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Open Access Journal of Pharmaceutical Research Research Article 15 min read

Numerical Modeling of Hepatitis B Dynamics with Vertical Transmission and Treatment

Razzaq A*, Raza A and Rafiq M
* Corresponding author
ISSN: 2574-7797  10.23880/oajpr-16000181  Received: June 25, 2019  Published: August 20, 2019
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Keywords
Hepatitis B disease Dynamical system Numerical modeling Convergence
Abstract

Numerical modeling of communicable disease is a device to understand the instrument of how disease blowouts and how it can be measured. It builds on our considerate of the spread process of a contagion in a population. In this thesis, we have studied the dynamics of hepatitis B with vertical transmission and treatment numerically. We formulate an unconditionally stable non-standard finite difference (NSFD) scheme for a mathematical model of Hepatitis B disease. The developed numerical scheme is bounded, dynamically consistent and preserves the positivity of the solutions. NSFD scheme shows convergence to the true equilibrium points of the model for any time step sizes. But Euler and RK-4 fail for large time step sizes.

Introduction

Hepatitis B is a virus who effects the liver [1]. We can say that Hepatitis B is a liver disease that emanates from the infection with Hepatitis B virus (HBV) [2, 3, 4, 5]. Its acute infection and chronic infection will be caused [6]. HBV is exposure to infectious blood or body fluids by transmitted. HBV can increase to other humans through the blood:

  • HBV arrived the time of birth.
  • HBV from contact with other people’s blood during childhood.
  • HBV interchange from person to person through blood.
  • HBV transfer child from mother during delivery.

Approximately 2 billion people infected this virus Hepatitis B, 360 million people effected the chronic HBV. 600,000 peoples die in each year with infection of HBV [5, 6, 7, 8, 9, 10, 11, 12, 13].

Mathematical Model

Variables and Parameters

Denoted by susceptible to chronic infection any time of . Denoted by chronic infection any time of . Denoted by chronic infection any time of . Denoted by susceptible to acute infection any time of . Denoted by acute infection any time of . Denoted by acute infection any time of . Denoted by treatment who infected by acute virus any time of . Denoted by adult females and juveniles.

Total population size. Denoted by HBV infection rate. Denoted by contact the rate with infection individuals. ˄ Denoted by susceptible adult female’s rate. Denoted by susceptible juvenile’s female’s rate. Denoted by increase susceptible juveniles by birth. Denoted by birth from HBV acute females are assumed to susceptible. Denoted by remaining birth are infected infants who are in acute virus. Denoted by acute status progress in chronic virus. Denoted by adult females and juveniles. Denoted by adult females treated rate.

Denoted by adult female’s recovery rate and become susceptible. Denoted by treated is not perfect and female may progress to chronic stage rate. Denoted by birth from treated adult females is assumed to be susceptible. Denoted by birth from treated adult females is assumed to be susceptible and remaining proportion is infected and join acute virus. Denoted by adult female experience natural death. Denoted by juvenile’s female experience natural death. Denoted by adult female death rate. Denoted by juvenile’s female death rate.

Figure 1: Hepatitis B Virus Disease Model. The Scheme of Nonlinear Differential Equations (DE) on behalf of the Typical remains specified by:
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Figure 1: Hepatitis B Virus Disease Model. The Scheme of Nonlinear Differential Equations (DE) on behalf of the Typical remains specified by:

Figure1: Hepatitis B Virus Disease Model. The Scheme of Nonlinear Differential Equations (DE) on behalf of the Typical remains specified by:

(1) We describe equilibrium points of system i.e Disease free equilibrium(DFE).

are stability facts of scheme (1), Where

recognized as Procreative integer who describes the usual number of inferior impurities introduced of the main impurity. is a beginning influence who describe the disease of the exit or persist? If we say that the scheme will observe disease Free Equilibrium (DFE) and iff the scheme to involvement Endemic Equilibrium (EE).

Numerical Modeling

Now we have conferred two standard finite difference structures to unravel the endless dynamical scheme (1) i.e. Euler’s Method and Runge-Kutta Method of Order 4.

Euler Method

The Forward Euler’s Structure for the unceasing model (1) certain through:

} { } { } {

} { }s { } {

Numerical Experiments

Now solve numerical tryouts by expending the values of given parameters Table 1 [6].

