key: cord-0800132-oi0dq0im
authors: Rathinasamy, A.; Chinnadurai, M.; Athithan, S.
title: Analysis of exact solution of stochastic sex-structured HIV/AIDS epidemic model with effect of screening of infectives
date: 2020-08-22
journal: Math Comput Simul
DOI: 10.1016/j.matcom.2020.08.017
sha: ea4d4c66c100f3ff1eb858a9d0f1873638814640
doc_id: 800132
cord_uid: oi0dq0im

The world of uncertainty motivates the study of stochastic perturbation in the mathematical models of real life. The main objective of this paper is to study stochastic sex-structured HIV/AIDS epidemic model with effect of screening of infectives. We have shown that the proposed stochastic epidemic model with boundedness and permanence has a unique global positive solution. The selection of suitable Lyapunov functions provides sufficient conditions for investigating persistence and extinction of disease. Based on numerical experiments, the theoretical findings of this paper have been verified.

One of the most deadly viruses is the human immunodeficiency virus (HIV), which causes the acquired immune deficiency syndrome (AIDS). It infects the immune system's cells and destroys the immune system's ability to fight infections and diseases during the process. AIDS refers to advanced HIV infection stages. According to recent estimates, about 34 million people worldwide are infected with HIV/AIDS and most are living in low and middle income countries. Such an epidemic can be effectively studied using mathematical modelling. Advanced information and precautions for an epidemic can be provided by a proper mathematical model. Several mathematical models have been studied to explain the dynamics of HIV spread due to the inherent strengths of the mathematical model. The mathematical approach to understanding the HIV epidemic and how to control the immune systems and their dynamics is discussed in [21, 30] . In Lin et.al. [22] , the role of infections and their effects has been created by dividing the infectious stage into 'r' classes. Workowski and Berman et.al. in [34] dealt with treatment and proper counseling that can control HIV transmission. Treatment of HIV/AIDS infected people may reduce transmission and infection rates. Cai et.al. [8] studied the importance of treatment in the transmission of HIV/AIDS has been discussed. Individuals who don't know about HIV can spread/transmit the disease to society unknowingly. In Tripathi et.al. [33] , this was discussed. In Kaur et.al. [17] , a stage-structured HIV model was created and analyzed by incorporating awareness and treatment effects.

In the real world, however, epidemic systems are often subject to environmental noise, and the effects of a fluctuating environment are not incorporated by deterministic models. Stochastic differential equation models therefore play an important role in different branches of applied science, including infectious dynamics, as they provide some additional degree of realism compared to their deterministic counterpart [3, 4] . Consequently, some authors have incorporated noise into HIV models and examined their dynamics. For example, by approximating one of the variables by a mean reverting process Dalal et.al. [9] analyzed a stochastic internal HIV model. Ji and Jiang [15] considered a cell-mediated immune response model of stochastic HIV-1 infection. They defined an appropriate condition in the large infection-free equilibrium for stochastic asymptotic stability. Since there is no infection equilibrium, they only explored the dynamics of the corresponding deterministic model around the two infection equilibria (one without triggering CTLs and the other with). Following the concept of [15] , both Liu [27] and Liu et.al. [27] demonstrated the asymptotic behavior of a nonlinear incidence stochastic delayed model of HIV-1 infection and a cell-to-cell model of HIV-1 infection. The asymptotic properties of a stochastic predator-prey system with functional response from Holling II was studied in Han et.al. [11] and the asymptotic behavior of stochastically perturbed DI SIR epidemic models with saturated incidence were analyzed in Liu et.al. [25] . A stochastic SIR model's threshold behavior was discussed in Cai et.al. [7] . Asymptotic behavior and stability of a stochastic AIDS transmission model have been discussed in Ding et.al. [10] . With time delays influenced by stochastic perturbations, Edoardo et.al. [5] discussed the epidemic model's stability. The threshold of an imperfect vaccination stochastic SIS model has been analyzed in Liu et.al. [23] .

