i W + s D + i D 1} Ω H = {(s H, i H, q H, r H ) R + 4 : s H + i H + q H + r H 1} The following steps are followed to establish the positive invariance of Ω (i.e., the solution in Ω remain inω for all t > 0). The rate of change of the avian human population (by adding the first four equations the last four equations of the model (1) (18)) is given dn B = β B β B n B + (d w + η)i W s W (d W + η)i W + (d w + η)i W + (d D +)i D s D (d D + η)i D + (d D + η)i D (4) dn H = β H β H n H + d H s H i H + θ H s H q H β H i H + θ H i H q H + d H i H β H q H + d H i H q H + θ H q H β H r H + d H i H r H + θ H q H r H (5) It follows from (4) (5) that 1

8 dn B β B β B N B (6) dn H β H β H n H (7) Integrating (6) with respect to t where the integrating factor, IF = e βb = e β Bt we have e βbt n B β B e βbt + C e βbt n B e βbt + C n B (t) 1 + Ce β Bt At t = 0, C = n B (0) 1 n B (t) 1 + (n B (0) 1)e β Bt n B (t) n B (0)e βbt + 1 e β Bt (8) Also integrating (7) with respect to t where the integrating factor, IF = e βh = e β Ht we have e βht n H β H e βht + C e βht n H e βht + C n H (t) 1 + Ce β Ht At t = 0, C = n H (0) 1 n H (t) 1 + (n H (0) 1)e β Ht n H (t) n H (0)e βht + 1 e β Ht (9) Applying the theorem of differential inequality (Birkhof Rota, 198) on equations (8) (9) we obtain 0 n B (t) 1 0 n H (t) 1 as t Thus, the region Ω is positively invariant. Hence it is sufficient to consider the dynamics of the flow generated by (1) (18) in Ω. In this region, the model can be considered as being epidemiologically mathematically well posed. Thus every solution of the model (1) (18) with initial conditions in Ω remains in Ω for all t > 0. This result is summarized below. Lemma1: The regionω = Ω B Ω H R + 4 R + 4 is positively invariant for the basic model (1) (0) with nonnegative initial conditions inr + 8.

9 3. Computation of The Basic Reproduction Number, R 0 The model in proportion given by equations (19) (3) has a unique disease free equilibrium state E 0 = (i W, i D, i H, q H, r H ) = (0, 0, 0, 0, 0) obtained by setting i W = 0, i D = 0, i H = 0, q H = 0, r H = 0. To compute the basic reproduction number, we rewrite the model equation which contribute to the transmission of infection, in this case the i W, i D i H classes. Thereafter write down matrix of infection rates F i the transition rate matrix V i which represents rates of appearance of new infections into infective class the transfer of individuals into out of this class by all other means respectively. The rate of appearance of new infection in compartments i W, i D i H are given by F(x) = ( α W (i W i W + i D i W i D ) (i W i W i D + i D i D ) ). α B (i W + i D )(1 i H q H r H ) While the remaining transfer terms in compartments i W, i D i H are given by (d W + β W + η)i W (d w + η)i W V(x) = ( (d D + β D + η)i D (d D + η)i D ). (ε + d H + β H + v + γ)i H θ H i H q H + d H i H Taking partial derivatives of F(x) with respect to i W, i D i H at the disease free equilibrium state E 0 = (i W, i D, i H, q H, r H ) = (0, 0, 0, 0, 0), to obtain 0 F x (E 0 ) = ( 0). α B α B 0 Similarly the matrix of partial derivatives of V(x) at the disease free equilibrium state E 0 = (i W, i D, i H, q H, r H ) = (0, 0, 0, 0, 0) is given by d W + β W + η 0 0 V x (E 0 ) = ( 0 d D + β D + η 0 ). 0 0 ε + d H + β H + v + γ V x 1 (E 0 ) = ( 1 d W + β W + η d D + β D + η ε + d H + β H + v + γ). Then F x (E 0 )V x 1 (E 0 ) = d W + β W + η d W + β W + η α B ( d W + β W + η d D + β D + η 0 d D + β D + η 0. α B d D + β D + η 0 ) The eigenvalues are determined by solving the characteristic equation det(f x (E 0 )V x 1 (E 0 ) λ) = 0 3

