Mathematical modeling of Avian Influenza epidemic with bird vaccination in constant population

The development of the industrial world and human life is increasingly modern and less attention to environmental sustainability causes the virus causes the epidemic has a high tendency to mutate so that the virus that initially only attack animals, is also found to have the ability to attack humans. The epidemics that lasted some time were bird flu epidemics and swine flu epidemics. The flu epidemic led to several deaths and many people admitted to the hospital. Strain (derivatives) of H5N1 virus was identified as the cause of the bird flu epidemic while the H1N1 strain of the virus was identified as the cause of the swine flu epidemic. The symptoms are similar to seasonal flu caused by H3N2 strain of the virus. Outbreaks of bird flu and swine flu initially only attacked animals, but over time some people were found to be infected with the virus.


Introduction
Perdue [1], Sedyaningsih [2], and Scoones & Forster [3] suggested that bird flu virus can be transmitted to humans and can cause death so that an outbreak occurs. In Yang et al [4] published in 2009, bird flu epidemic then erupted swine flu epidemic. The flu epidemic led to several deaths and many people admitted to the hospital. Strain (derived) H5N1 virus preserved as the cause of the epidemic of bird flu virus H1N1 virus as a cause of swine flu epidemic. Symptoms caused by seasonal flu are caused by H3N2 strain of the virus. Jansen et al [5] mentioned influenza viruses responsible for the number of deaths and people who are sick in the hospital. In Widiasih [6] the sense of presence in poultry infected with this virus with a very large number of economic terms very.
de Jong et al [7] mentioned that the influenza A subtype H5N1 virus with substitution of an amino acid in neuramiside isolated from 2 patients undergoing therapy/treatment and known the virus is immune to a given drug. Both of these patients died because of this viral infection. Gooskens [8] also mentioned that there is a mutation of influenza A virus that produces a virus that is immune to oseltamivir. The mutated virus is contagious pathogenic and lethal for high-risk patients. The ability of the H5N1 virus to mutate is so high that it is necessary to watch out for the spread of this virus in the poultry population so that some precautions have been taken such as the destruction of infected poultry and quarantine for infected humans.

Methods
The first step to do this research was literacy study. In this step, we study the fact and some assumptions from various scientific literacies. After that, we complete the facts with some assumptions to build the model. The second step was building and analyzing mathematics model. In this step, we build the mathematics model and then analyze it to determine the equilibrium points and their stability. The third step was making simulation with parameters value which was gotten from other paper.

Mathematical Model
From literature review we got: in Tuncer & Martcheva [9], it were stated that the avian influenza virus of subtype H5N1 can infect humans and cause death in human and bird population. In Tuncer & Martcheva [9] and Bourouiba [10], there was stated that vaccination in poultry are still being implemented. In Bourouiba [10], it was stated that vaccinated poultry which is even free of clinical signs, should not be traded to avoid all risk of silent shedding and transmission. Vemula et al [11] used several different approaches that are currently available for diagnosis of influenza infections in humans. These are used to diagnosis of influenza virus infections following natural infection and vaccination in humans.
In this paper, we assume that the population is constant so the death rate of infected human and infected bird were assumed have same value with natural death rate in every population. We also assumed that death in infected humans and infected birds only occurs due to viral infection and the probability of infectious contacts of bird-human and human-human are same. Transfer diagram of AI epidemic is given at Figure 1.  where is the total human population, is total number of susceptible person, is total number of infected person, is total number of recovered person, is the total bird population, is total number of susceptible bird, is total number of infected bird, and is total number of vaccinated bird. The meaning of parameter in model were given in Table 1.
From Fig. 1 we construct te system of ordinary differential equation as System (1).
We assumed that = and = then we get System (2).

Simulation
Simulation was done for three cases like three conditions in Theorem 2. Value of some parameter followed from Kharis & Arifudin [12]. Value of parameter were given in Table 2.

Simulation for
< and < In this case, we used the value = 0,08 and = 0,5. From the formula 0 and 0 in Theorem 1, we got 0 = 0,713 < 1 and 0 = 0,816 < 1. From Theorem 1, There is only one equilibrium point i.e. 0 = ሺ , , , ሻ = ሺ6000,0,39.06,0ሻ. The graphs for this simulation were given on Figure 2. From Figure 2, it can be seen that the solutions that is near from 0 converge to 0 . These simulations were similar with Theorem 2.

Conclusion
From analysis above, we get the dynamic of mathematics model of AI-epidemic with vaccination on bird population especially for constant population. We also got the formula of reproduction number ሺ 0 and 0 ሻ which can be used to determine whether the epidemic spread widely or vanish. For the formula of 0 , we got that the proportion of vaccinated susceptible bird can change the value of 0 . If this proportion increase then 0 decrease. It means we can prevent the spreading of this epidemic in bird population by increasing the proportion of vaccinated susceptible bird. For human population, we can prevent the spreading of this epidemic by reducing the probability of infectious contact between infected people and susceptible people. It can be done by quarantine infected people. For the next research, we propose to make the mathematics model for non-constant population.