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Fractional view analysis of sexual transmitted human papilloma virus infection for public health – Scientific Reports


In this section of the paper, our focus is on the numerical solution of our system (4) of the infection. To do this, we consider the below stated Atangana–Baleanu derivative system

$$\begin{aligned} ^{ABC}_0 D^{\upsilon }_t g(t)= f(t,g(t)), \end{aligned}$$

(17)

transform the previously stated equation into the subsequent form according to34:

$$\begin{aligned} g(t)-g(0)= \frac{1- \upsilon }{AB(\upsilon )} f(t,g(t))+ \frac{\upsilon }{AB(\upsilon ) \Gamma (\upsilon )} \int _0^t f(\theta ,g(\theta )) (t-\theta )^{(\upsilon -1)} d\theta , \end{aligned}$$

(18)

the above at \(t_{r+1}=(r+1)\Delta t\) can be stated as

$$\begin{aligned} g(t_{r+1})-g(0)= \frac{1- \upsilon }{AB(\upsilon )} f(t_r,g(t_r))+ \frac{\upsilon }{AB(\upsilon ) \Gamma (\upsilon )} \int _0^{t_{r+1}} f(\theta ,g(\theta )) (t_{r+1}-\theta )^{(\upsilon -1)} d\theta , \end{aligned}$$

(19)

this can be further transformed into:

$$\begin{aligned} g(t_{r+1})= g(0)+ \frac{1- \upsilon }{AB(\upsilon )} f(t_r,g(t_r))+ \frac{\upsilon }{AB(\upsilon ) \Gamma (\upsilon )} \Sigma _{\imath =2}^{r} \int _{t_\imath }^{t_{\imath +1}} f(\theta ,g(\theta )) d\theta . \end{aligned}$$

(20)

In the subsequent phase, we employ the Newton polynomial method to estimate f(tg(t)) as follows

$$\begin{aligned} P_r (\theta )= & {} f(t_{r-2},g(t_{r-2}))+ \frac{f(t_{r-1},g(t_{r-1}))-f(t_{r-2},g(t_{r-2}))}{\Delta t} (\theta -t_{r-2})\nonumber \\{} & {} + \frac{f(t_{r},g(t_{r}))- 2 f(t_{r-1},g(t_{r-1}))+f(t_{r-2},g(t_{r-2}))}{2(\Delta t)^2} \times (\theta -t_{r-2})(\theta -t_{r-1}). \end{aligned}$$

(21)

Utilizing the above stated polynomial in (20), we get that

$$\begin{aligned} g^{r+1}= & {} g^0+ \frac{1- \upsilon }{AB(\upsilon )} f(t_r,g(t_r))\nonumber \\{} & {} + \frac{\upsilon }{AB(\upsilon ) \Gamma (\upsilon )} \sum _{\imath =2}^{r} \int _{t_\imath }^{t_{\imath +1}} \bigg ( f(t_{\imath -2},g^{\imath -2}) \nonumber \\{} & {} + \frac{f(t_{\imath -1},g^{\imath -1})-f(t_{\imath -2},g^{\imath -2})}{\Delta t} (\theta -t_{\imath -2})\nonumber \\{} & {} + \frac{f(t_{\imath },g^{\imath })-2f(t_{\imath -1},g^{\imath -1})+f(t_{\imath -2},g^{\imath -2})}{2 (\Delta t)^2} (\theta -t_{\imath -2})(\theta -t_{\imath -1}) \bigg ) (t_{r+1}-\theta )^{\upsilon -1} d\theta . \end{aligned}$$

(22)

Moreover, we get

$$\begin{aligned} g^{r+1}= & {} g^0+ \frac{1- \upsilon }{AB(\upsilon )} f(t_r,g(t_r))\nonumber \\{} & {} + \frac{\upsilon }{AB(\upsilon ) \Gamma (\upsilon )} \sum _{\imath =2}^{r} \bigg (\int _{t_\imath }^{t_{\imath +1}} f(t_{\imath -2},g^{\imath -2}) (t_{r+1}-\theta )^{\upsilon -1} d\theta \nonumber \\{} & {} + \int _{t_\imath }^{t_{\imath +1}} \frac{f(t_{\imath -1},g^{\imath -1})-f(t_{\imath -2},g^{\imath -2})}{\Delta t} (\theta -t_{\imath -2}) (t_{r+1}-\theta )^{\upsilon -1} d\theta \nonumber \\{} & {} + \int _{t_\imath }^{t_{\imath +1}} \frac{f(t_{\imath },g^{\imath })-2f(t_{\imath -1},g^{\imath -1})+f(t_{\imath -2},g^{\imath -2})}{2 (\Delta t)^2} (\theta -t_{\imath -2})(\theta -t_{\imath -1}) (t_{r+1}-\theta )^{\upsilon -1} d\theta \bigg ), \end{aligned}$$

