MATHEMATICAL MODELLING APPROACH FOR THE STUDY OF NIPAH VIRUS DISEASE TRANSMISSION DYNAMICS

. Ever since the ﬁrst outbreak of Nipah virus disease, which occurred both in human and non-human primates in developing countries in Far East Asian between 1998 and 1999 which led to a majority of deaths, with the effect of such occurrence still witnessed up till date. We studied the spread of Nipah virus and obtained a system of equations comprising of ten equations which effectively described the transmission of Nipah Virus in a population where control measures were incorporated and two major sources of contacting the disease which are coming in contact with contaminated foods by infected bats from crop farming and pig farming, these were also incorporated. We investigated the local stability of the disease-free equilibrium using the Jacobian Matrix approach and the disease-endemic stability using the center manifold theorem. We also investigated the global


INTRODUCTION
Nipah virus (NiV) is a single-stranded RNA virus that causes severe disease both in humans and animals.It has two distinct strains, Nipah virus Malaysia (NiV M ) and Nipah virus Bangladesh (NiV B ).These two strains differ in transmission pattern, researches have shown that (NiV B ) could be more pathogenic than (NiV M ), Mire et.al [1].These two viruses are known to cause Nipah virus disease (NiVD) in humans.
Human beings can contract NiVD through but not limited to the following means: direct contact (such as broken skin, saliva, urine, or mucus membrane in the eyes, nose, or mouth) with blood or body fluids of a person who is sick with Nipah disease, or a non-human infected with the virus, Goh et.al [2].One can equally get the disease through objects (such as syringes and needles) that have been contaminated with the virus from a person sick with Nipah disease or the body Skowron et.al [3].
Humans have been seen to be infected by direct contact with infected pigs, probably through mucous membranes, but possibly also through skin abrasions, or consumption of poorly cooked infected pig meat.In a Nipah-like outbreak in the Philippines Banerjee et.al [4], most patients had been involved in slaughtering sick horses or had eaten undercooked horsemeat from sick horses Chua et al [5].In Bangladesh, human cases have been linked to drinking unpasteurized date palm sap (juice).Experimental infections have shown that urine is a significant route of henipavirus (HeV) excretion from fruit bats Mire et.al [1].Therefore, the persistence of henipaviruses (HeVs) in fruit bat urine was examined Skowron et.al [3].NiVD infection cluster was originally first identified in September 1998 in Perak, Malaysia, and this was closely followed by a second and third cluster in the state of Negri Sembilan, with cases occurring primarily among adult men in contact with swine Chua et.al [5]; Abang et.al [6].
Observations from several clinical data for NiV outbreaks reveal some key differences between patients infected with NiV M and NiV B .Some of these differences are: (1) NiV B has a shorter average incubation period and a more narrow range for the incubation period than NiV M , (2) most of the recorded cases of NiV B included respiratory symptoms while few patients infected with NiV M presented with respiratory symptoms, (3) Rear cases of patients during the Bangladeshi and Indian NiVB outbreaks reported myoclonus, while a significant proportion of patients from the Malaysian outbreak presented with segmental myoclonus as well as the fatal cases in the Philippines presenting with an acute encephalitis Mire et.al [1]; Pallister et.al [7], (4) During the Bangladeshi outbreaks, sources revealed that the virus is either unknown in some cases or has been traced to be from consumption of contaminated fruit or date palm sap (or palm-wine), which was then followed by human-to-human transmission and nosocomial spread, whereas the source of the virus in the Malaysian outbreak is known to be from pigs, which served as an amplifying host Mire et.al [1]; Sultana et.al [8].On the issue of host to the virus, it is speculated that bats are the host for Nipah virus Mondal et.al [9]; Biswas et.al [10].Direct transmission from reservoirs or secondary infected animals may occur.nondomesticated animals may transfer the virus to humans when an infected and humans comes in contact with their infected and infectious body fluids or when their meat is eaten by humans uncooked or under-cooked.In this situation, the virus then spreads in the human population either through human-to-human, pig-to-human, or bat-to-human transmission.Airborne spread of the disease has not been detected or discovered in the observed Nipah virus outbreaks Weingartl et.al [11].