In this paper, motivated by the referred works, we will study the dynamics of the epidemic model of stochastic sex-structured HIV/AIDS with effect of screening of infectives which is considered in [2] :

where

Γ 1 is the rate of recruitment in the susceptible males class, Γ 2 is the rate of recruitment in the susceptible females class, µ is the rate of natural death rate, β 1 is the death rate due to infection in the male class, β 2 is the death rate due to infection in the female class, β 3 is the death rate due to AIDS, γ m is the fraction of total infected males, who are screened, γ f is the fraction of total infected females, who are screened, ξ 1 is the AIDS progression rate in treated class, ξ 2 is the AIDS progression rate in untreated class, τ is the fraction of aware infectives taking treatment, α is the rate of transmission of HIV and S m (t), S f (t), I m (t), I f (t), A(t) denote susceptible males, susceptible females, infected males, infected females, AIDS-class respectively. Since the variable A does not appear explicitly in the first four equation of the system (1). So, we omit the last equation. We assume that stochastic perturbations are white noise type which are directly proportional to S m (t), S f (t), I m (t) and I f (t). Then, the deterministic system (1) will be extended to the following system of stochastic differential equations of the form:

The paper is organized as follows: In section 2, we proved the existence and uniqueness of the global positive solutions, the stochastic boundedness and permanence of the system (2). In section 3, the extinction of the proposed stochastic model is presented and in section 4, the disease persistence of the stochastic system (2) is studied. We investigated the suitable numerical simulations are presented in section 5 to illustrate the theoretical results. Finally, the conclusion of this paper is discussed.

Let (Ω, F, P) be the complete probability space with a filtration (F t ) t≥0 satisfying the usual conditions, (i.e, it is right continuous and increasing while F 0 contains all P-null sets). Let R 4

, I m (t), I f (t)) and denote C 2,1 ((0, ∞) × R 4 ; R + ) as the family of all non-negative functions V(t, Y) defined on (0, ∞) × R 4 such that they are continuously twice differentiable in Y and once in t.

We consider the following differential operator L associated with 4-dimensional stochastic differential equation of the form

where

If L acts on a function V ∈ C 2,1 (R 4 × (0, ∞)), then

be the domain containing the line y = y * and assume there exist a function V(t, Y) two times continuously differentiable in U which is the positive define in Lyapunov sense and satisfies LV ≤ 0 for y y * . Then the solution Y(t) = y * of stochastic differential equation (3) is stable in probability.

In this section, we prove that the existence and uniqueness of the solution of the stochastic HIV/AIDS model (2) . Next, we discuss one of the important concept of population dynamics, that is the stochastic ultimate boundedness of the solution of the model (2) . Also, we investigate the long time survival in a population dynamics based on the concept of stochastic permanence. In accordance with the approaches indicated in [16, 28, 31, 32] . Now let's begin with the existence and uniqueness of the solution of the stochastic HIV/AIDS model (2) .

where τ n is the explosion time, because the stochastic epidemic model (2) satisfies the locally Lipschitz continuous conditions. We need to show that the solution of the stochastic model (2) is global. We need only to prove that τ n = ∞ almost surely.

Choose a sufficiently large positive constant l 0 > 0 such that S m (0), S f (0), I m (0) and I f (0) are belong in [ 1 l 0 , l 0 ]. Consider the following sequence of stopping times for each integer l ≥ l 0 as

For the the empty set ∅, we set inf ∅ = ∞. Since τ l is non-decreasing as l → ∞, we have

Then τ ∞ ≤ τ n a.s. Now, we have to show that τ ∞ = ∞ a.s. If not, then there exist T > 0 and δ ∈ (0, 1) such that

There is an integer

We define a function V :

Hence

This completes the proof.

of the stochastic epidemic model (2) is said to be stochastically permanent, if for any ∈ (0, 1), there exists a pair of positive constants θ and Ψ such that for any initial value (S m (0), S f (0), I m (0), I f (0)) ∈ R 4 + , the solution Y(t) to stochastic model (2) has the properties

Theorem 2.3. Let µ < (Γ 1 + Γ 2 ) and for any positive initial value (S m (0),

where p > 0 be a constant satisfying

6 J o u r n a l P r e -p r o o f in which η > 0 be a constant satisfying

By choosing a constant p > 0 that satisfies (15) and using Itô's formula, we have

where

and m 1 , m 2 are already defined in the theorem. Then

Therefore

consequently,

, then the solutions of stochastic epidemic model (2) are stochastically permanent.

Proof. From theorem 2.2, we have P {|Y(t)| > θ} ≤ ,

This implies that lim t→∞ inf P {|Y(t)| ≤ θ} ≥ 1 − .

For any > 0, let Ψ = p p , then

The proof is completed.

