On solutions of time fractional order random HIVAIDS modelling

Abstract

Keywords: Fractional HIV/AIDS Model; Expected Value; Variance; Confidence Intervals

Introduction

AIDS was first discovered in the USA in 1981. HIV (human immunodeficiency virus) factor causes AIDS by crashing the immune system in humans. The HIV virus, known to have emerged for the first time in 1960, was first seen in monkeys. In 2017, 940,000 HIV-related deaths and

Materials And Methods

The concept of a fractional derivative was first introduced in the latter half of the 17th century and was developed by Leibniz and Newton. As a matter of fact, the concept of a fractional derivative predates that of an integer derivative. Due to the fact that the fractional derivative is really just an expanded version of the integer derivative. The Rieman-Liouville and Caputo definitions are the ones that are most frequently discussed in the literature [9]. For more than 300 years, the fractional derivative has drawn interest in mathematics. On this topic, scientists have conducted a number of investigations. Due to this circumstance, various definitions have occasionally emerged [10]. The fractional derivative has major advantages over the integer derivative in applied fields such as medicine, biology, and bioengineering. These advantages, which are supported by a number of models, not only improve the convenience of our daily lives but also present persuasive arguments for solutions to issues such as the spread of diseases [11]. Fractional differential equations in mathematical models have been more and more common in recent years due to their advantages. However, the problem’s structure also influences the analytical solution of these equations [12]. Preliminaries Definition 1. The Riemann-Liouville fractional derivative of order 0 < α < 1 of a function is defined as [13]

for all real numbers t ≥ 0 if the integral exists. Definition 5. [14-15] The Laplace transform of the transform of the Caputo fractional derivative is defined as (4) where α > 0, m - 1 < α < m and ℒ denotes the Laplace transform operator.

Differential Transform Method

The basic definitions of differential transform are introduced as follows. Let u(t) be analytic in a domain D and let t = t0 represent any point in D. The function u(t) is then represented by one power series whose center is located at t0. The differential transform of the k-th derivative of a function u(t) is defined as follows:

In (1), u(t) is the original function and U(k) is the transformed function[16]. The differential inverse transformation of U(k) is defined as follows:

(1) Definition 2. The Riemann-Louville fractional integral of order 0 < α < 1 of a function is defined as [13] (2)

Sigma J Eng Nat Sci, Vol. 42, No. 6, pp. 1899−1906, December, 2024

The fundamental theorems of the one-dimensional differential transform[16-18] are: Theorem 1. If z(t) = u(t) ± v(t), then Z(k) = U(k) ± V(k). Theorem 2. If z(t) = cv(t), then Z(k) = cV(k), where c is constant. Theorem 3. If

Table 1. Parameter descriptions and values [19] Parameter Explanation

In real applications, the function u(t) is expressed by a finite series and (3) can be written as The

Fractional Time Derived (Random) HIV/AIDS Model This section presents the results of solving the random fractional order HIV/AIDS modeling using the differential transformation method (DTM) and provides examples of the variances, confidence intervals, and expected values of various probability distributions of these solutions [1921]. There are now initial conditions that include randomly effective terms with various probability distributions. In recent years, the mean square calculation has been used to solve a few first-order random differential equations and models [22-32]. The deterministic form of the nonlinear HIV/AIDS model [19] is following: (5)

(6) In this model, those infected are denoted by l(t). Those who are infected can transmit the disease to susceptible persons indicated by S(t). Also, when infected persons in our model are sick, they are denoted by A(t). The parameters of our model are μ, β, δ and d and take values in the range of [0, ∞). The Table 1 below displays the values and descriptions for these parameters. If DTM is applied with the fractional HIV/AIDS model given in (5) and the initial conditions given in (6),

is in the form above. The initial conditions in our model are S0 ~Exp(λ = 2) exponential distribution, I0 ~N(μ = 3, σ2 = 9) normal distribution and A0 ~Beta(k = 3, l = 3) beta distribution, chosen from three different absolute continuous distributions. The parameters of the normally distributed random variable X are X~N(μ, σ2). Using the moment generating function of the normal distribution, we get

from (17), the 1st and 2nd moment of the random variable X~N(μ, σ2) are,

is calculated as. If the basic properties of the expected value for the X and Y independent random variables are used, the expected value of equation (16) is

Sigma J Eng Nat Sci, Vol. 42, No. 6, pp. 1899−1906, December, 2024

where Y is a random variable. If there is a probability density function of the form, it has a standard Beta distribution and is denoted by Y~Beta(k, l). Expected value for this distribution

The expectations can be compared to the deterministic results of equation (1) above (Figure 1) in a single graph (Figure 1). The following are the maximum and minimum values of the expected values of the random variables: S(t) has a maximum value of 265.1 when t = 2 and a minimum value of 2 when t = 0.

is in the form above. The moment generating function of the exponential distribution;

is indicated by X~(λ). Expected value and variance[20]: E(I(t))

Figure 2. Time-dependent variation of the expected value of infected individuals. For comparison with the deterministic outcomes of the equation above, expectations can be presented in a single graph (Figure 2). The following methods are used to determine the maximum and minimum values of the expected values of the random variables: At t = 0 and t = 2, respectively, l(t) takes the maximum value of 3 and the minimum value of -138.3.