PRELIMINARIES
Having deliberated in detail the process and mode of transmission of this virus in the community and noting the peculiar features in these areas in question that aided the spread of the virus, we now develop a mathematical model representing this information that can enable us to carry out some analyses and provide vital information for the medical health officials about the disease caused by the virus.To do this successfully, we have our model assumptions as: 1.Those recruited into the system are not infected or contacted the disease.
2. We focused on a single strain, either the Nipah virus Malaysia (NiV M ) strain or Nipah virus Bangladesh (NiV B ) strain.
3. There is an intensive supportive care for those infected with the virus that goes for treatment.
4. The treatment given to those infected is a combinational therapy method, because at the time of this study there is no single specific approach or method for treating Nipah virus disease.
5. Nipah virus is seen to be a highly emerging disease and no single drug is available yet for its treatment.
6. Humans can contact the disease by contact with infected bats, infected pigs, or infected humans.
7. The infected individual can recover either by care giving/combinational therapy or naturally.
8. One can recover from the infection without treatment.9. Once recovered, a person or animal cannot get the same strain of the virus again.10.We assume the latent class is merged with the infectious class.
11.In the nearest future, a globally acceptable vaccine for humen in the treatment of Nipah virus would be produced.
12. Pigs get infected by contact with infectious pigs or infectious bats.
13.People or pigs can die naturally or as a result of the disease.
14. Bats are the natural carriers of Nipah virus disease.
15.The pigs are caged so that they do not roam to eat defecation from humans that are infected.
Considering the above assumptions, the total human, animal, and bird populations were grouped based on the disease or non-disease status of those involved.Which leads us to the following: The human individuals that can readily contact the disease if in contact with an infectious person, infectious animal, or infectious bird, which we call the susceptible individuals, S H ; we have ϕ H to be the rate of vaccination form the infection; we present (ϕ H S H ), to be the proportion of the susceptible population that have been vaccinated; (1-ϕ H ) S H is the proportion of susceptible population that are yet to be vaccinated;(I H ) Infected humans not yet going for treatment; (I HT ) Infected humans population that are currently under-going for treatment; (R H ), human population that have recovered from the virus.
The pig population compartment is also divided into four mutually disjoint compartments, which are: (S P ), susceptible pigs; (I P ), infected pigs; (I PT ), virus infected pig population that has been taken for treatment; (R P ), the population of pigs that has recovered from the virus.
The bat population is also divided into two compartments, which are: susceptible bats (S B ), infected bats (I B ).
The model tracks how each individual group communicates between each other with respect to time by the application of differential equations.
We obtained the following mathematical system of equations from our above model assumptions: (1)

MAIN RESULTS
Having developed the model equations, we check to see the well-posedness of the model by showing that the model satisfied the conditions on the positivity of the variables of the model as well as the continual remaining of the variables in the region of existence of the variables Castillo-Chavez and Song [12].Thus, we have the theorem: Theorem 1.Let us consider the region defined as: We show that the solutions to the system of equations (2.1 − 2.11) are positive for all t > 0 given that the initial conditions are positive.For the system of equations (2.1 − 2.11), the region Ω 1 is positively invariant, and the solution starting in Ω 1 remains in Ω 1 .Obasi and Mbah [13].

PROOF
Suppose that all initial conditions of the variables are all positive, then, from (2): Next, we solve using the integrating factor method comparing equation ( 13) with the standard first order equation given as: ( 14) Which solution is of the following form: where the integrating factor a(X): Given the following from ( 13) then, ( 16) Therefore, the solution for equation ( 16) becomes: Divide through by (e a 1t ) and then apply the initial condition at t = 0, we have: The solution is bounded by zero below and is positive for all t ≥ 0 We can continue using this same approach to prove for the remaining variables.Therefore, in general, we have that: By this, we have shown that the solutions of the equations in the system (equations 2.2 -2.11) are positive and in R 10 + , provided that the initial conditions are positive.This implies that Ω 1 is positively invariant, which indicates that the model is mathematically and biologically well posed and reasonable enough to be used in the study of the transmission of Nipah virus disease.
Next, in our analytics agenda, we move to our other point which is the continuous existence and non-existence of this infection in the population.This is done by analyzing to get the equilibrium points of the disease at the disease free and endemic states.At these points, it is expected that the virulence of the disease does not change, which means we try not to study different viral strains of the virus.Thus, we study the equilibrium state as follows: 3.1.Disease free equilibrium (DFE).Let E(S H , I H , I HT , R H , S P , I P , I PT , R P , S B , I B ) be the equilibrium point of the displayed system (equations 2.2-2.11).Then, at the equilibrium state, we have that: This then implies that the right-hand side of equation (2.2 -2.11) will be equated to zero, therefore, solving the resulting equations from our proposed model, we obtain the equilibrium point, E n as: , 0, 0, 0, Therefore, the disease-free equilibrium (DFE) for our Niv model is achieved when all the disease classes of humans, pigs, and bats are zero.

