a pandemic avian influenza mathematical model

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Chapter XLIIIA Pandemic Avian Influenza

Mathematical ModelMohamed Derouich

Faculté des sciences Oujda-Morocco, Morocco

Abdesslam BoutayebFaculté des sciences Oujda-Morocco, Morocco

Copyright © 2009, IGI Global, distributing in print or electronic forms without written permission of IGI Global is prohibited.

abstract

Throughout the world, seasonal outbreaks of influenza affect millions of people, killing about 500,000 individuals every year. Human influenza viruses are classified into 3 serotypes: A, B, and C. Only influenza A viruses can infect and multiply in avian species. During the last decades, important avian influenza epidemics have occurred and so far, the epidemics among birds have been transmitted to humans; but the most feared problem is the risk of pandemics that may be caused by person-to person transmission. The present mathematical model deals with the dynamics of human infection by avian influenza both in birds and in humans. Stability analysis is carried out and the behaviour of the disease is illustrated by simulation with different parameters values.

INtrODUctION

Worldwide, seasonal outbreaks of influenza (also known as flu) affect millions of people, killing about 500,000 individuals every year (WHO, 2005). Human influenza viruses are classified into three sero-types: A, B and C. Only influenza A viruses are known to infect and multiply in avian species. �hese�hese viruses present 16 HA (haemagluttinin) and 9 NA (neuraminidase) subtypes (H1N1, H2N2, H3N2, H5N1, H7N7,....) (Ale�ander, 200�).Ale�ander, 200�).

At the domestic poultries, the infection by viruses of avian influenza provokes two main forms of illness characterized by an e�tremely weak and e�tremely elevated virulence. �he first form weakly pathogen only provokes some benign symptoms (tousled feather, less frequent punter) and can pass

795

A Pandemic Avian Influenza Mathematical Model

easily unobserved. �he second form pathogen has consequences well more serious. It propagates very quickly in raisings and of which the mortality rate can approach 100%, the death often occurring in the �8 hours.

�he wide spread of influenza in poultry and wild birds during the last decade and the occurrence of human influenza infection has raised the question of pandemics. For a pandemic to start, three condi-tions are required: a novel influenza virus subtype must emerge against which the general population has in its majority no immunity; the virus infects humans and causes serious illness; and the new virus should have a high rate of person-to-person transmission (WHO 2005; Ferguson, 200�). �hree major pandemics occurred during the last century. In 1918, the Spanish flu has killed an estimated �0-50 mil-lion people, in 1957 the pandemic (Asian flu) killed about 2 million people and the Hong Kong flu killed an estimated one million people in 1968. Although influenza pandemics are considered inevitable, the avian epidemics that occurred during the last decade, starting in 1997 (Hong Kong), have not engendered pandemics. Studies have shown that direct contact with diseased poultry was the source of infection and found no evidence of person-to-person spread of the virus. However, due to the potential for cross-spe-cies of avian and human influenza viruses and the possibility of viruses reassortment, the high rates of mortality among the few cases observed recently (�able 1) could lead to devastating pandemics (Yuen, 2005; Kuiken, 2006; Smith, 2006). Consequently, the risk of pandemics and its corollaries remains on the agenda of national and international health bodies.

Country 2003 200� 2005 2006 2007 �otal

cases deaths cases deaths cases deaths cases deaths cases deaths cases deaths

Azerbaijan 0 0 0 0 0 0 8 5 0 0 8 5

Cambodia 0 0 0 0 � � 2 2 1 1 7 7

China 1 1 0 0 8 5 13 8 3 2 25 16

Djibouti 0 0 0 0 0 0 1 0 0 0 1 0

Egypt 0 0 0 0 0 0 18 10 16 � 3� 1�

Indonesia 0 0 0 0 20 13 55 �5 2� 21 99 79

Iraq 0 0 0 0 0 0 3 2 0 0 3 2

Lao People’s Democratic

Republic0 0 0 0 0 0 0 0 2 2 2 2

Nigeria 0 0 0 0 0 0 0 0 1 1 1 1

�hailand 0 0 17 12 5 2 3 3 0 0 25 17

�urkey 0 0 0 0 0 0 12 � 0 0 12 �

Viet Nam 3 3 29 20 61 19 0 0 0 0 93 �2

�otal � � �6 32 98 �3 115 79 �7 31 310 189

Table 1. Cumulative number of confirmed human cases of avian influenza A/(H5N1) reported to WHO (last update: 6 June 2007). Total number of cases includes number of deaths. WHO reports only labora-tory-confirmed cases. All dates refer to onset of illness. (WHO, 2007).