J o u r n a l P r e -p r o o f

There is a disease free equilibrium E 0 Γ 1 µ , Γ 1 µ , 0, 0 of the deterministic HIV/AIDS epidemic model (1) and it is globally stable if R 0 = a 1 a 2 Γ 1 Γ 2 (µ+β 1 +b 1 )(µ+β 2 +b 2 )µ 2 < 1. This means that the disease will die out after a certain period of time. Hence, studying the disease-free equilibrium to control infectious disease is interesting. But the stochastic HIV/AIDS epidemic model (2) does not have disease free equilibrium and the stochastic solutions do not converges to E 0 . Therefore, we estimate the fluctuation around the disease free equilibrium E 0 in this section to analyze whether or not the disease is going to die out. + . If R 0 ≤ 1, σ 2 S m + (2µ+β 1 +b 1 ) 2 6(µ+β 1 +b 1 ) < µ, σ 2 S f + (2µ+β 2 +b 2 ) 2 6(µ+β 2 +b 2 ) < µ, σ 2 I m < (µ + β 1 + b 1 ), σ 2 I f < (µ + β 2 + b 2 ) and the system satisfies the following condition

where c ∈ a 2 Γ 2 µ(µ+β 1 +b 1 ) , µ(µ+β 2 +b 2 ) a 1 Γ 1 is positive number, i.e the disease will die out with probability one.

Proof. For our proof we will use some ideas from [1, 6, 13, 29, 35] . Define a C 2 -functions V 11 , V 12 , V 13 , V 14 , V 15 and V 16 defined for S m , S f , I m , I f ∈ R 4 + by

By applying Itô's formula, we have

where

Similarly, we get

J o u r n a l P r e -p r o o f

Because of the basic reproduction number R 0 ≤ 1, we get a 2 Γ 2 µ(µ+β 1 +b 1 ) ≤ µ(µ+β 2 +b 2 )

, then we choose a positive number c,

Combining (22), (23) , (24) , (25) , (26) and (27), we get

Integrating (28) from 0 to t and taking expectation, we get

Thus, taking the limit t → ∞ the above equation we get,

This completes the proof.

Remark 3.1. Theorem 3.1 shows that the solution of the system (2) fluctuates around the certain level which is relevant to E 0 . The value of σ S m and σ S f decreasing, then the solution of the stochastic system (2) will be close to the disease free equilibrium E 0 of the system (1) for most of the time. Besides, if σ 2 S m = σ 2 S f = 0, then E 0 is also the disease free equilibrium of the stochastic model (2) . We can get the proof of Theorem 3.1

If the basic reproduction number R 0 ≤ 1, µ > (2µ+β 1 +b 1 ) 2 6(µ+β 1 +b 1 ) , µ > (2µ+β 2 +b 2 ) 2 6(µ+β 2 +b 2 ) , σ 2 I m < (µ + β 1 + b 1 ) and σ 2 I f < (µ + β 2 + b 2 ), then we get LV 1 ≤ 0. Consequently, the model solution (2) is stochastically asymptotically stable most of the time.

We're interested in two things while studying epidemic dynamic disease models. One is when the disease dies, as seen in the above section, the other is when the disease prevails. We assume that R 0 > 1, then there is a unique endemic equilibrium

a 1 µ(µ+β 1 +b 1 )(µ+β 2 +b 2 )+a 1 a 2 Γ 1 (µ+β 2 +b 2 ) for the deterministic model (1) is globally stable but there is no endemic equilibrium in stochastic model (2) . Therefore, in this section, we demonstrate the existence of a unique ergodic stationary distribution for the stochastic model (2) , which shows that the disease will persist. We need the following Lemma before discussing the ergodic properties of the stochastic model (2) . h,k=1 a hk ρ h ρ k ≥ Λ|ρ| 2 , ∀y ∈Ū and ρ ∈ R 4 + . C 2 : There exist a neighborhood D and a non-negative C 2 −function V such that LV is negative for any y ∈ R 4 + \ D. Then, the markov process Y(t) has a stationary distribution π(.) with density in R 4 + such that for any Borel set B ⊂ R 4 + , lim t→∞ P (t, y, B) = π(B) and

J o u r n a l P r e -p r o o f ∀y ∈ R 4 + , where f (y) is a function integrable with respect to the probability measure π. Theorem 4.1. Let S m (t), S f (t), I m (t), I f (t) ∈ R 4 + be the solution of the stochastic model (2) with any given initial value (S m (0), S f (0), I m (0), I f (0)). 

where

Proof. We will use some methodology in our proof from [1, 6, 13, 29, 35] . When R 0 > 1, there is a unique endemic equilibrium point 