endemic disease equilibrium (EE).
In this section, we discuss the endemic equilibrium state of the disease; this is the state where the disease is not totally absent in the various compartments of the population.We can say that the disease may persist in the population, such that the susceptible class, the infectious class, the infectious treated class, the recovered class, and others must not all be zero at this equilibrium state Gurley et.al [14].In other, to obtain the endemic equilibrium points of the model, we solve equations (2.2 -2.11) simultaneously.By equating : Therefore, the endemic equilibrium points for our proposed model equation (2.2 -2.11) exists, which is given as: = (0, 0, 0, 0, 0, 0, 0, 0, 0, 0) Having produced an acceptable equilibrium points, we need to verify whether these equilibrium points are stable or not.But first we need to generate the basic reproductive number for our studied disease, then we would study the per capita disease growth rate, stability of the disease free and endemic states of the studied disease at the equilibrium points stability: the local and global stability of the disease states 3.3.The Basic Reproductive number (R 0 ) for the study.To derive the basic reproduction number for our proposed model system as described in equations (2.2 -2.11).We then form the next generation matrix which is F V −1 , where F is the new infection term and V is the matrix of all the transferred terms.Also, F is the rate of appearance of new infection; V is the difference between the transfer rate of individual out of compartment by other means and the transfer rate of individuals into the same compartment by other means Shah et.al [15].
Then, we let We then computed FV −1 , and evaluated all its eigenvalues.Where we got the following to be the R 0 , and it is the same as computed by Abang et.al [6] (24) Infected population per capita growth rate.In this section, we try to show the relationship between R 0 and the rate at which the product of the proportion of susceptible individuals in the population who are yet to be infected or have not been vaccinated times the transmission rate of the disease (this is known as per capita growth rate for the disease).From equation (3), we have that: this equation is simplified, thus, the following: we then have (25), which is the rate at which infections occur in the community per infected individual.It helps us determine whether the disease will spread and grow in the population or eventually fade away.

Local stability.
Here, we conduct the local stability analysis for our proposed model.
Local stability analysis helps us to determine whether an equilibrium point of a dynamic system is stable or unstable Tyagi et.al [16].We have the local stability analysis in the disease free and endemic state.In the disease-free state.

3.4.1.
Local stability for Nipah virus single variant disease free state.
From the obtained eigenvalue results, it can be seen that all the eigenvalues have negative real parts, which implies that the disease-free equilibrium state is locally asymptotically stable.

3.4.2.
Local stability for Nipah virus single variant Endemic Equilibrium.For the diseaseendemic state, we studied the local stability using the Centre Manifold theorem by Castillo-Chavez and Song [12].
We define the transformation of equations ( 1) to (11) as such that and Then, we can further define equations ( 1) to (11) as: where Next, we applied the central manifold theorem and obtained the bifurcation coefficients a < 0 and b < 0 Ahman et.al [17]; Atokolo and Mbah et al [18].This shows that the endemic equilibrium state is locally asymptotically stable and at the same time exhibits forward bifurcation.
This implies that we can reduce the reproduction number to be less than one by increasing our educational campaign.This is enough to restrain the spread of the Nipah virus disease, irrespective of the initial sizes of the infected populations.Now we can study the global stability.Proof.In order to establish the global stability of this equilibrium point (E 0 ), we construct the Lyapunov function Weingartl et.al [11]; Ahman et.al [17]; Atokolo and Mbah [18].Figure 1a shows the various dynamics of Nipah virus in the human population, during the disease outbreak.