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Mathematical models have been used for infectious diseases in general and for influenza in particular (Ale�ander, 200�; Ferguson, 200�; Hethcote, 2000; Mena-Lorca, 1992; Derouich, 2006). In the case of avian influenza, deterministic models were used for comparing interventions aimed at preventing and controlling influenza pandemics (Ferguson, 200�; Carrot, 2006) and stochastic models were proposed to model and predict the worldwide spread of pandemic influenza (Colizza, 2006; Colizza, 2007).

In the present chapter, we propose a mathematical model to study the dynamics of human infection by avian influenza. �he model deals with both infections (avian and human). Stability analysis is given and simulation is carried out with different parameters values. �he model illustrates in particular, the importance of parameters such as the average number of adequate contacts of a human susceptible with infected human and the average number of adequate contacts of a human susceptible with infected birds in determining the incidence of the disease and consequent preventive strategies.

FOrMULatION OF tHE MODEL aND stabILItY aNaLYsIs

Parameters of the Model

Let N and N0 denote the human and bird population size. In this model death is proportional to the population size with rate constant μ and we suppose that N and N0 are constant.

�he human population (respectively bird population) of size N (resp. N0) is formed of Susceptible S, of Infective I and of Removed R (resp. S0 and I0).

0 0

0

( )II SN N

+ is the human incidence, i.e. the rate at which susceptible individuals become infectious.is the human incidence, i.e. the rate at which susceptible individuals become infectious.

If the time unit is days, then the incidence is the number of new infection per day. �he daily contact rate β is the average number of adequate contacts of a human susceptible with infected human per day, �he daily contact rate β0 is the average number of adequate contacts of a human susceptible with infected birds per day,

IN

is the infectious fraction of the human population and 0

0

IN is the infectious fraction of

the bird population. �ime units of weeks, months or years could also be used.Similarly

'0 0

00

I SN is the bird incidence and '

0 is the average number of adequate contacts of a bird

susceptible with other birds per day. �he man life span is taken equal to 25 000 days (68.5 years), and the one of the bird is about 2500 days. �he other parameters used in the model are birth rate constant (μ); contact rate, human to human (β); effective contact rate, bird to human (β0); effective contact rate, bird to bird ( '

0); human life span (1/μ); and host infection duration (1/(μ + γ).

Equations of the Model

A schematic representation of the model is shown in Figure 1.In human we consider SIRS compartmental model that is to say that human susceptible individuals

become infectious then removed with temporary immunity after recovery from infection and susceptible when again immunity fades away, in bird population we consider SI compartmental model.

�he model is governed by the following equations:

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Human population:

S I− +

0 0

0

0 0

( )

( ) ( )

( )

IdS IN S Rdt N N

IdI Idt N NdR I Rdt

dN N Idt

= − + + +

= + + = − + = Λ − −

Bird population:

'0 0 0

0 0 0 00

'0 0 0

0 0 00

( )dS IN Sdt N

dI I S Idt N

= − +

= −

Introducing the proportions: ; ; ;S I Rs i rN N N

= = = 00

0

SsN

= ; and 00

0

IiN

= .

So with the conditions s+i+r=1 and s0+i0= 1 i.e r=1-(s+i) and s0=1 –i0, the two previous systems become:

0

R

0

NI /0'0 NII /)( 00+

S

I

R

0 0

S0

I0

Figure 1.

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− +

0 0

0 0

'00 0 0 0 0

( )

( ) ( )

( )

(1 )

ds i i s rdtdi i i s idtdr i rdtdi i i idt

= − + + + = + + = − + = − −

Equilibrium Points

Theorem 1

Let 0

0

R = and =+ +

, then the previous system admits the following equilibrium points:

• If R ≤ 1 there are two equilibrium points:1. �he trivial state E1(1,0,0,0) is the only equilibrium which is locally asymptotically stable for

σ ≤ 1.2. �he endemic equilibrium

+ + + ×

+ +'1

( ), ( ( )) , ,0( )( ) ( )

E i+− +

+ − +

which is locally asymptotically stable for σ ≤ 1.• If R > 1 then an endemic equilibrium ( )2 , , ,h h v hE s i i n will be the equilibrium point that is locally

asymptotically stable.

