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Edited by: Mustafa Inc, Firat University, Turkey

Reviewed by: Ilyas Khan, Ton Duc Thang University, Vietnam; Yilun Shang, Northumbria University, United Kingdom

This article was submitted to Mathematical Physics, a section of the journal Frontiers in Physics

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

This study is concerned with finding a numerical solution to the delay epidemic model with diffusion. This is not a simple task as variables involved in the model exhibit some important physical features. We have therefore designed an efficient numerical scheme that preserves the properties acquired by the given system. We also further develop Euler's technique for a delayed epidemic reaction–diffusion model. The proposed numerical technique is also compared with the forward Euler technique, and we observe that the forward Euler technique demonstrates the false behavior at certain step sizes. On the other hand, the proposed technique preserves the true behavior of the continuous system at all step sizes. Furthermore, the effect of the delay factor is discussed graphically by using the proposed technique.

One of the greatest threats that the world is facing is Human Immunodeficiency Virus (HIV) and its development into Acquired Immune Deficiency Syndrome (AIDS). This is the disease that attacks the immune system of the human body. There are specific type of white blood cells that fights against disease. HIV interrupts the body's ability to fight the organism, causing disease. When the CD4 cells are decreased up to the certain level, the immune system becomes too weak to defend the body against the infection. The role of mathematical models in studying infectious diseases, such as Hepatitis C and HIV/AIDS, is very significant [

Various researchers have pointed out the difficulties and discrepancies in the classical models. They improved the existing models by introducing some key facts, such as diffusion, the time delay, and other related factors. For reference we can see that Pan-Ping Lin discussed epidemic models with diffusion and delays and used Neumann boundary conditions. Many researchers pointed out that in some infectious diseases, such as Hepatitis, AIDS, Cholera etc. [

The dynamical behavior of epidemic models with the time delay is the most common and interesting topic that has sparked the interest of researchers nowadays. It is a matter of fact that infection does not spread with the same speed in all parts of the world. In some areas it communicates rapidly, and in others it spreads slowly. This situation can be handled by considering the reaction–diffusion term in the model. In this study, a time-delayed epidemic model for HIV/AIDS infection is studied with reaction diffusion.

Mathematical modeling is a useful tool for the study of the dynamical system relating to difference physical phenomenon [

Unlike some other viruses, the human body cannot get rid of HIV completely. Once you have HIV, you have it for life. Furthermore, HIV can lead to the disease AIDS if left untreated. No effective cure for HIV currently exists, but, with proper care, treatment, and medical aid, HIV can be controlled. Li and Ma [

μ = Death rate

β = Rate at which susceptible individuals will become Infected

α_{0} = Death Rate of Infected population

α_{1} = Death Rate of Recovered population

and

where τ is the incubation period. This is a time during which an infected individual will become infectious, i.e., they can spread the infection further. The incidence rate β^{−μτ} appearing in the second equation of system (1) represents the rate at time ^{−μτ} follows on from the assumption that the death of individuals follows a linear law given by the term −μτ [

The second order partial derivatives in the model with respect to x (the space variable) describe the diffusion in space. The term β^{−μτ} in the second equation of model addresses the incidence rate at moment (^{−μτ} reveals that the death of an individual at time τ obeys the linear law as expressed by −μτ. This fact can be followed from the assumption made in the model as discussed [

With the initial set of conditions being

and homogeneous Neumann boundary conditions.

The epidemic models exhibit two types of steady states: the infected steady state and uninfected steady state. The steady states employed by the delay epidemic model of HIV/AIDS dynamics are given as Uninfected steady (US) state _{1} = _{2} = _{3} = 0 is the reproductive number of the HIV/AIDSS epidemic model. When _{0} < 1 the disease is going to be at an end and when _{0} > 1 then disease will spread further.

