Analysis of the Dynamics of SI-SI-SEIR Avian Influenza A(H7N9) Epidemic Model with Re-infection

The spread of Avian influenza in Asia, Europe and Africa ever since its emergence in 2003, has been endemic in many countries. In this study, a non-linear SI-SI-SEIR Mathematical model with re-infection as a result of continuous contact with both infected poultry from farm and market is proposed. Local and global stability of the three equilibrium points are established and numerical simulations are used to validate the results.


Introduction
Avian Influenza is a virus majorly spread in birds but can also be transmitted to humans with the human infections acquired through direct contact with infected animals or contaminated environment. Human are infected with avian influenza Nomenclature S h (t) Susceptible human Λ h recruitment rates of human E h (t) exposed human Λ f recruitment rates of poultry I h (t) infected human µ h natural mortality rates of human R h (t) recovered human µ a natural mortality rates of poultry S f (t) susceptible poultry in farm µ d disease-related death rates of infected human I f (t) infected poultry in farm µ f disease-related death rates of infected poultry S m (t) susceptible poultry in market α 0 proportion of poultry from farms to markets I m (t) infected poultry in market r recovery rate of infected human has similar disease presentation with the Influenza virus; they are both zoonotic. Both the influenza virus and coronavirus cause respiratory disease ranging from mild illness to severe disease and death. The two viruses are transmitted by contact or droplets. For this reason, they can both be prevented by proper hand hygiene and good respiratory etiquette [1,2,3].
Iwami et al. [4] proposed a mathematical model that considered two types of avian influenza outbreak which may occur if humans fail to stop the spread of the disease. Liu and Fang [5] developed a dynamical model of avian influenza A(H7N9) to check the effect of the spread between poultry and poultry, poultry and human, and human and human and it was established that the likelihood of human-to-human transmission of the avian influenza A(H7N9) is low. Though the probability is low, the possibility of human-to-human transmission should still be considered. Che et al. [6] furthered the research by including saturated contact rate in the model for a highly pathogenic avian influenza epidemic. Chen and Wen [7] diversified the model by treating a bilinear disease incidence case with mutant avian influenza A(H7N9) virus, meanwhile, the highly pathogenic avian influenza epidemic model in the presence of vertical transmission function in poultry was proposed in [8]. The model proposed and investigated in [9] describes the transmission dynamics of avian influenza A(H7N9) between human and poultry. The model in [9] was further extended in [10] by introducing the possibility of re-infection. The results of the research indicated that a recovered individual who continue to have contact with an infected poultry may be re-infected with the disease. More recently, the spread of the avian influenza has attracted the attention of many researchers who working vehemently to unravel the dynamics of the transmission and as a result, bring an end to the spread of the disease. Advancement in the study of avian influenza include the models with Vaccination and Seasonality, the study of the Antigenic Variant, effects of other illnesses on patients infected with Avian Influenza A (H7N9) Virus, spatiotemporal variation and hotspot detection of the Avian Influenza A(H7N9) Virus, specificity, kinetics and longevity of antibody responses to avian influenza A(H7N9) [11,12,13,14,15,16]. The studies have been very successful so far.
It is important to note that despite the intense concentration on the dynamics of the avian influenza A(H7N9) virus, the SI-SI-SEIR model with re-infection has not been explored. In this paper, as a further advancement to [10], an SI-SI-SEIR model for the transmission of an avian influenza A(H7N9) virus with re-infection is proposed. The dynamics of the transmission is unraveled by the available mathematical tools. This paper is organized as follows: Section 2 presents the SI-SI-SEIR model with re-infection to study avian influenza A(H7N9) transmission, the reproduction number and the existence of the equilibria points are established in Section 3 and Section 4 shows that the equilibria are locally, and globally asymptotically stable using the Lyapunov functions. We present the numerical simulation to validate our results in Section 5 and give the conclusion in Section 6.
Considering the work of [10] and [9], the novelty of this work is included in the following four assumptions; 1) the natural and the disease-induced death rate are the same for the poultry population. 2) the human population is classified with the inclusion of Exposed class to the susceptible, infected and the recovered compartments. 3) every bird is moved to market at once, if the progression rate of susceptible poultry from farm to market is α 0 , then the progression rate of infected poultry to market will be (1 − α 0 ). 4) human get infected not only in market but also in farm. 5) Human re-infection with the disease is expressed as We propose the epidemic dynamics model of avian influenza A(H7N9) virus as 1) 3)
(2.9) The reproduction number 0 is defined as the expected number of secondary cases produced in a totally sensitive population by a typical infective individual during infectious period at a disease free equilibrium. The effective reproduction number is used to ascertain the transmission ability of a disease. The reproduction number is affected by the rate of contacts in the host population, the probability of infection transmission during contact and the contagious duration, hence, we obtain the reproduction number using the next generation matrix proposed by [17] (2.11) where