ProofEquilibrium Points�he equilibrium points satisfy the following relations:

0 0( ) 0i i s r− + + + = (1)0 0( ) ( ) 0i i s i+ − + + = (2)

( ) 0i r− + = (3)'0 0 0 0 0(1 ) 0i i i− − = (�)

From the equation (�) we have: i0 = 0 or 0 00 ' '

0 0

1 ( 1)i R= − = − where '0

0

R = .

From the equation (3) we have: r i=+

.

1. For i0 = 0:From the equation (2) we have: i = 0 or s

+ +=

• If i = 0 so r = 0 and s = 1. �hen the equilibrium point is E1(1,0,0,0)• If s

+ += from the equation (1) we have:

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+ −+ ++ +

( )( ( ))( )( )

i += − × . So i>0 for 1= >

+ +

In this case the equilibrium point is:

+ −+ + + ×

+ +'1

( ), ( ( )) , ,0( )( ) ( )

E i+− +

+ .

2. For 00 '

0

1i = − :

From the equation (1) we have:

1

( )s i+ += − −

+

From the equation (2) we have:

0)(

)()( 0000

2 =+

+

−++

−++−+

+

−++

− iiii

Since 0

)(1 ≥

+

−++

−= is and i ≥ 0 then

−+++

+∈

))(()(,0i .

We have 0'0

(0) ( 1)Q R= − and

0)()(

))(()))((

)(( <+++

−+++−=

−++++Q

where

00002

)()(

)()( iiiiiQ +

+

−++

−++−+

+

−++

−=

When R ≤ 1 the value of the polynomial Q(i) is negative at the end points of the interval

−+++

+))((

)(,0 therefore there are no roots in this interval.

If R > 1 then Q(0) > 0 therefore there e�ists a unique root in the interval which implies the e�istence of a unique equilibrium points: ( )2 , , ,h h v hE s i i n □

Stability Proof�he matri� of linearization (Jacobian matri�) is given by:

−−

+−−++−+−−++−

=

)21(000

0)(00)(

)(

00

'0

0

00

00

iR

siissii

J

800


1. For the point E1 the matri� J becomes:

−+−

−++−−−−

=

)1(0000)(0

0)(0

0

0

0

R

J

So E1 is stable if and only if R ≤ 1.2. For the point '

1E the matri� J becomes:

−+−

−++−−−+−

=

)1(0000)(0

0)()(

0

0

0

R

sissi

J

�hus the eigenvalues of matri� J are λ1 = –μ(1–R) and the root of the polynomial:

( )( )siiiP

isiiP223

2

))()(())()(()())()(()()(

++++++++++++++−=

+++++−++−++−=

( ))))((())(( −+++++++− i

If we put

)()( ++++++= iAiB )()( +++++=

)))((())(( −+++++++= iC

So: AB > C , A > 0 , B > 0 and C > 0 then following Routh-Hurwitz conditions for the polynomial P, the state E2 is locally asymptotically stable for R ≤ 1.

3. �he local stability of E2 is demonstrated in the same way as for '1E . □ □□

rEsULts aND DIscUssION

Stability analysis and values of the threshold were obtained. Simulation was carried out with different values of the parameters and illustration was given by Figures 2-5.

Figure 2 illustrates the behaviour of the solutions in the case of endemic equilibrium.(R > 1).Figure 3 and Figure � give a comparison of the human infectious according to different values of

the effective contact rate β (birds to human).Figure 5 shows the typical behaviour of the solutions indicating that the rate of (human) susceptible,

infectious and removed, as well as the avian infectious approaches, asymptotically, the trivial equilib-rium (R ≤ 1).

�he wide spread of avian influenza in birds poses two main risks for human and birds health. �he first risk of direct infection in birds and human when the virus passes from birds to birds and to human.