10 det ( d W + β W + η λ d W + β W + η α B d W + β W + η d D + β D + η d D + β D + η λ 0 = 0 α B 0 λ d D + β D + η ) 0 That is (0 λ) [( d W + β W + η λ) ( d D + β D + η λ) d W + β W + η d D + β D + η ] = 0 (0 λ) [λ ( d W + β W + η + d D + β D + η ) λ] = 0 (0 λ)λ [λ ( d W + β W + η + d D + β D + η )] = 0 λ = 0 or λ = + d W +β W +η d D +β D +η The maximum eigenvalue of F x (E 0 )V x 1 (E 0 ) is given as: λ = d W + β W + η + d D + β D + η Thus, the basic reproduction number is given as: R 0 = + d W +β W +η d D +β D +η (30) This leads us to the following result. Proposition 1: R 0 is a strictly decreasing function of η ε (0,1). Proof Taking the partial derivative of R 0 with respect to η (0,1) to obtain R o η = ( (d D + β D + η) + α W (d W + β W + η) ) Therefore R 0 is a strictly decreasing function of η ε (0,1). 4

11 Figure : Graph of The Basic Reproduction Number, R 0 as a function of the culling rate, (η [0,1]). The simulation in Figure shows that by increasing the culling rate η, the value of the basic reproduction number R 0 decreases. At threshold, R 0 = 1, corresponding to η = 0.6. Any control programme with culling of infected birds (η > 0.6) will be effective. 3.3 Existence And Local Stability of The Disease Free Equilibrium (DFE) State As stated in Section 3., the model given by equations (19) (3) has a unique disease free equilibrium state E 0 = (i W, i D, i H, q H, r H ) = (0, 0, 0, 0, 0) obtained by setting i W = 0, i D = 0, i H = 0, q H = 0, r H = 0. To establish the local stability of the disease free equilibrium (DFE) state, the associated Jacobian of (19) (3) is evaluated at the DFE state. The Jacobian matrix of the system (3.4.1) (3.4.16) evaluated at the disease free equilibrium state at J(E 0 ) is given by α W (d W + β W + η) α W (d D + β D + η) J(E 0 ) = α B α B (ε H + d H + β H + v + γ) ε H (v + θ H + γ + β H ) γ γ (σ + β H ) 5

12 The disease free equilibrium state is locally asymptotically stable if only if all of the eigenvalues of the Jacobian matrix J(E 0 ) have negative real part (Benjah, 007). The eigenvalues are determined by solving the characteristic equation det(j λi) = 0. α W (d W + β W + η) λ α W (d D + β D + η) λ = 0 α B α B (ε H + d H + β H + v + γ) λ ε H (v + θ H + γ + β H ) λ γ γ (σ + β H ) λ That is [ (σ + β H ) λ][ (v + θ H + γ + β H ) λ][ (ε H + d H + β H + v + γ) λ] [[α W (d W + β W + η) λ][ (d D + β D + η) λ ] α W ] = 0 or [ (σ + β H ) λ][ (v + θ H + γ + β H ) λ][ (ε H + d H + β H + v + γ) λ] [λ [α W (d W + β W + η) + (d D + β D + η)]λ + (α W (d W + β W + η)) ( (d D + β D + η)) α W ] = 0 Thus λ = (σ + β H ), (v + θ H + γ + β H ), (ε H + d H + β H + v + γ) [λ [α W (d W + β W + η) + (d D + β D + η)]λ + (α W (d W + β W + η)) ( (d D + β D + η)) α W ] = 0 (31) Clearly, three eigenvalues are negative. We further need to show that equation (31) has negative eigenvalues. Now equation (31) is the characteristic equation of sub matrix J 1, where J 1 = ( α W (d W + β W + η) α W (d D + β D + η) ) We shall use the trace determinant method to show that sub matrix (J 1 ), has negative eigenvalues. The sub matrix (J 1 ) satisfy Re(λ j ) < 0 i = 1,, if only if only trace of (J 1 ) < 0 detj 1 > 0 (Benjah, 007). The trace of (J 1 ) = α W (d W + β W + η) + (d D + β D + η) α W = (d W + β W + η) [ d W + β W + η 1] + (d D + β D + η) [ d D + β D + η 1] = (d W + β W + η) [(R 0 d D + β D + η ) 1] + (d D + β D + η) [(R 0 d W + β W + η ) 1] = (d W + β W + η) [(R 0 1) d D + β D + η ] + (d D + β D + η) [(R 0 1) d W + β W + η ] α W α W < 0 if R 0 < 1. The detj 1 = (α W (d W + β W + η))( (d D + β D + η)) α W = (d W + β W + η)(d D + β D + η) α W (d D + β D + η) (d W + β W + η) = (d W + β W + η)(d D + β D + η) [α W (d D + β D + η) + (d W + β W + η)] α W = 1 [ d W + β W + η + d D + β D + η ] 6

