Modeling Saturated Diagnosis and Vaccination in Reducing HIV / AIDS Infection

and Applied Analysis 3 ?̇? 3 (t) = f 3 = λ (x 1 + (1 − ε) x 2 ) − (σ 1 + q 1 + hx 3 + μ)x 3 , ?̇? 4 (t) = f 4 = qx 3 1 + hx 3 − (σ 2 + μ) x 4 , ?̇? 5 (t) = f 5 = σ 1 x 3 + σ 2 x 4 − (ψ + μ) x 5 , (6) where λ = β((x 3 + η 1 x 4 + η 2 x 5 )/N). If β is taken as the bifurcation parameter and we consider the case R 0 = 1, solving for β gives β = β; that is, β ∗ m(1 + η 1 q σ 2 + μ + η 2 μ + ψ ( σ 2 q σ 2 + μ + σ 1 )) = σ 1 + q + μ. (7) First of all, observe that the eigenvalues of the Jacobianmatrix J(E 0 ) at β = β∗ [13], that is, J(E 0 )| β=β ∗ , J (E 0 ) 󵄨󵄨󵄨󵄨β=β∗ = ( ( (


Introduction
Acquired immunodeficiency syndrome (AIDS) is spreading rapidly in the world ever since it was firstly detected in 1981 and continues to threaten the health of human seriously, especially among sex workers and injecting drug users.Furthermore, AIDS also influences the economy of many countries which has attracted great attention of governments.For such a severe scenario, the governments have taken intervention measures to reduce HIV transmission.
Mathematical models play a vital role in gaining a quantitative insight into HIV transmission dynamics and suggesting the effective control strategies.In order to study the effect of various intervention strategies on HIV transmission, extensive mathematical models have been formulated.Traditionally, models of HIV/AIDS dynamics often incorporate staged progression (see, e.g., [1][2][3]), but these did not include any control measures.Hyman et al. [4] extended these models to consider screening and contact tracing and discussed which strategy would slow infectiousness.Compartmental models with staged progression that incorporate the imperfect vaccine were constructed in [5] to predict HIV epidemic, but they did not consider diagnosis.Elbasha and Gumel [6] considered that a proportion of new recruits are vaccinated and upon becoming infected with HIV, susceptible and vaccinated individuals enter the classes of infected and vaccine infected people, separately.They showed the existence of backward bifurcation via numerical simulations.Sharomi et al. [7] explored the role of the choice of incidence function in HIV models formulated in [6] and obtained that the phenomenon of backward bifurcation can be removed by substituting the standard incidence function with a mass action incidence.In South Africa, testing and screening campaign was launched for HIV; Nyabadza and Mukandavire [8] analyzed their effects by developing HIV models.More recently, a model of HIV/AIDS with diagnosis was presented in [9].The authors estimated parameter values and predicted its transmission in China in the next few years.
The majority of mathematical models consider only one control strategy, vaccination or diagnosis, for instance [5,9]; however, curbing HIV/AIDS infection needs comprehensive strategies, since, under the serious threat of HIV, it may be more rational to adopt various measures for different high risk groups.These motivate us to consider two combined intervention measures, vaccination and diagnosis.Furthermore, due to the limited resources, we then choose a nonlinear function which can be used to describe saturation effect.We use a parameter ℎ, representing the half saturation 2 Abstract and Applied Analysis constant, in the diagnosis function to measure the effect of HIV individuals being late for diagnosis [10].When the number of infected individuals  is low, the number of actual per capita diagnosed individuals is proportional to , whereas when the number of infected individuals  is sufficiently large, there is a saturation effect which makes the number of diagnosed individuals approach to be constant due to the limitation of human and economic power.The number of new diagnosed cases per unit time is saturated with the total infected population.
The paper is organized as follows.The model is formulated in Section 2. The existence of backward bifurcation and the stability of the disease-free equilibrium are discussed in Section 3. In Section 4, uniform persistence of the model is investigated.Numerical simulation results are concluded in Section 5.In Section 6, we give a brief summary.
The equations of the model are where the incidence rate () = ((() +  Since the model monitors change in the human population, the variables and parameters are assumed to be nonnegative for all  ≥ 0. The system will be analyzed in a suitable feasible region Ω ⊆  5  + , where Ω = {(, , , , ) ∈  5  + |  +  +  +  +  ≤ Π/}.We can easily prove that the solutions of system (1) with nonnegative initial conditions remain nonnegative, and the feasible region Ω is positively invariant and attracting with respect to system (1) for all  > 0. (2)
Denote by  0 the basic reproduction number as that is, the spectral radius of the next generation matrix  −1 .Biologically speaking,  0 is the average number of new secondary infections generated by a single HIV infected individual, introduced into a susceptible population in which some individuals have been vaccinated.
If  is taken as the bifurcation parameter and we consider the case  0 = 1, solving for  gives  =  * ; that is, First of all, observe that the eigenvalues of the Jacobian matrix ( 0 ) at  =  * [13], that is, are given by The other two eigenvalues satisfy the following equation: where Substituting ( 7) into  and , we find Clearly,  and  are positive.Equation (10) has two roots with negative real parts.Hence,  3 = 0 is a simple zero eigenvalue and all other eigenvalues have negative real parts.The assumptions in [12] are satisfied.Therefore, the center manifold theory can be used to analyze the dynamics of system (1) near  =  * (or, equivalently,  0 = 1).The Jacobian matrix of system (1) at  =  * has a right eigenvector , given by  = ( 1 ,  2 ,  3 ,  4 ,  5 )  .And it can be computed from the system (( 0 )| = * ) ⋅  = 0; that is, from ( 13), we derive the following solutions: The left eigenvector of ).And it can be computed from the system with the following solutions: The local bifurcation analysis near  =  * ( 0 = 1) is then determined by the signs of two associated constants, denoted by  and , respectively, as The computations of  and  are done as follows: for system (6) the associated nonzero partial derivatives of  at the disease-free equilibrium are Substituting ( 18) into (17), we get From [14,15], we know that if  > 0,  > 0, there exists a backward bifurcation.Since the bifurcation coefficient, , is always positive, then we establish the following result.Theorem 1.If  > 0, system (1) exhibits a backward bifurcation when  0 = 1.
Due to existence of backward bifurcation we know that, for positive , there exists another critical value   , which is less than unity, for model (1).Moreover, there is no endemic equilibrium for  0 <   ; there are two distinct endemic equilibria for   <  0 < 1, and a unique endemic equilibrium exists for  0 =   < 1 or  0 > 1. Numerical studies will confirm this in the end of this subsection.
We now analyze the endemic equilibrium of model (1).The equilibrium of model ( 1) can be obtained as follows: where Substituting (20) into the third equation of system (1), it is easy to derive the following equation: where Clearly,  * = 0 is a fixed point, which corresponds to the disease-free equilibrium  0 .For  = 0, we can obtain Define From model (1), it can be shown that if  * is a positive solution of () = 1, then  * ,  * ,  * , and  * are positive.Thus, the equilibrium is biologically relevant.Unfortunately, it is hard to solve the equation () = 1 analytically; in the following we numerically show that this equation can have two positive roots, which confirms the existence of backward bifurcation.
In Figure 1 below which the disease-free equilibrium is unique equilibrium.

