Stochastic Modeling for HIV and AIDS Epidemics With Viral Load Detectability

In this paper, we investigated on the stochastic epidemic model by incorporating viral load detectability. We derived HIV and AIDS stochastic model from the deterministic counterpart model and presented a stochastic threshold in terms of stochastic basic reproduction number. We showed that Rs 0 < 1 then the disease dies out exponentially and when R s 0 > 1 the disease persists in the population. We further derived the existence and uniqueness, extinction and persistence properties of the stochastic model models, then the numerical simulation is done by using Milsten’s numerical scheme. The finding shows that the random perturbation introduced in the stochastic model suppresses disease outbreak as compared to its deterministic counterparts which provide some useful control strategies to eradicate the disease.


Introduction
As in the early stages of an epidemic, the number of infectious will be very small, randomness alone can be the source of an epidemic to perish out, and it is much more significant particularly to include perturbation into models. Stochastic models possibly will be an additional suitable technique for modelling disease epidemics in many circumstances which gives a better understanding of transmission dynamics derived from its deterministic counterpart. It has similarly been revealed that some stochastic epidemic models can deliver an extra degree of realism in contrast to the deterministic model. Explicitly, the stochastic model suits well for disease extinction and disease persistence (Van Herwaarden and Grasman, 1995).
There is a fair quite a number literature about stochastic models, for example, authors such as Cai et al. (2016) and Mao et al. (2002) studied deterministic and stochastic models for ratio dependents transmission rates with the main focus on how environmental perturbations particularly affects the dynamics of the diseases and others focus on the environment with media effect. Also, Zhang et al. (2014), developed and gave an excellent clear investigation of non-linear incidence rate, and analyzed the extinction, persistence, and stationary distribution of stochastic disease epidemic model. Similarly, Song et al. (2018) developed SIRS stochastic model and particularly investigated extinction and persistence with saturated incidence rate, most importantly they showed that large noise in the model reduces disease outbreak. Stochastic models could be a more suitable way of modelling epidemics in many situations and many realistic stochastic models can be deriving from their deterministic counterparts (Fan et al., 2017;Ji and Jiang, 2017;Zhang and Zhou, 2019;Zhao and Yuan, 2016;Witbooi, 2013).
There are numerous ways of deriving a stochastic model from the deterministic model: parametric perturbation stochastic model which introduces diffusion coefficient of the specific parameter(s) of interest to be perturbed, while on other hands non-parametric perturbation stochastic model involves adding Brownian processes to each differentia equation and assume that each compartment has uncertainty (Nsengiyumva et al., 2013;N'zi and Kanga, 2016;Miao et al., 2017). Several authors have undertaken various stochastic models for HIV and AIDS including authors such as (Mbogo et al., 2013;Zhang and Zhou, 2019;Fan et al., 2017). However, none of these models have considered stochastic models for HIV and AIDS particular by incorporating viral load detectability in the model, and the stochastic model is derived from deterministic counterpart as considered by Tengaa et al. (2020). This paper is arranged as follows:-In section 2, we formulated a stochastic model from deterministic as considered by Tengaa et al. (2020). In section 3, we investigate the quantitative analysis of the model such as deriving the existence and uniqueness, extinction, and persistence properties of the stochastic model models. Section 4 shows the numerical solution of the model and lastly, in section 5 is the concluding remark.

Description of the Model
The model is divided into five compartmental models which includes Susceptibles population (S t ), Infected population(I t ), AIDS population on treatment(A t ), Detectable(D t ) and Undetectable(U t ) viral load population. The frequency dependent transmission is assumed with force of infection λ = β I N(t) , where β is the rate of transmission. I t can be increase by the proportion (1 − ρ) from infected mothers who give birth to infected children, also I t can progress to A t with a rate of θ 1 . Moreover, A t after failure of therapy they can either join D t with detectable viral load failure rate of ψ or can join U t with a rate of (1 − ψ). In turn with HIV and AIDS viral load test or drug resistance tests while also HIV replicates at an extremely rapid rate then individuals in D t can be treated with a rate θ 2 and individuals in U t can be treated with a rate of θ 3 to move back to A t on treatment class, whilst µ is the natural mortality rate while α is the death due to AIDS individuals. The normalized deterministic model as considered by Tengaa et al. (2020) subject to condition s + i + a + d + u = 1. All the feasible solutions of system (2.1) enters the region of biological interest defined by Ω = (s, i, a, d, u) ∈ 5 + : s + i + a + d + u = 1 that is positive-invariant solution. We consider the dynamics of the flow generated by system (2.1) in Ω. In this region, therefore the model (2.1) is considered to be both biologically and mathematically well posed with a deterministic reproduction number given as R d 0 = βρ (θ 1 + Λ) as used by Tengaa et al. (2020).
Therefore, the stochastic model for nondimensionalized system of equations (2.1) becomes: where B i (t)(i = 1, 2) are independent standard Brownian Motions and σ i (i = 1, 2) are constant intensities of environmental fluctuations respectively. We define bounded set Ω as Throughout this paper, let (Ω, F , P) be a complete probability space with filtration {F t } t∈ + satisfying the usual conditions(right continous and increasing while F 0 contains all P-null sets). We denote as x(t) = (s, i, a, d, u) := (x 1 (t), x 2 (t), x 3 (t), x 4 (t), x 5 (t)) and denote C 2,1 ( 5 × (0, ∞) : + ) as the family of all nonnegative functions V(x, t) defined on 5 × (0, ∞) such that they are continously twice differentiable in x and differentiable once in t.

