The HIV Epidemic: What kind of vaccine are we looking for?

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1 adia Abuelezam MATH 181 May 5, 2008 The HIV Eidemic: What kind of vaccine are we looking for? 1 Problem Background 33.2 million eole are living with HIV and AIDS worldwide [4]. Human Immunodeficiency Virus (HIV) is a virus that attacks the immune system and renders it weak to infection. HIV can rogress to Acuired Immune Deficiency Syndrom (AIDS) once the number of T cells in the immune system have been significantly reduced [2]. Peole living with HIV are rone to infections and illnesses including neumonia, Kaosi s sarcoma skin cancer, and yeast infections. Although treatments are available to rolong life for HIV infected eole, such anti-retroviral theraies are extremely exensive and atients can acuire drug resistance if the medication is not taken everyday [2]. HIV is transmitted through unrotected sex, needle exchange, the birthing rocess, and breast feeding. In 2007, 2.1 million eole died from HIV/AIDS related comlications, while 2.5 million eole were newly infected with HIV worldwide [4]. The extent to which the disease has sread and the devastation it has left behind emhasizes the need for useful and effective reventative measures against the sread of the disease. There are many revention strategies used to minimize the sread of HIV and AIDS but none have been successful at eradicating the disease. Since the disease is rimarily transmitted sexually, the only effective revention strategies currently in use to curb the sread of HIV are abstinence and condoms [2]. Even with these revention methods in use, HIV is still sreading raidly, esecially in develoing countries. Scientists and researchers therefore need to look towards more otent revention strategies such as vaccines. A significant ortion of HIV/AIDS research funding is being sent on vaccine research. Thirty clinical trials have been initiated on 30 vaccines since the early 1990s. one of the vaccines have graduated ast the human testing (stage III), as their efficacy has been hard to rove [2]. Additionally it is extremely difficult to roduce a vaccine that is able to adat to the varying nature of the disease itself which mutates u to one million times each day. Drug resistance is common, and it is currently believed that any vaccine that is roduced will be imerfect, and would wane in efficacy over time. In order to redict what effect a vaccine will have on the current eidemic, we roduce a mathematical model and study the dynamics of its solutions to make redictions for the number of infections and deaths that might be revented by a vaccine. The first section of this aer will describe the euations and arameters of the differential euations model I chose to study. The second section outlines the ultimate goal of the study and the study s larger imlications. This discussion is followed by a descrition of the steady state euilibrium of the model and how it was characterized comutationally. Section four outlines the results from two sensitivity analyses. The aer is concluded with a brief descrition of the conclusions and ideas for future work to extend the roject. 1.1 The Model Although there are many mathematical models in the literature that investigate the dynamics of the sread of HIV in the resence of a vaccine, I chose to focus on an ordinary differential euation (ODE) comartment model. This model, develoed by Elbasha, Gumel et al. in 2006, describes the sread of HIV through five distinct oulation grous (Figure 1). These subgrous are: unvaccinated suscetible individuals (X), vaccinated suscetible individuals (V), unvaccinated infected individuals (Y), vaccinated infected individuals (W), and individuals with AIDS (A). The model assumes that individuals are vaccinated before they enter the sexually active oulation, and that individuals cannot be infected by HIV at birth. The model takes into 1

2 Table 1: A list of variable state descritions. Variables X(t) V (t) Y (t) W (t) A(t) Descrition Unvaccinated suscetible individuals Vaccinated suscetible individuals Unvaccinated infected individuals Vaccinated infected individuals Individuals with AIDS Table 2: A list of arameter descritions and ranges. Parameters Descrition Range U.S. Estimate Λ Rate of recruitment (birth rate) (0, 1] 0.01 (UAIDS) β Transmission coefficient (0, 1] 0.04 (UAIDS) Vaccination rate (0, 1] 1 Degree of rotection (0, 1] s Relative risk of infection (0, 1] 0.5 θ Modification arameter (0, 1] µ Removal rate (death rate) (0.02, 0.07] (CIA) γ Rate of waning immunity (0, 0.5] σ Progression rate to AIDS (0, 1] 0.03 (UAIDS) o Poulation size 298,213,000 (UAIDS) account recruitment into the sexually active oulation, deaths, waning immunity rate, and the imerfection of the vaccine. The model can be reresented by the following ordinary differential euations: dx dt dv dt dy dt dw dt da dt = (1 )Λ µx λx + γv = Λ µv λv γv = λx (µ + σ)y = λv (µ + θσ)w = σy + θσw (µ + α)a, where the variable states and arameters are defined in Tables 1 and 2. Here λ is the infection rate at which suscetible individuals are infected with HIV and is reresented by the following uantity: λ = βy + sβw o. (1) General ranges for each arameter are known. We can also estimate various arameters by referring to data banks like United ations Programme on HIV/AIDS (UAIDS) and the World Health Organization 2