Stability Analysis of Equilibria.
First, we have the following result on the local stability of  0 .
Theorem 2. The disease-free equilibrium  0 of system (1) is locally asymptotically stable if  0 < 1 and unstable otherwise.
Proof.By checking the Jacobian matrix of system (1) evaluated at  0 , we know that the characteristic equation for ( 0 ) has two eigenvalues as and the others satisfy the following equation: If the real parts of the roots of the equation ℎ() = 1 are nonnegative, that is, R() ≥ 0, then [17] |ℎ ()| ≤ ℎ (0) =  0 . (30) showing that there are no solutions to ℎ() = 1 with positive real part.Hence,  0 is locally asymptotically stable if  0 < 1.This proof is completed.

Numerical Simulations
which implies that an increase of vaccination rate and vaccine efficacy leads to the basic reproduction number decline, as shown in Figure 2(a), in which the contour plots of  0 versus vaccine efficacy  and vaccination rate  were plotted.It also shows that the basic reproduction number is more sensitive to vaccine efficacy than vaccination rate.Figure 2(b) shows the contour plot of  0 with diagnosis rate and vaccination rate, which implies a decrease in  0 with increasing diagnosis rate  and vaccination rate .Furthermore, when 50% of HIV individuals are diagnosed, vaccination level of at least 60% would be needed to achieve  0 < 1.This suggests that the strategies of diagnosis and vaccination should be stringent enough to reduce  0 .Next, we consider the effect of different transmission rate, vaccination rate, vaccine efficacy, and recruitment rate on transmission of HIV/AIDS.We take the year 2004 as starting time; since then the policy of diagnosis is consistent.In [21], we get that the number of diagnosed HIV-positive individuals and AIDS patients in Yunnan province was 27168 and 1223 in year 2004, respectively.Besides, 22.6% of these HIV individuals were transmitted by share injectors [22].Hence, (0) = 27168×22.6%= 6140, (0) = 1223×22.6%= 276.Note that the diagnosis rate is estimated to be 0.304 [9]; then we have (0) = (0)/0.304= 20197.We have no reliable data on the number of susceptible individuals, that is, number of IDUs in Yunnan province.However, we know  that 3.2 million blood samples were tested in Yunnan in [23].
We then assume that in these blood samples the fraction of share injectors is the same as fraction of transmission via share injectors (i.e., 22.6%).Then the number of susceptible individuals who share injectors is (0) + (0) = 3.2 × 10 6 × 22.6% = 723200.If the vaccination rate is assumed to be 0.
Figure 3 shows the variation in the number of HIV infected individuals with different transmission rates, vaccination rates, vaccine efficacy, and recruitment rates.It follows from Figure 3(a) that decreasing transmission rate could lead to the number of HIV-positive individuals decline.The effect of increasing vaccination rate on HIV transmission is shown in Figure 3(b) and it is seen that the number of HIVpositive individuals becomes much smaller if vaccination rate increases more.Figure 3(c) illustrates that, with increasing vaccine efficacy, the number of HIV-positive individuals decreases.Figure 3(d) shows that if the inflow of susceptible individuals into the community is restricted due to education, the disease spread will slow down.