ParametersValues
β0.5
Λ0.4
c10.5
b0.8
γa0.7
γc0.054
θ0.88797
p0.6
τ0
π0.03
μc0.4
δa0.47
δc0.04
μ0.4
ε0
η0
α0

Table 1: Numerical Tryouts.

Time(Years) h=0.1

Fourth Order Runge-Kutta Scheme

For Stage-1
$$K_1 = h[bs^n_a + pbU^n_a - (\pi + \mu_c)s^n_a]$$
$$L_1 = h[(1 - p)bU^n_a - (\mu_c + \gamma_c)U^n_a]$$
$$M_1 = h[\gamma_cU^n_a - (\delta_c + \mu_c)I^n_a]$$
$$N_1 = h[\lambda + \pi s^n_a - (\lambda + \mu_c)J^n_a]$$
$$O_1 = h[\lambda s^n_a - (\mu + \gamma_a + b)U^n_a]$$
$$P_1 = h[\gamma_aU^n_a - (\delta_a + \mu)I^n_a]$$

For Stage-2
$$K_2 = h[b(s^n_a + \frac{N_1}{2}) + pb(U^n_a + \frac{O_1}{2}) - (\pi + \mu_c)(s^n_a + \frac{K_1}{2})]$$
$$L_2 = h[(1 - p)b(U^n_a + \frac{O_1}{2}) - (\mu_c + \gamma_c)(U^n_a + \frac{L_1}{2})]$$
$$M_2 = h[\gamma_c(U^n_a + \frac{L_1}{2}) - (\delta_c + \mu_c)(U^n_a + \frac{M_1}{2})]$$
$$N_2 = h[\lambda + \pi(s^n_a + \frac{K_1}{2}) - (\lambda + \mu)(s^n_a + \frac{N_1}{2})]$$
$$O_2 = h[\lambda(s^n_a + \frac{N_1}{2}) - (\mu + \gamma_a + b)(U^n_a + \frac{O_1}{2})]$$
$$P_2 = h[\gamma_a(U^n_a + \frac{N_1}{2}) - (\delta_a + \mu)(U^n_a + \frac{P_1}{2})]$$

For Stage-3
$$K_3 = h[b(s^n_a + \frac{N_2}{2}) + pb(U^n_a + \frac{O_2}{2}) - (\pi + \mu_c)(s^n_a + \frac{K_2}{2})]$$
$$L_3 = h[(1 - p)b(U^n_a + \frac{O_2}{2}) - (\mu_c + \gamma_c)(U^n_a + \frac{L_2}{2})]$$
$$M_3 = h[\gamma_c(U^n_a + \frac{L_2}{2}) - (\delta_c + \mu_c)(U^n_a + \frac{M_3}{2})]$$
$$N_3 = h[\lambda + \pi(s^n_a + \frac{K_2}{2}) - (\lambda + \mu)(s^n_a + \frac{N_3}{2})]$$
$$O_3 = h[\lambda(s^n_a + \frac{N_3}{2}) - (\mu + \gamma_a + b)(U^n_a + \frac{O_2}{2})]$$
$$P_3 = h[\gamma_a(U^n_a + \frac{N_3}{2}) - (\delta_a + \mu)(U^n_a + \frac{P_3}{2})]$$

For Stage-4
$$K_4 = h[b(s^n_a + N_3) + pb(U^n_a + O_3) - (\pi + \mu_c)(s^n_a + K_3)]$$
$$L_4 = h[(1 - p)b(U^n_a + O_3) - (\mu_c + \gamma_c)(U^n_a + L_3)]$$
$$M_4 = h[\gamma_c(U^n_a + L_3) - (\delta_c + \mu_c)(U^n_a + M_3)]$$
$$N_4 = h[\lambda + \pi(s^n_a + K_3) - (\lambda + \mu)(s^n_a + N_3)]$$
$$O_4 = h[\lambda(s^n_a + N_3) - (\mu + \gamma_a + b)(U^n_a + O_3)]$$
$$P_4 = h[\gamma_a(U^n_a + O_3) - (\delta_a + \mu)(U^n_a + P_4)]$$