Figure 1. Time-dependent variation of the expected value of individuals susceptible to the virus.

The expectations can be presented in a single graph (Figure 3) for comparison with the deterministic outcomes of equation (1) above (Figure 3). The following methods

Sigma J Eng Nat Sci, Vol. 42, No. 6, pp. 1899−1906, December, 2024

Figure 3. Time-dependent variation of the expected value of sick individuals.

Figure 4. Time-dependent variation of the variance value of individuals susceptible to the virus.

of the expected values of the random variables: At t = 0 and t = 2, respectively, A(t) takes the maximum value of 1.167

and the minimum value of -120.7. In our model, variance values were calculated according to the values and distributions given above.

(17) The S(t) variances are shown above (Figure 4). The following is how the extreme variances of the random variables are obtained: Maximum max[Var(S(t))] = 10720 at time t = 2 and minimum min[Var(S(t))] = 0.25 at time t = 0.

Figure 5. Time-dependent variation of the variance value of infected individuals. The variances of l(t) is given above (Figure 5). Extremum values of the variances of the random variables are obtained as follows: min[Var(I(t))] = 2 at t = 0 and max[Var(I(t))] = 9788 at t = 2. (21)

Sigma J Eng Nat Sci, Vol. 42, No. 6, pp. 1899−1906, December, 2024

Figure 7. Time-dependent variation of % 99 C.I. intervals of virus-susceptible individuals.

Figure 6. Time-dependent variation of the variance value of sick individuals. The variances of A(t) is given above (Figure 6). Extremum values of the variances of the random variables are obtained as follows: min[Var(I(t))] = 0.04 at t = 0 and max[Var(I(t))] = 51230 at t = 2. Confidence intervals for random variable expected values,

is equal, which can be determined using standard deviations. For K = 3, this formula yields a % 99 confidence interval(C.I.) for the expected value of a normally distributed random variable [21]. Figure 7 depicts a % 99 C.I. plotted with MATLAB (2013a).This popular rule, known as the three sigma rule, states that 99.73% of values for a normally distributed variable are within three standard deviations of the mean. Thus, using the appropriate parameters, we will compare the variations of the results for two continuous distributions with limited and unlimited support, respectively. Nearly all potential values for the random effects for both distributions will come from the same range if the parameters are chosen appropriately. The confidence intervals of S(t) are given in Figure 7. The extremum values of the confidence intervals are as follows: min(E(S(t)) - 3std(S)) = -45.48 at t = 2 and max(E(S(t))

Figure 8. Time-dependent variation of %99 C.I. of infected individuals Confidence Interval

Figure 9. Time-dependent variation of % 99 C.I. of patients.

Sigma J Eng Nat Sci, Vol. 42, No. 6, pp. 1899−1906, December, 2024

+ 3std(S)) = 575.6 at t = 2. Here, K = 3 gives an approximate %99 confidence interval. The confidence intervals of l(t) are given in Figure 8. The extremum values of the confidence intervals are as follows: min(E(I(t)) - 3std(I)) = -1077 at t = 2 and max(E(I(t)) + 3std(I)) = 800.2 at t = 2. Here, K = 3 gives an approximate %99 confidence interval. The confidence intervals of l(t) are given in Figure 9. The extremum values of the confidence intervals are as follows: min(E(A(t)) - 3std(A)) = -799.7 at t = 2 and max(E(A(t)) + 3std(A)) = 558.3 at t = 2. Here, K = 3 gives an approximate %99 confidence interval.

Conclusion

In this study, random fractional ordinary differential equations were solved using the fractional differential transformation method. The motivation of this study is to determine the probability characteristics of a randomized fractional-order HIV/AIDS Modelling under random effects. Normal, Beta, and exponential distributions are used to select the initial conditions or coefficients of random fractional ordinary differential equations. Expected value, variance, and confidence intervals from probability properties were found and graph for the analysis of random effect are presented accordingly. Examining standard deviations, variations, and confidence intervals for expected values reveals how virus transmission dynamics change over time. By selecting different probability distributions from many epidemic models in the literature, the behavior under random effects can be examined. We believe that this study will be an important component in mathematical modeling studies on the transmission of the HIV/AIDS virus. The results are guaranteed to be useful for the random differential equation system if the deterministic and random analysis results are comparable. The stochastic model can simulate the dynamics of disease transmission just as well as the deterministic model, but it can also yield information about the model’s variability, including coefficients of variation and standard deviations. We observe that random expectations are also consistent with the deterministic model’s results, which are comparable to the numerical results from the relevant study. Results for the expected values’ confidence intervals are also provided by the random model. Any deterministic compartment disease model can be created using this method of creating random models from deterministic models. Using actual data for parameter variations rather than speculative values can improve the results. Research on the stability and ideal management of diverse mathematical models of other illnesses, like COVID-19 and tuberculosis, can also be utilized to explore the dynamics of disease dissemination.

Data Availability Statement

The authors confirm that the data that supports the findings of this study are available within the article. Raw data that support the finding of this study are available from the corresponding author, upon reasonable request.

Conflict Of Interest

The author declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Ethics

There are no ethical issues with the publication of this manuscript.

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