Global Stability Analysis for our
In Figure 1b, the susceptible S H decreased as time increased due to the outbreak of Nipah virus without knowledge of the kind of virus that emerged.We discovered that the susceptible population never got to zero due to various factors such as natural immunity to the virus by those that have suffered from disease similar to Nipah virus.Figure 2a depicts a scenario where the susceptible bat population grows as time increases, which is inline with our model assumptions and our model of equation 10.In figure 2b we say a graphical display of the level which must be vaccinated for the population to reach an immunity threshold, where the Nipah virus disease can no longer be a threat in the observed society.
Figure 3 expresses the graphs that display the phase diagram for our proposed model 1 to equation 11 and the per capita growth for the Nipah virus disease is also shown in Figure 3. Figure 3a indicates the extent of how serious each outbreak of Nipah virus disease would be if an outbreak occurs, it shows that every subsequent outbreak would not be as devastating as the one before, but the disease might be endemic in the society.In Figure 3b the graph shows that as Nipah virus per capita growth increases, the number of susceptibles reduces, so in other for us to maintain a better healthy society, we need the disease per capita growth to be low.

CONCLUSION
In our study of Nipah virus, we were able to formulate a clear mathematical model that contained the majority of the disease features obtainable in Nipah virus outbreak that instigates the broad and rapid transmission of Nipah Virus disease within an endemic population.We included the issue of vaccine coverage in the community as one of the major sources for achieving population immunity and reducing contacting and spreading of the Nipah virus disease.
In the analysis of equations ( 1) to (11), we obtained the equilibrium points of the transmission of the disease and further analyzed these equilibrium points to obtain the conditions for the local and global stabilities of the disease free and endemic equilibriums of the disease transmission dynamics.
It can be seen from our graph that early intervention and reducing the disease contact rate is the best option for managing and treating an infectious people in order to reduce the spread and transmission of the disease, which would further lead to a healthy and more productive society.
In addition, from our graphs, the population immunity threshold was identified in the population for effective administering of vaccine to a large number of the vulnerable community as it has a great effect on the disease spread and control.
In our further publications, we are going to include the susceptible population of palm-wine drinkers, how they can alter the disease dynamics and carry out sensitivity analyses to see the effect of the control, transmission, and treatment parameters on the spread of the disease in the population.

2. 1 .
Model Assumptions.The development of the model is based on the following assumptions: Nipah virus disease model.Theorem 3. If W > 1, the non-negative equilibrium point (E 0 ) of the model (1) to (11) is globally stable and convergence to the equilibrium point (E 0 ).
But at the Nipah virus-free equilibrium,I H = I HT = R H = I P = I PT = R P = I Bwe expand (38) and collect the positive and negative terms separately, where the positive term is N and and the negative term is M, thendW dt = N − MThe graphical findings below were developed through python programming software applying the parameter values in 2 and 4, with also the initial values.

4. 2 .
Simulation Figures.Figure1depicts the Nipah virus dynamics in the human population and that of the susceptible human population S H , S v , over a period of 100 days.

2 .
FIGURE 1.Nipah virus disease dynamics in human population and vaccinated popupulation

3 .Figure 4
FIGURE 3. Phase plot of Susceptible, Infected population and Nipah per capita growth rate

Figure 4b displays a
Figure4bdisplays a graphical visualization of the treatment effect of treatment on our susceptible population regardless of that at any point in time there must be someone infected with the virus, but once the right treatment is administered coupled with positive vaccine penetration, the Nipah virus disease could be control.

FIGURE 4 . 5 .
FIGURE 4. Vaccine and treatment rate on Nipah dynamics population

TABLE 1 .
1.1.Model Parameters.This is shown in table 1. Model Parameters with description.Rate which bats die as a result of human and other activitiesThe continuation for some of the parameters used and their descriptions are shown in table 2.
B Proportion of new entry into the bat society ϕ H Rate of vaccination in the human population µ H Rate of natural death in the human population µ P Rate of natural death of pig µ B Rate of natural death of bat β 1 Contact rate between infected classes and humans β 2 Contact rate between infected classes and pigs β 3 Contact rate between infected classes and bats τ 1 Rate of infected human goes for treatment τ 2 Rate of infected pig that is taken for treatment δ 1 Rate which infected humans dies of the disease

TABLE 2 .
Model Parameters with description.Continuation H Force of infection for the human class λ P Force of infection for the pig class λ B Force of infection for the bat class

TABLE 3 .
Numerical values for Parameters adopted for implementation.

TABLE 4 .
Continuation of Numerical values for Parameters adopted for imple-These parameters in 2 and 4 describe the transmission dynamics of the Nipah virus disease and the effects of different proposed control interventions.The parameters are given specific values that are based on the characteristics of the disease being modeled.