Existence of Equilibria
We prove the theorem on the existence of equilibria. S where then we can produce that = b 2 − 4ac > 0 if 02 > 1, thus, I * * * m has a unique positive root The other equilibrium points are obtained from Eq.(2.5-2.8) as Therefore, the endemic equilibrium is established whenever 01 > 1 and 02 > 1.
2. Now, we consider the condition I f = 0, therefore, the model (2.1-2.8) reduces to
3. Again, we consider a case where all the means of transmission of the disease between the poultry in market is negligible including all infected poultry in farm considered killed such that only susceptible poultry goes to market i.e. I m = 0 and (1 − α 0 ) I f = 0. The model (2.1-2.8) will then reduce to 17) We obtain from Eq.(3.18) and Eq.(3.17) that On solving Eq.(3.19) and substituting S * * f , Therefore, the boundary equilibrium is established whenever 01 > 1 and 02 < 1.

Stability of Equilibria
We have established the existence of disease-free and positive equilibria. We further investigate the stability of these equilibria.

Stability of the disease-free equilibrium
Proof. The characteristics equation of the Jacobian matrix at the disease-free equilibrium U 0 is . Clearly, all eigenvalues have negative real parts if 01 < 1 and 02 < 1 and consequently, 0 = max{ 01 , 02 } < 1. Thus, the disease-free equilibrium U 0 is locally asymptotically stable if 0 < 1 but unstable if 0 > 1.
Proof. We shall construct this proof by taking the three subsystems one after the other Poultry subsystem in farms: Define a Lyapunov function for the poultry subsystem in farms It follows that the derivative of L 11 is Thus, is globally asymptotically stable [18,19].
Poultry subsystem of markets: The poultry subsystem of markets with the avian components of farms already at the disease-free steady state is we define a Lyapunov function as It follows that Thus, which according to Lassale's invariace principle, U 0 m is globally asymptotically stable [18,19].
Human subsystem: Finally, we consider the human subsystem with the avian components already at the disease-free steady states.
we define a Lyapunov function then, it follows that the derivative of L 13 along the solution of Eq.(4.6) is Thus, which according to Lassale's invariace principle, U 0 h is globally asymptotically stable [18,19].

Stability of the boundary equilibrium and the endemic equilibrium
The characteristics equation of the Jacobian matrix of 2.1 is obtained as Theorem 4. For system (2.1 -2.8), the boundary equilibrium U * is locally asymptotically stable whenever 01 < 1 and 02 > 1, the boundary equilibrium U * * is locally asymptotically stable whenever 01 > 1 and 02 < 1 and the endemic equilibrium U * * * is locally asymptotically stable whenever 01 > 1 and 02 > 1.
Proof. We have the proof as follows: four of the eigenvalues are obtained from the two quadratic equations, and the other three eigenvalues are obtained from the cubic equation from which, if 01 < 1, 02 > 1, all the eigenvalues have negative real parts.
2. For the boundary equilibrium U * * = S * * f , I * * f , S * * m , I * * m , S * * h , E * * h , I * * h , R * * h , one of the eigenvalues is four of the eigenvalues can be obtained from the following two quadratic equations 12) The remaining three eigenvalues are obtained from the cubic equations from which, if 01 > 1, 02 < 1, all the eigenvalues have negative real parts.