(23)

the following result is achieved after simplification

$$\begin{aligned} g^{r+1}= & {} g^0+ \frac{1-\upsilon }{AB(\upsilon )}f(t_r,y(t_r))\nonumber \\{} & {} + \frac{\upsilon }{AB(\upsilon ) \Gamma (\upsilon )} \sum ^{r}_{\imath =2} f (t_{\imath -2},g^{\imath -2}) \Delta t \int _{t_\imath }^{t_{\imath +1}} (t_{r+1}-\theta )^{\upsilon -1} d\theta \nonumber \\{} & {} +\frac{\upsilon }{AB(\upsilon ) \Gamma (\upsilon )} \sum ^{r}_{\imath =2} \frac{f (t_{\imath -1},g^{\imath -1})-f (t_{\imath -2},g^{\imath -2})}{\Delta t} \int _{t_\imath }^{t_{\imath +1}} (\theta -t_{\imath -2}) (t_{r+1}-\theta )^{\upsilon -1} d\theta \nonumber \\{} & {} +\frac{1}{\Gamma (\upsilon )} \sum ^{r}_{\imath =2} \frac{f (t_{\imath },g^{\imath })-2f (t_{\imath -1},g^{\imath -1})+f (t_{\imath -2},g^{\imath -2})}{2 (\Delta t)^2} \nonumber \\{} & {} \times \int _{t_\imath }^{t_{\imath +1}} (\theta -t_{\imath -2})(\theta -t_{\imath -1}) (t_{r+1}-\theta )^{\upsilon -1} d \theta , \end{aligned}$$

(24)

the integrals above can be evaluated using the following method

$$\begin{aligned} \int _{t_\imath }^{t_{\imath +1}} (t_{r+1}-\theta )^{\upsilon -1} d \theta= & {} \frac{(\Delta t)^\upsilon }{\upsilon } \bigg ( (r-\imath +1)^\upsilon -(r-\imath )^\upsilon \bigg ) \nonumber \\ \int _{t_\imath }^{t_{\imath +1}} (\theta -t_{\imath -2}) (t_{r+1}-\theta )^{\upsilon -1} d \theta= & {} \frac{(\Delta t)^{\upsilon +1}}{\upsilon (\upsilon +1)} \bigg ( (r-\imath +1)^\upsilon (r-\imath +3+2 \upsilon ) \nonumber \\{} & {} – (r-\imath )^\upsilon (r-\imath +3+3 \upsilon ) \bigg ) \nonumber \\ \int _{t_\imath }^{t_{\imath +1}} (\theta -t_{\imath -2}) (\theta -t_{\imath -1}) (t_{r+1}-\theta )^{\upsilon -1} d \theta= & {} \frac{(\Delta t)^{\upsilon +2}}{\upsilon (\upsilon +1)(\upsilon +2)}\nonumber \\{} & {} \times \bigg [(r-\imath +1)^\upsilon V_1 -(r-\imath )^\upsilon V_2 \bigg ], \end{aligned}$$