The problems of the most real world aren't deterministic. The stochastic effects that take place in the deterministic model give us a more practical way to build epidemic models. In this paper, we have studied the stochastic sexstructured HIV/AIDS epidemic model with effect of screening of infectives. Firstly, we have proved some qualitative properties, such as the existence of global positive solutions, boundedness and permanence solution of the proposed stochastic model (2) . Secondly, by constructing suitable Lyapunov functions and applying Itô's formula, we've found that the stochastic model (2) has a disease free equilibrium point E 0 and it is globally asymptotically stable when the reproduction number R 0 does not exceed a critical level. We also shown that, when the white noise intensity is sufficiently small and the reproduction number R 0 is larger than a critical level, the stochastic model (2) has a unique stationary distribution, and any solution of the stochastic model (2) in the distribution concentrates around the unique endemic equilibrium E 1 . Finally, to validate our theoretical studies, some computer-numerical simulations are provided. We conclude that the stochastic model (2) shows that disease extinction and persistence depends on the magnitude of the intensity of white noise as well as the parameters involved in the stochastic model (2) . In addition, our analysis shows that screening with appropriate counseling is an effective way to reduce the prevalence of HIV/AIDS infections. Increasing the screening rates reduces the basic reproduction number and the magnitude of the infectious individuals in the population. We conclude that efforts should be made, by screening with appropriate counseling, in order to prevent the disease.

Furthermore, some other topics of interest are worthy of further consideration. The approach used in this paper can also be used to investigate other epidemic models, such as TB, malaria, dengue, the current COVID-19 pandemic model, etc. We are interested to study analysis of extinction and persistence of the disease for stochastic HIV-TB co-infection epidemic model. In addition, this paper only analyzes the effect of white noise on the deterministic model, we can also add colored noise to the deterministic model and analyze the existence of an ergodic stationary distribution of the positive solutions to the model being considered. We are planning to study these in our future work.

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andand andAccording to the fact that the arithmetic mean is greater than or equal to the geometric mean, it follows that 2 − S m (34) and (35),and taking (33) and (40), we haveSimilarly, we can obtain

J o u r n a l P r e -p r o o f

ThenBy using the inequality a 2 = 2(a − b) 2 + 2b 2 , ∀a, b ∈ R, we havewhich can be simplified intowhere κ 1 , κ 2 , κ 3 , κ 4 and κ are defined in Theorem statement. Integrating (45) from 0 to t and taking expectation, we getThus, taking the limit t → ∞ the above equation we get,lies entirely in R 4 + , thus there exists a positive constant z > 0 and a compact set U ⊂ R 4 + such that, for any y ∈ R 4 + \ D,By taking (45) into account, for any y ∈ R 4 + \ D, LV 2 ≤ −z, which implies that the property (C 2 ) in lemma 4.1 is satisfied. The diffusion matrix of the stochastic model (2) is given byThere is a positive constantThen, according to Lemma 4.1, we obtain the ergodic property of the stochastic model (2) . This completes the proof.the solution of the stochastic HIV/AIDS model (2) fluctuates around the endemic equilibrium. Particularly, when the intensities are equal to zero, the stochastic HIV/AIDS model (2) degenerated into the corresponding determinisitc HIV/AIDS model (1) . If the value of intensities σ S m , σ S f , σ I m , σ I f decreasing, then the difference between the solution of the stochastic HIV/AIDS model (2) and the endemic equilibrium E 1 is small to reflect that the disease will persist.

We perform some numerical examples to illustrate the analytical results of stochastic model (2) . Then the system of the equations (2) can be rewritten as the following discretization equations: are satisfied and shows that the stochastic model (2) disease free equilibrium point E 0 (3125, 2500, 0, 0) is stochastically asymptotically stable. Fig.1 shows that the deterministic model (1) and the stochastic model Fig.2 . We choose the parameter values Γ 1 = 50, Γ 2 = 40, µ = 0.016, α = 0.00003, γ m = 0.5, γ f = 0.56, σ 1 = 0.005, σ 2 = 0.006, τ = 0.85, µ 1 = 0.0008, µ 2 = 0.0009, µ 3 = 0.003 with intensities of white noise σ S m = 0.03, σ S f = 0.03, σ I m = 0.02, σ I f = 0.03. Note that the basic reproduction number for these parameter valuesand the deterministic model (1) has a unique endemic equilibrium point 1867.857, 1356.648, 898.9622, 815.8059) .Additionally,Then, Theorem 4.1 conditions are fulfilled. Therefore the stochastic model (2) (3062.931, 2400.129, 44.38456, 72 .05186) which is stable (see Fig.8 ). However, the disease still die out for the stochastic model (2), hence which indicates that the conditions given in Theorem 4.1 are sufficient but not necessary.The relationship between the parameters γ m , γ f , ξ 1 , ξ 2 and basic reproduction number is shown in the Fig.9 . In addition, from Fig.5 to 8 shows that the HIV screening parameters male and female increases, the equilibrium of the male and female HIV infections may be reduced. This is shown more clearly in Fig.10 . In Fig.11 shows path simulations of male and female infectives of the different rate of aware of infectives taking treatment.