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Figure 2. 002.0,1.0,25.0,0000�.0,035.0,000�.0 00 ======

Figure 3. 002.0,1.0,25.0,0000�.0,035.0,000�.0 00 ======

802


Figure 4. 002.0,1.0,25.0,0000�.0,035.0,0�.0 00 ======

Figure 5. 075.0,002.0,1.0,25.0,0000�.0,035.0,0�.0 00 =======

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�he second risk is that the virus may change into a form that is highly infectious for birds and/or for humans. However, as stressed in the introduction section, the most crucial case is the eventual occur-rence of a pandemic, caused by person-to-person spread of the virus. As indicated by the simulation of different patterns, the dynamics of the disease is mainly determined by the average number of adequate contacts of a human susceptible with infected human and the average number of adequate contacts of a human susceptible with infected birds. �hese two parameters constitute essential keys to preventive strategies against pandemics.

rEFErENcEs

Ale�ander, M. E., Bowman, C., Moghadas, S. M., Summers, R., Gumel, A. B. & Sahai, B. M. (200�). A vaccination model for transmission dynamics of influenza. SIAM Journal of Applied Dynamical Systems, 3(�), 503-52�.

Bradt, D. A. & Drummond, C. M. (2006). Avian influenza pandemic threat and health systems response. Emergency Medicine Australia, 18(5-6), �30-��3.

Carrot, F., Luong, �., Lao, H., Sallé, A. V. , Lajaunie, C. & H. Wadernagel. (2006). A small-world-like(2006). A small-world-like model for comparing interventions aimed at preventing and controlling influenza pandemics. Biomedi-cal Central Medicine, 4, 26-28

Colizza, V., Barrat, A., Barthelemy, M., Valleron, A. J., & Vespignani, A. (2006). �he modelling of global epidemics: stochastic dynamics and predictability. Bulletin of Mathematical Biology, 68, 1893-1921.

Colizza, V., Barrat, A., Barthelemy, M., Valleron, A. J., & Vespignani, A. (2007). Modelling the worldwide spread of pandemic influenza: Baseline case and containment intercations. Plos Medicine, 4(1) e13

Derouich, M., & Boutayeb, A. (2006). Dengue fever: Mathematical modelling and computer simulation. Applied Mathematics and Computation, 177, 528-5��.

Ferguson, N. M., Fraser, C., Donnelly, C. A., Ghani, A. C., & Anderson, R. M. (200�). Public health risk from the avian H5N1 influenza epidemic. Science, 304, 968-969.

Heatcote, H. W. (2000). �he mathematics of infectious diseases. SIAM Review, 42(�), 599-653.

Jaime Mena-Lorca, & Heatcote, H. W. (1992). Dynamic models of infectious diseases as regulators of population sizes. Journal of Mathematical Biology, 30, 693-716.

Kuiken, �., Holmes, E. C., McCauley, J., Rimmelzwaan, G. F., Williams, C. S., & Grenfell, B. �. (2006). Host species barriers to influenza virus infections. Science, 312, 39�-397.

Smith, D. J. (2006). Predictability and preparedness in influenza control. Science, 312, 392-39�.

WHO. (2007). Avian influenza (“bird flu”)-Fact sheet. Retrieved June 06, 2007, from http://www.who.int/csr/disease/avian\_influenza

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WHO. (2007). Avian influenza: assessing the pandemic threat. Retrieved June 06, 2007, from http://www.who.int/csr/disease/influenza/H5N1-9reduit.pdf

Yuen, K. Y. & Wong, S. S. M. (2005). Human infection by avian influenza A H5N1. Hong Kong Medi-cal Journal, 11, 189-199.

KEY tErMs

Avian: Related to birds.

Immunity: Inherited, induced or acquired (by vaccine) resistance to infection by a specific patho-gen.

Incidence: �he number of new cases of a specific disease occurring during a given period (a year in general).

Influenza: An acute contagious viral infection characterized by inflammation of the respiratory tract and by fever, chills and muscular pain.

Mathematical Model: An abstract model using equations to describe the behaviour of a system (biological, physical, …).

Pandemic: Epidemic affecting a large proportion of a population over a wide geographical region.

Susceptible: Lacking immunity and resistance and consequently at risk of infection.

Stability: �he condition of being resistant to changes and perturbations.

a pandemic avian influenza mathematical model

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