Sensitivity Analysis.
In this section, we use sensitivity analysis method [24] to investigate the impact of various intervention measures on HIV transmission in Yunnan province, China.We hope that these results obtained here could improve the knowledge of the effects of different interventions.
Figures 4(a) and 4(b) show the comparison of sensitivity coefficients of new cases and prevalence against parameters , , , , and Π, separately.Note that the sensitivity coefficient of new cases and prevalence can be interpreted as the percentage change in the number of new cases and prevalence for 1% decline in the parameters  and Π or 1% increase in , , and , respectively [25].In particular, let function  be new cases or prevalence; the sensitivity coefficients (SC) of new cases and prevalence are given by SC =  (perturbed variables)− (original variable)  (original variable) ×100%.
It follows from Figure 4 that a decrease in transmission coefficient  causes new cases and prevalence decline substantially.Besides, an increase in vaccine efficacy  and vaccination rate  can lead to a decrease in new cases and prevalence, whereas the change of both diagnosis rate  and recruitment rate Π slightly affects the new cases or prevalence.Thus, new cases and prevalence are sensitive to transmission coefficient, vaccine efficacy, and vaccination rate.Then, reducing the transmission coefficient and increasing the vaccine efficacy and vaccination rate can greatly reduce new cases and prevalence.

Conclusion
In this paper, we established an epidemic model to investigate effects of saturated diagnosis and vaccination on HIV transmission.It proved that backward bifurcation occurs by employing center manifold theory, which causes the diseasefree equilibrium to be locally asymptotically stable instead of globally asymptotically stable for  0 < 1.Thus, making the basic reproduction number less than unity is not enough to eliminate the HIV infection.We note that  0 < 1 is equivalent to

× ( (𝜎
which means that only the vaccination rate is greater than   ; HIV infection might be eliminated, depending on initial data.There exists the critical threshold   , which cannot be explicitly expressed due to nonlinearity, such that when  0 < min{  , ( 1 + )/( 1 +  + )} < 1, the diseasefree equilibrium is globally asymptotically stable.However, if  0 > 1, the disease uniformly persists.
It is interesting to note that if the diagnosis is described linearly backward bifurcation does not happen.This implies that nonlinear diagnosis due to limited medical resources leads to backward bifurcation, and consequently complete elimination of HIV infection becomes difficult.That is, HIV infection might be extinct only by improving integrated interventions, which ensures that  0 is less than   and ( 1 + )/( 1 +  + ).  1.
Since several candidate HIV vaccines are in development, it is useful to study the effectiveness.Moreover, the detection of HIV-positive individuals is limited due to medical resources.We then applied the proposed model with nonlinear diagnosis and vaccination to examine HIV infection among IDUs in Yunnan province, China.Sensitivity analysis shows that new cases and prevalence are sensitive to transmission rate, vaccine efficacy, and vaccination rate, whereas diagnosis rate and recruitment rate slightly affect both of them.Therefore, enlarging vaccination rate, improving vaccine efficacy, and lowering transmission rate by reducing sterile injecting equipment are beneficial to reduce transmission of HIV infection.In order to efficiently reduce HIV transmission, combined intervention strategies are suggested to be implemented simultaneously.
Effective antiretroviral therapy (ART) is an important strategy to slow down the progression to AIDS due to great reduction in viral loads and is not included in our model.Note that when HIV infected individuals are diagnosed and CD4 T cell counts decrease to 350 copies/L, they will accept treatment.We will include treatment strategy to construct HIV/AIDS models to investigate the transmission of HIV/AIDS in the future work and provide policy makers with effective suggestions.
(a), () is plotted versus  for different values of  and all other parameters are fixed.Figure1(a)shows that an increase in  would lead to curve () becoming tangent to line 1 and defining a critical value ( * )| =  = 1 and   ( * )| =  = 0 hold true.Figure1(b)shows the occurrence of the backward bifurcation as parameter  varies.We write  0 () as the threshold value to indicate  as the bifurcation parameter while all other parameters are fixed.Define[16]

Figure 2 :
Figure 2: Contour plots of  0 versus (a) vaccine efficacy  and vaccination rate  and (b) diagnosis rate  and vaccination rate .

Figure 3 :
Figure 3: Variation in the number of HIV-positive individuals with different (a) transmission coefficient, (b) vaccination rate, (c) vaccine efficacy, and (d) recruitment rate.

Figure 4 :
Figure 4: Sensitivity coefficients of new cases (a) and prevalence (b) on , , , , and Π over time .All other parameters are shown in Table1.

Table 1 :
Parameter description and values.