Finally
$$s^n^{n+1} = s^n_c + \frac{1}{6}[K_1 + 2K_2 + 2K_3 + K_4]$$

Non-Standard Finite Difference Model

Now we show an unreservedly convergent non-standard finite difference (NSFD) numerical model which be there describe on non-standard finite difference modeling concept introduced by Micken’s [3]. Now show the convergence scrutiny of the suggested structure. The NSFD model for the incessant dynamical system is given by:

$$s_c^{n+1} = \frac{s_c^n + h(bs_c^n + pbU_c^n)}{(1 + h(\pi + \mu_c))} U_c^{n+1} = \frac{U_c^n + h(1 - p)bU_c^n}{(1 + h(\mu_c + \gamma_c))} I_c^{n+1} = \frac{I_c^n + h\gamma_c U_c^n}{(1 + h(\delta_c + \mu_c))} S_a^{n+1} = \frac{S_a^n + h(\lambda + \mu)}{\U_c^n + h\lambda S_a^n} U_a^{n+1} = \frac{U_a^n + h\lambda S_a^n}{(1 + h(\mu + \gamma_a + b))} I_a^{n+1} = \frac{I_a^n + h\lambda S_a^n}{(1 + h(\delta_a + \mu))}$$

Convergence Analysis of NSFD Scheme

Let us define

$$F = \frac{s_c^n + h(bs_c^n + pbU_c^n)}{(1 + h(\pi + \mu_c))} G = \frac{G^n + h(1 - p)bG^n}{(1 + h(\mu_c + \gamma_c))} H = \frac{H^n + h\gamma_H^n}{(1 + h(\delta_c + \mu_c))} I = \frac{I^n + h(\lambda + \mu)}{\U_c^n + h\lambda I^n} J = \frac{J^n + h\lambda J^n}{(1 + h(\mu + \gamma_a + b))} K = \frac{K^n + h\gamma_K^n}{(1 + h(\delta_a + \mu))}$$

Now the Jacobian Matrix is given by

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Comparison Analysis

In this section, we see the comparison among of two standard difference schemes and non-standard difference

Results and Discussion

The model of transmission dynamics of Hepatitis B virus disease consumes introduced expending PSIT Model. (i.e. Threatened, Susceptible, Infected and Treated). The constancy of solid positions i.e. the Disease free equilibrium(DFE) deliberated numerically. We describe an unqualifiedly constant Non-Standard Finite Difference (NSFD) structure aimed at the incessant dynamical system. The suggested structure exists dynamical consistent, numerically steady and holds all the athetic assets of the incessant model. The outcomes equaled well known standard finite difference schemes i.e. Euler’s and Runge-Kutta method of order 4 (RK-4). The Euler and RK-4 are provisionally convergent and diverge of the assured ethics of step size 'h' while the constructed NSFD scheme for every assessment used to residues convergent [14, 15, 16, 17, 18, 19, 20].

Conclusion

The non-standard finite difference scheme created for the communication dynamics of HBV is unconditionally convergent. Inappropriately the abovementioned schemes like Euler and RK-4 are unsuccessful they depend on step size. So, Euler and RK-4 are conditionally convergent. When we intensify the step size, the graph of Euler and RK-4 are divergent and from time to time give variation in solution. The new advanced numerical scheme like non-standard finite difference scheme is independent on step size. Uncertainty we intensify the step size in hundreds and thousands then NSFD motionless convergent. The NSFD scheme is informal implement that gives mathematically stable, positivity, bounded-ness and shows an equal behaviour of the continuous model and discrete model.

References

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@article{razzaq2019,
  title   = {Numerical Modeling of Hepatitis B Dynamics with Vertical Transmission and Treatment},
  author  = {Razzaq A, Raza A and Rafiq M},
  journal = {Open Access Journal of Pharmaceutical Research},
  year    = {2019},
  volume  = {3},
  number  = {3},
  doi     = {10.23880/oajpr-16000181}
}
Razzaq A, Raza A and Rafiq M (2019). Numerical Modeling of Hepatitis B Dynamics with Vertical Transmission and Treatment. Open Access Journal of Pharmaceutical Research, 3(3). https://doi.org/10.23880/oajpr-16000181
TY  - JOUR
TI  - Numerical Modeling of Hepatitis B Dynamics with Vertical Transmission and Treatment
AU  - Razzaq A, Raza A and Rafiq M
JO  - Open Access Journal of Pharmaceutical Research
PY  - 2019
VL  - 3
IS  - 3
DO  - 10.23880/oajpr-16000181
ER  -