Existence and Uniqueness of Stochastic Model
To investigate the behaviour of the system (2.2) we need to consider the system is positive and ensure the existence of global behaviour. In order to get stochastic model which has unique global solution(no explosion in finite time) for a given initial values, the coefficient of the equation are required to satisfy the linear growth condition and the Lipschitz condition. But according to system (2.2), the linear growth condition is not satisfied, but the system is locally Lipschitz and the solution of the system (2.2) may explode at finite time.
Proof. Since the coefficients of the equation are locally Lipschitz continuous for any given initial value where τ e is the explosion time. To show that the solution of the system (2.2) is global, we need to show that τ e = ∞ a.s. Let k 0 ≥ 0 be sufficiently large such that (s(0), i(0), a(0), d(0), u(0)) lies within the interval [ 1 k 0 , k 0 ] . For each integer k ≥ k 0 we define the stopping time as follows Note that according to definition, τ k is increasing as k −→ ∞, by setting τ ∞ = lim s. But if this condition is false, then there exist a pair of constants T > 0 and ∈ (0, 1) such that P {τ ∞ ≤ T } > Therefore, there is an integer k 1 ≥ k 0 such that Let C 2,1 be twice continuously differentiable in x and once continuously differentiable in t and let lyapunov function candidate be V : 5 We define the differential oparetor L associated with 5-dimensional stochastic differential Similarly the quadratic variation becomes; From equation (3.4), substituting (3.5) and (3.6) and considering the drift part we get where, M is a positive constant and K is a constant which independent of variables, therefore from (3.4) we have; Integrating both sides of (3.7) from 0 to Applying expectation both sides gives Set Ω k = {τ k ≤ T } for k ≥ k 1 and by (3.2), P(Ω k ) ≥ , and note that for every ω ∈ Ω k , there is atleast one of s(τ k ∧ ω), i(τ k ∧ ω), a(τ k ∧ ω), d(τ k ∧ ω) and u(τ k ∧ ω) that equals to either k or 1 k and thus V( We must therefore have τ ∞ = ∞ almost surely.
Using the above lemma (3.3) we showed that the solution of system (2.2) is stochastically ultimately bounded. For any > 0, and upon setting δ = δ 2 1 2 , then by using Chebyshev's inequality we get,

Stochastic Extinction of the Disease
We will focus on the conditions which guarantee the extinction of disease i(t). We are going to explore the conditions which lead to the extinction of diseases of the system (2.2) under a white noise stochastic disturbance. Consider the following theorem regarding the extinction of disease for stochastic model.
Proof. From the first equation of the system (2.2), we have which means that By using Itô formula, from (2.2) , let consider infected it is C 2,1 ( 5 × (0, ∞) : Let M = σ 1 t 0 sdB 1 (r). Then, M is a martingale Khasminskii (2011). This implies that quadrartic variation of stochastic integral becomes: by the Strong Law of Large Numbers for martingales, we have Upon dividing both sides by t we have, Taking the limit superior on both sides leads to lim sup which indicates that lim t−→∞ i(t) t = 0 then disease will go to extinction exponentially with probability one, then (R s 0 − 1) < 0 implying that R s 0 < 1. We conclude that the number of infected population will also tend to zero exponentially almost surely.