(25)

where \(V_1=2(r-\imath )^2+(3\upsilon +10) (r-\imath )+2\upsilon ^2+9 \upsilon +12,\) and \(V_2=2(r-\imath )^2+(5\upsilon +10) (r-\imath )+6\upsilon ^2+18 \upsilon +12\). After simplification, we get that

$$\begin{aligned} g^{r+1}= & {} g^0+ \frac{1-\upsilon }{AB(\upsilon )} f (t_r,g(t_r)) \nonumber \\{} & {} +\frac{\upsilon (\Delta t)^\upsilon }{AB(\upsilon )\Gamma (\upsilon +1)} \sum ^{r}_{\imath =2} f (t_{\imath -2},g^{\imath -2}) [(r-\imath +1)^\upsilon – (r-\imath )^\upsilon ] \nonumber \\{} & {} +\frac{\upsilon (\Delta t)^\upsilon }{AB(\upsilon )\Gamma (\upsilon +2)} \sum ^{r}_{\imath =2} [f (t_{\imath -1},g^{\imath -1}) – f (t_{\imath -2},g^{\imath -2}) ] \nonumber \\{} & {} \times \bigg ( (r-\imath +1)^\upsilon (r-\imath +3+2 \upsilon ) – (r-\imath )^\upsilon (r-\imath +3+3 \upsilon ) \bigg ) \nonumber \\{} & {} +\frac{\upsilon (\Delta t)^\upsilon }{2AB(\upsilon )\Gamma (\upsilon +3)} \sum ^{r}_{\imath =2} [f (t_{\imath },g^{\imath })-2f (t_{\imath -1},g^{\imath -1}) + f (t_{\imath -2},g^{\imath -2}) ] \nonumber \\{} & {} \times \bigg [(r-\imath +1)^\upsilon V_1 -(r-\imath )^\upsilon V_2 \bigg ]. \end{aligned}$$

(26)

We will employ the aforementioned approach to depict the time series of the proposed infection model. Time series analysis holds significant importance in comprehending, monitoring, and managing diseases. It furnishes valuable insights into the dynamics of the disease, aids in the early detection of outbreaks, and enables the assessment of intervention effectiveness. This, in turn, contributes to more informed and targeted public health initiatives. The numerical values of system parameters and state variables will be assumed for computational purposes. Various simulations will be conducted to illustrate how these parameters impact the infection system.

In the initial simulation, illustrated in Figs. 2 and 3, we scrutinized the impact of the fractional parameter \(\upsilon\) on the dynamics of HPV. In Fig. 2, we consider the values of \(\upsilon\) to be 1.00, 0.95, 0.90,  and 0.85, while in Fig. 3, the value of \(\upsilon\) is varied as 0.80, 0.70, 0.60,  and 0.50. This systematic exploration of diverse values for the input parameter \(\upsilon\) allows us to thoroughly investigate the characteristic solution pathways of the system. The outcomes of these simulations unequivocally highlight the substantial influence exerted by the fractional parameter on the dynamics of the infection. Notably, \(\upsilon\) emerges as a promising tool for effectively managing the spread of the infection within the community. Therefore, we strongly advocate for a more in-depth exploration and analysis of this fractional parameter by policymakers to enhance their understanding of its potential in mitigating the impact of the infection on public health. This comprehensive investigation can contribute valuable insights for developing targeted strategies in the control and prevention of the infection. Figure 4 depicts the impact of the input parameter \(\beta\) on the dynamics of HPV infection. In this simulation, we considered \(\beta\) values of 0.20, 0.40, 0.60, and 0.80. Our observations highlight the crucial role of this parameter, indicating a direct association with an increased risk of the infection.

In Figs. 5 and 6, we have illustrated the biological implications of varying input parameters \(\rho\) and \(\theta\) on the dynamics of HPV. In Fig. 5, we explored the effects of different values of \(\rho\) (0.45, 0.55, 0.65, and 0.75), while maintaining \(\theta\) at values of 0.2, 0.3, 0.4, and 0.5 in Fig. 6. Our investigation specifically focuses on discerning how changes in these parameters influence the behaviors of asymptomatic and infected individuals within the HPV system. In the conclusive simulation, depicted in Fig. 7, we investigated the impact of the input parameter \(\eta\) on the solution pathways of HPV infection. For this analysis, we considered values of \(\eta\) as 0.25, 0.30, 0.35, and 0.40. The observation centered on understanding how variations in \(\eta\) contribute to the dynamics of the asymptomatic and infected classes within the model. These insights hold significant relevance for informing public health strategies, intervention measures, and the formulation of effective control policies aimed at managing and mitigating the repercussions of infectious diseases on populations. Understanding the intricate relationships between input parameters and the dynamics of HPV infection is essential for the development of targeted and efficient approaches to tackle such public health challenges.



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