Numerical modeling involves the study of methods to find approximate solutions to differential equations. In recent times, numerical solutions of delay differential equations are of great import due to the versatility of the modeling processes in different fields [

The Finite Difference Method is a numerical method that deals in mathematical sciences to solve differential equations by discretization. The Forward Euler technique is a widely used, well-known numerical technique in which derivatives are discretized by finite differences. In this approximation method, forward difference is used for the time derivative and central difference is implemented on the space derivative. By applying this technique on the system (4) and (5), we have

where

As for as stability of the forward Euler technique for where the system under study is concerned, we first implement the Von-Neumann stability within the scheme (6). The Von Neumann stability method is very efficient and widely used method to see the stability of the finite difference schemes. First, we consider the scheme (6),

Put

The difference scheme is stable if amplification factor, |ξ| ≤ 1. We thus get

Put

as |ξ| < 1 and ξ^{−m} < 1 we have

The non-standard finite difference scheme is made up of a general set of methods in numerical analysis that provide numerical solutions for differential equations by constructing a discrete model [

and

and

We will prove this theorem by using a mathematical induction method. By ensuring

which shows that clearly

which implies that

and

Now we need to prove that the result is true for

and

By using the values of

□

The Von Neumann technique is once again applied to the proposed scheme to check the stability. For this we have

Put

which implies that

In a similar fashion, we substitute

After simplification, we have

which implies that ξ < 1 as ξ^{−m} < 1.

While considering the series expansion purposed by Taylor in our proposed FD method, we produced a consistency analysis:

Similarly, for the Forward Euler technique we obtained the expression

Now, putting the values

Put

The proposed technique for a susceptible compartment is

Now, putting the values

Put ^{3} with

The forward Euler technique for an infectious compartment is

Now, putting the values of

Putting

The proposed technique for an infectious compartment is

Now, putting the values of

letting ^{3} above exactly as (5).

In this section, we present the example related to the model and furnished the numerical simulations with some suitable conditions as mentioned below.

with _{1}(_{2}(

Susceptible population and infected population with the help of Euler method (forward), where dynamics of S and I as shown in _{1} = _{2} = 0.15, τ = 1.5.

As in

Susceptible population and infected population dynamics by our proposed FD technique with the time step _{1}, and _{2} kept equal and selected as 0.15, τ = 1.5.

Once again, the forward Euler technique could not retain the actual behavior of the diffusive epidemic model of HIV/AIDS, as shown in

Graphs of susceptible population and infected population using Euler method for forward difference, with _{1} = _{2} = 0.15, τ = 1.5.

As compared to the forward Euler technique, the proposed technique approached the IS state and sustained the positive solution of the continuous delayed reaction–diffusion epidemic system, as depicted in

Sketch of susceptible population and infected population, giving the selection of values as _{1} = _{2} = 0.15, τ = 1.5 (with proposed scheme).

In

Dynamics of susceptible population and infected population considering h as 0.1 and _{1} = _{2} = 0.15, τ = 6.

Graphs _{1} = _{2} = 0.15, τ = 9.

Mesh graphs for the susceptible population and infected population by our proposed method, giving the parametric values as _{1} = _{2} = 0.15, τ = 12.

Discretized graphs of the susceptible population and infected population for our designed FD scheme, giving _{1} = _{2} = 0.15, τ = 15.

In this study, we have proposed the use of the delay epidemic model of HIV/AIDS dynamics with diffusion and applied two numerical techniques to assess the behavior of the solution of the continuous reaction–diffusion system with a delay factor. The techniques used to solve the proposed epidemic system are the well-known forward Euler technique and proposed unconditionally positivity preserving technique. The reaction–diffusion delayed HIV/AIDS epidemic model revealed a positive solution as the variables involved were the population sizes. The forward Euler technique, however, provided the negative values of the solution, which were not the part of the solution of the model under study. On the other hand, the proposed technique exhibited the same behavior as the solution of the HIV/AIDS delayed reaction–diffusion epidemic model. It preserved the positivity of the solution unconditionally and showed the convergence toward the true steady states of the HIV/AIDS reaction–diffusion system with a delay, as demonstrated in the simulations. The control of the spread of disease with the help of a delay factor was verified by the simulations. In future the proposed method will be applied to multi-dimensional delayed reaction diffusion epidemic models. Similarly, structure-preserving numerical techniques may be designed for fractional reaction diffusion systems with and without a time delay as well as stochastic models associated with diffusion and time delay. Furthermore, the Lie algebra approach may also be investigated for the above mentioned models [

MR and NA: conceptualization. MJ and NA: methodology. NA: software. DB and MR: validation. MJ and NA: formal analysis. MR and MAR: investigation. MAR: resources. MR: data curation. MJ: writing–original draft preparation. DB, MR, and MAR: writing–review and editing. MR, NA, and MJ: visualization. MAR and MR: supervision and project administration.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

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