13 = 1 R 0 > 0 if R 0 < 1. Thus we proved the following lemma. Lemma : The DFEs of the model (3.4.1) (3.4.16), given by E 0, is locally asymptotically stable (LAS) if R 0 < 1 E 0 is unstable if R 0 > : Conclusion. The stability analysis of the model showed that the existing domain Ω is positively invariant attracting. In this region, the model can be considered as being epidemiologically meaningful mathematically well defined. Thus every solution of the basic model with initial conditions in Ω will remains in Ω for all t > 0. Crucial to the stability analysis is the basic reproduction number, R 0. R 0 is an important threshold parameter used to determine the threshold between disease eradication outbreak. We computed the basic reproduction number, R 0 using the next generation method. Further analysis shows that the basic reproduction number, R 0 is affected by the culling rate (η)for infected birds as shown in proposition 1 figure. The result shows that increasing culling of infected birds can reduce the basic reproduction number below unity. From the computation for R 0 it was obvious that R 0 is not affected by isolation rate for infected humans. We further ascertain the stability for the disease free equilibrium states (DFEs) using linearization method, taking R 0 as the threshold parameter. The result in Lemma shows that if R 0 < 1, the DFEs is locally asymptotically stable. Lemma implies that a small influx of new infective will not generate large outbreaks avian flu can be eliminated from the avian-human population (when R o < 1) if the initial sizes of the populations of the avian-human model are in the basin of attraction of the DFE, E o. REFERENCES Alexer, D. J. (000), A review of influenza in different bird species. Vet Microbiol. 7 pp3 13 Arora, D. R. Arora, B. (008), Text book of Microbiology(3 rd Edition) CBS Publishers & Distributors, New Delhi. Benyah, F. (007). Epidemiological Modelling Analysis. A paper presented at the 13 th Edward A. Bouchet/Abdus Salam Workshop held at University of Ghana, Legon, Accra. 9 th 13 th July, 007. Bodewes, R., Osterhaus, A. D. M., Rimmelzwaan, G. F. (010) Targets for the induction of protective immunity against influenza A viruses. Virus,, pp Birkhoff, G. Rota, G. (198). Ordinary Differential Equations. (3 rd Ed). John Wiley Sons, New York. De Jong, M. D. Hein, T. T. (006), Avian influenza A (H5N1), Journal of Clinical Virol., 35 pp 13 Derouich, M. Boutyeb, A. (008), An avian influenza mathematical model. Applied Mathematical Sciences, (36) pp Ferguson, N. M., Cummings, D. A. T., Fraser, C., Riley, S., Meeyai, A.,Iamsirithaworn, S. & Burke, D. S.(005), Strategies for containing an emerging influenza pemic in South east Asia. Nature, 437 pp Fusaro, A., Joannis, T. M., Monne, I., Salviato, A., Yabub, B., Meseko, C., 7

14 Oladokun, T., Fassina, S., Capua, I. Cattoli, G.(009), Introduction into Nigeria of a distinct genotype of avian influenza virus (H5N1). Emerging Infectious Disease, 15(3), Iwani, S., Takeuchi, Y. Liu, X. (007). Avian- human influenza epidemic model Math. Bioscs, 07, 1 5. Le Menach, A., Vergu, E., Grais, R. F., Smith, D. L. Flahault, A. (006), Key strategies for reducing spread of avian influenza among commercial poultry holdings: lessons for transmission to humans. Proceedings of Biological Sciences, October, 7; 73(1600) pp Longini, I. M., Nizam, A., Xu, S., Ungchusak, K., Hanshaoworakul, W., Derek, T. C., Halloran, M. E.,(005), Containing Pemic Influenza at the source. Science 309 pp doi: /science Monne, I., Joannis, T. M., Fusaro, A., De Benedictis, P., Lombin, L. H., et al, (008), Reassortant avian influenza virus (H5N1) in poultry, Nigeria, 007. Emerging Infectious Disease 14(6), Okosun, K. O., Yusuf, T. T. (007), Numerical Simulation of Bird flu Epidemics. Research Journal of Applied Sciences (1) pp 9 1 Srinivasa, A. S. R. (008), Modeling the rapid spread of avian influenza (H5N1) in India. Mathematical Biosciences Engineering, 5(3) pp The Writing Committee of the World Health Organisation (WHO) Consultation on Human influenza A/H5, (006). Avian influenza H5N1, Thail, 004. Emerging Infectious Diseases, 11 pp [Pub Med] Todar, K. (008) The Microbial World. Lectures in Microbiology, Department of Bacteriology, University of Wisconsin Medison. WHO (004), Influenza A (H5N1): WHO Interim Infection Control Guidelines for Health Care facilities, WHO 10 March 004. WHO (006), Avian influenza situation in Nigeria update EFR March 9,

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