SEI -MODEL FOR TRANSMISSION OF NIPAH VIRUS

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Introduction
Mathematical modeling has become an important tool for analysing the spread as well as control of infectious diseases.In recent years, epidemiological modeling of infectious disease transmission has had an increasing influence on the theory and practice of disease management SEI-MODEL FOR TRANSMISSION OF NIPAH VIRUS and control.Epidemiology is the study of the distribution, determinants of health-related states or events in specified populations and the application of epidemiology is to control of health problems.The application of mathematical modeling to the spread of epidemics has a long history, which was been initiated by Bernoulli [3] work on the effect of cowpox inoculation on the spread of smallpox.
Nipah virus (NiV), belongs to the genus Henipavirus, a new class of virus in the Paramyxoviridae family, has drawn attention as an emerging zoonotic virus in southeast and south-Asian region.This emerging infectious disease has become one of the most alarming threats of the public health mainly due to its periodic outbreaks and the high mortality rate.NiV was been first noted in Malaysia in 1998 in pigs and pig farmers [7].In 2001, NiV outbreaks have had been reported in Meherpurin Bangladesh and Siliguri in India.Of which, the highest mortality has occurred in Bangladesh, where the outbreak was typically noticed in winter season.In 2003-2005, the outbreak again appeared in Naogaon, Manikganj, Rajbari, Faridpur and Tangail districts [5].In Bangladesh, the outbreaks were again been reported in subsequent years [9].Recently, an outbreak has had been reported in the Kozhikode district of Kerala, India, wherein seventeen deaths were recorded, including one healthcare worker, which was declared to be officially ceased on June 10, 2018 [2].The natural host of NiV is fruits bats [18].Antibodies versus NiV have identified in fruit bats wherever they have tested including Cambodia, Thailand, India, Bangladesh and Madagascar [12], [16], [8], [9], [10].Though NiV has caused a few outbreaks, it infects a wide range of animals and causes severe disease and death in people, making it a public health concern.Treatment is mostly symptomatic and supportive as the effect of antiviral drugs is not satisfactory.Therefore, the high mortality addresses the need for its control and mitigation.Hence, the present analysis deals with application of SEI-model for NiV transmission.
NITA H. SHAH, ANKUSH H. SUTHAR, FORAM A. THAKKAR, MOKSHA A. SATIA Wenzel have conducted a Markov Chain Monte Carlo simulation to estimate the unknown parameters of transmission of NiV [17].Biswas have investigated the disease propagation and control strategy of NiV infections using SIR type mathematical model [4].Sultana and Podar studies the optimal use of intervention strategies to mitigate the spread of NiV using optimal control technique [15].Allen et al. review some mathematical models developed for the study of viral zoonoses in wildlife and identify areas [1].
In this paper, mathematical model for transmission of NiV is been formulated using system of non-linear ordinary differential equations in second section.Further, in third section, reproduction number and three equilibrium points are been derived from the system of differential equations.Local and global stability of the equilibrium points is been calculated in section four and analysis is completed by discussing numerical simulation.   1

Mathematical modeling
where  is positively invariant.i.e. every solution of model ( 1), with initial condition in  remains SEI-MODEL FOR TRANSMISSION OF NIPAH VIRUS there for all 0 t  .

Reproduction number and equilibria
By solving system of non-linear differential equations (1), we get three equilibrium points: i) Infection free equilibrium , 0, , 0, 0 , where ( ) We will find the basic reproduction number 0 R by the method of next generation matrix method [6].Let Thus, the next generation matrix is ( ) )   are given by the polynomial, ( ) ( )
Proof.Evaluating the Jacobian matrix for system (1)  .

Numerical simulation
The changes in different compartments under influence of other compartments and different parameters can be analysed through numerical simulations.

Theorem 3 . 1 .Theorem 3 . 2 .
(Local stability of 0 E ) The infection free equilibrium point 0 Evaluating the Jacobian matrix for model (1) at point 0 E gives SEI-MODEL FOR TRANSMISSION OF NIPAH VIRUS (Local stability of 1 E ) The infectious bats free equilibrium point 1 Evaluating the Jacobian matrix for system (1) for point 1 E ,

Figure 4 . 1 .
Figure 4.1.demonstrates the change in compartments of susceptible bats and infected bats with respect to time.It can observe from the above figure, that population of susceptible bats decreases continuously and become negligible in two weeks.And in three days, the population of infected bats decreases up to almost 70% after which it gradually increases.

Figure 4 . 2 .
Figure 4.2.depicts that population of susceptible humans' increase to some extent and then gets vanished in one week, after four weeks slight improvement in the class is observed.Highly improvement in exposed class is observed during first week, later it decreases gradually.Improvement in infected individuals is observed during NiV explosion for the parametric values

Figure 4 . 6 .
Figure 4.6.Impact on infected individual due to change in 

Table 2 .
1. Notations and parametric values used in the model