For the boundary equilibrium
one of the eigenvalues is four of the eigenvalues can be obtained from the two quadratic equations The remaining three eigenvalues are obtained from the cubic equations h is globally asymptotically stable whenever 01 < 1, 02 > 1.
Proof. We consider the global stability of the boundary equilibrium and the endemic equilibrium.
(a) Consider the poultry subsystem in farms and define a Lyapunov function then the derivative of L 21 along the solution of Eq.(2.1) and Eq.(2.2) is which according to Lassale's invariace principle, U * f is globally asymptotically stable [18,19].
(b) Next, we consider the poultry subsystem of markets with the avian components of farms already at the disease-free steady state we define a Lyapunov function and then the derivative of L 22 along the soluion of Eq.(4.17) and Eq.(4.18) is Sm ≤ 0 if 02 > 1, then L 22 ≤ 0, and thus which according to Lassale's invariace principle, U * m is globally asymptotically stable [18,19].
(c) Finally, we are considering the human subsystem with the avian components of markets already at the endemic steady state.
2. The boundary equilibrium U * * (a) We firstly consider the poultry subsystem in farm and define a Lyapunov function then the derivative of L 31 along solutions of system (3.17) is obtain as which according to Lassale's invariace principle, U * * f is globally asymptotically stable [18,19].
(b) Next, we consider the poultry subsystem of markets with avian components of farm at the disease-free steady state We define a Lyapunov function which according to Lassale's invariace principle, U * * m is globally asymptotically stable [18,19].
(c) Lastly, considering the human subsystem with the avian components already at the endemic steady state We define a Lyapunov function and then, the derivative of L 33 along the solutions of system (4.30) is L 33 ≤ 0, and thus, is globally asymptotically stable [18,19]. In conclusion, if 01 < 1, 02 > 1, the boundary equilibrium U * * is globally asymptotically stable.
and define a Lyapunov function then take the derivative of L 33 along the solution of the system (4.37 -4.40) if 02 > 1, then L 43 ≤ 0 and thus, According to Lassalle's invariance principle, U * * * h is globally asymptotically stable.

Numerical Simulations
In this section, we present numerical simulations of model (2.1 -2.8) by considering the parameters in the following examples to obtain the stability of the disease free-equilibrium, the boundary equilibria and the endemic equilibrium represented as a time-series diagram. It is discovered that the disease-free equilibrium U 0 is globally asymptotically stable whenever 0 < 1.   It is discovered that the boundary equilibrium U * is globally asymptotically stable whenever 01 < 1 and 02 > 1.   It is discovered that the boundary equilibrium U * * is globally asymptotically stable whenever 01 > 1 and 02 < 1.   It is discovered that the endemic equilibrium U * * * is globally asymptotically stable whenever 01 > 1 and 02 > 1.       However, in order to reduce the spread of avian influenza, the following measures as indicated in Table 1, Table 2 and Table 3 can be taken; 1. increasing µ f by killing infected poultry, 2. reduce β f , β m and β h by closing down farms and markets where there is infected poultry to avoid continuous contact or transmission to human, 3. increase α 0 , this is achieved by increased number of susceptible poultry that move to market. If the susceptible poultry are higher than the infected poultry in the market, the chances of human having contact with infected infected poultry will reduce.

Conclusion
In this paper, we combined human and poultry to developed an SI-SI-SEIR dynamic model of avian influenza A(H7N9) with the inclusion of re-infection and transmission of the disease occurring both in farm and market. The reproduction number we obtained is sufficient to establish the following; 1. there exist the disease-free equilibrium U 0 which is globally asymptotically stable whenever 01 < 1 and 02 < 1, hence, the disease dies out (see Figure  5.1).