Stochastic Persistence of the Disease
In this subsection we will show the condition when the disease prevails, we shall investigate the persistence of the disease. > 1 then for any initial value (s(0), i(0), a(0), d(0), u(0)) ∈ 5 + , the disease i(t) of the model system (2.2) has the property that That is to say, the disease will prevail ifR S 0 > 1 Proof. We employ the stochastic Lyapunov method, let set V 1 = −a 1 log s − log i − log a − log d − log u as our Lyapunov function, where positive constant a 1 to be determined later.
From the generalized Itô formula we have, Upon simplifying we have, Adding and substracting σ 2 1 2 , and let a 1 = βρ 2Λσ 2 2 2 integrating both sides and dividing by t, we obtain We compute i in terms of others by the strong law of large numbers for martingale from (3.26) and (3.27) it follows that Regarding the quadratic variations of the stochastic integral, this implies that the quadrartic variation is given as: = a 2 1 σ 2 1 i 2 − a 2 1 σ 2 1 s 2 + σ 2 2 − ( by strong law of large numbers for martingales that is to say, the disease will prevail ifR s 0 > 1, which is the required assertion. Remark 3.6. It is noted that R s 0 < R d 0 means the dieases dies out. Similarly, ifR d 0 > 1 then R d 0 > 1 means the disease persists .That is to say that, if for stochastic model the disease persists, then it also persists for a deterministic model.

Numerical Simulations
To illustrate the various theoretical results presented above, the system (2.2) is simulated for various sets of parameters. In this section, we give some numerical simulations to show the effect of noise on the dynamics of the S IADU models by using the Milstein numerical Method as mention by Higham (2001).

Disease Extinction
For the stochastic model, we have stochastic reproduction number as: That is to say that, I(t) will tend to zero exponentially with probability one as observed in figure (1). Therefore the stochastic model (2.2) has disease extinction with choise of stochastic noise intensities σ 1 = 0.9 and σ 2 = 0.8 showing that white noise is helpful for disease control.

Disease Persistence
To see the disease dynamics of model (2.2) we decrease the noise intensity say (σ 1 = 0.2,σ 2 = 0.3), θ 1 = 0.1 and Λ = 0.4 while keeping other parameters unaltered. Then from theorem (3.4) we have stochastic reproduction number as: Therefore, the condition of theorem (3.4) is not satisfied. In this case, numerical simulations suggest that I(t) is stochastically persistent and the disease will prevail as in figure (2). Dynamic behaviaral comparisons between deterministic and the stochastic S IADU model by investigating the basic reproduction number. For stochastic model system in (2.2) with parameter and the initial values as (s(0) = 0.9, i(0) = 0.7, a(0) = 0.5, d(0) = 0.15, u(0) = 0.35) ∈ 5 + we obained R s 0 = 0.1877 < 1 while for the deterministic model (2.1) by using the same parameters, we obtained the reproduction number as R 0 = 0.2462 < 1. This shows that the stochastic model (2.2) supresses the disease outbreak.

Concluding Remark
In this paper, we discussed the properties of a stochastic model for HIV and AIDS by incorporating the viral load detectability, we derived from its deterministic counterpart. We investigated analytically the existence, uniqueness, and positivity of the stochastic model, similarly, we established sufficient conditions for stochastic disease extinction and persistence for the HIV and AIDS model. We used Milstein numerical method to obtain numerical results in order to illustrate the main analytical results, and by using stochastic basic reproduction number shown in theorem (3.4), we investigated the stochastic extinction as shown in figure (1) which exhibit diseases extinction with R 0 = 0.4223 < 1, showing that the disease dies out exponentially if the condition is met. Meanwhile using theorem (3.5) we showed that figure (2) exhibits diseases persistence with R 0 = 1.2115 > 1 that the disease will prevail in the population. Moreover, figure (3) and (4) clearly shows the stochastic trajectories and time evolution for the deterministic model. Additionally, with initial values (s(0) = 0.9, i(0) = 0.7, a(0) = 0.5, d(0) = 0.15, u(0) = 0.35) we investigated on the effects of noise strength intensity level, and the results revealed that the high noise intensity level results into disease suppression whilst on other hands low level of noise strength into the model does not suppress diseases.
Finally, the study concurs to the studies done by (Zhang and Zhou, 2019;Fan et al., 2017;Zhao and Yuan, 2016) showing that stochasticity in the model suppresses the outbreak of diseases. Therefore, as the reproduction number affected by random perturbation is smaller as compared to the reproduction number for the deterministic case, then the stochasticity present in the model (2.2) suppresses disease outbreak, hence useful for disease control and eradication.