3 Figure 1: A ictoral deiction of the five comartment ODE model describing the sread of HIV through a oulation [3]. Individuals are exected to enter the sexually active oulation as either suscetible or vaccinated. Both of these grous can then become infected with HIV and rogress to AIDS. Parameter descritions can be found in Table 2. (WHO) (U.S. estimates are resented in Table 2). The model is a non-linear set of five ODEs with a large number of arameters which makes working with the model comutationally intensive. In the aer that examines the model described above, the authors sought to characterize the bifurcations that took lace when different incidence functions, λ, were used [6]. For the uroses of this aer, I will consider the basic model with standard incidence (Euation 1). The stability and characterization of the disease-free euilibrium state, when no one in the oulation is infected with HIV (Y = W = 0) is thoroughly described and examined by the authors. The authors also describe the basic reroductive number of the system. The basic reroductive number (R ) describes the number of secondary infections that would result from one infected erson entering the oulation. The existence of endemic euilibria, in which a roortion of the oulation is infected with HIV (Y, W 0) is roven. The endemic euilibrium deends on whether R is greater than unity. The basic reroductive number for this model is given by: β (γ(µ + θσ) + µ ((s + 1) µ + (s θ + θ) σ)) R =. (2) (γ + µ)(µ + σ)(µ + θσ). Elbasha and Gumel rove the following lemma in their analysis that gives conditions on the arameters such that an endemic euilibria exists [3]. Lemma: Given B 1 = [µ + θµ + (1 θ)σ] ( ) ( ( )) β β B 2 = (γ + µ + σ)(µ + θσ) s(µ + σ) 2 + (µ + σ)[(µ + θσ)(1 ) 1 + µ] µ + σ µ + σ B 3 = (γ + µ)(µ + σ)(µ + θσ)[1 R ], the vaccination model has (i) a uniue endemic euilibrium if B 3 < 0 R > 1; (ii) a uniue endemic euilibrium if B 2 < 0 and B 3 = 0 or B 2 2 4B 1 B 3 = 0; (iii) two endemic euilibria if B 3 > 0, B 2 < 0, and B 2 2 4B 1 B 3 > 0; 3

4 (iv) no endemic euilibrium otherwise. o analysis is done by the authors to determine the stability of the endemic euilibria or examine their roerties. 2 Project Goals and Descrition The general goal of this aer is to determine the range of vaccination arameters that will create a diseasefree euilibrium. Ultimately, by erfoming a sensitivity analysis on the endemic euilibria, we enforce bounds on the vaccination arameters,, γ and θ such that a disease-free euilibrium is guaranteed. This information can then be used to understand better the conditions under which a vaccine will have a ositive effect on the rogression of HIV and AIDS in society. The first ste of the roject is to verify the disease-free euilibrium results given by the authors. After the basic reroductive number and the Jacobian are roduced for a general euilibrium oint, I will then focus on finding the conditions under which the endemic euilibria exists, and the sensitivity of the endemic euilibrium oints on the arameter values. The sensitivity of the endemic euilibrium to arameters will be determined using a Latin Hyercube Samling method described in Section 4. Because of the comutationally intensive nature of the roblem, I will be erforming all calculations and simulations using Wolfram s Mathematica. 3 Steady State Euilibria The majority of the initial work on this roject was sent verifying the results obtained in the aer regarding the disease-free euilibrium state: ( ) [γ + (1 )µ]λ Λ (X DFE, V DFE, Y DFE, W DFE ) =, µ(µ + γ) µ + γ, 0, 0, (3) as well as the Jacobian matrix for the disease-free euilibrium. This Jacobian determines the dominant eigenvalue, which verifies R (Euation 2). The general endemic euilibrium oints (for R 1) are defined to be X = (γ + (1 )(λ + µ))λ (λ + µ)(γ + λ + µ) V Λ = γ + λ + µ Y = (γ + (1 )(λ + µ))λ ΛD u (λ + µ)(γ + λ + µ) W λ ΛD v = γ + λ + µ, with λ = ( B2+ B2 2 4B1B3) 2B 1, D v = 1 µ+θσ and D u = 1 µ+σ. We see that these endemic euilibria deend only on arameter values and can therefore be easily solved once arameter values are chosen. 4 Sensitivity Analysis In order to determine the sensitivity of the endemic euilibria to changes in arameter values, a sensitivity analysis was erformed on the model. Sensitivity analyses are usually done on one of three levels [5]. The first tye is called screening and is usually erformed by testing all ossible combinations of arameters in the 4

5 sace and observing the effect on the model. This rocess is not only time-consuming but comutationally taxing, since our model has ten arameters we would like to vary. Local sensitivity analysis is also used to determine the sensitivity of the model on a local level and is usually done with a method called differential analysis. The last method is global sensitivity analysis, which describes the model s sensitivity to arameter values throughout arameter and variable sace. The most common tye of global sensitivity analysis is the Monte Carlo method. The method I chose to use in my analysis is described by Blower and Dowlatabadi [1]. This method, which is called Latin Hyercube Samling (LHS), samles the arameter sace efficiently while simulataneously allowing us to observe changes in the endemic euilibia. 4.1 Latin Hyercube Samling LHS is a tye of stratified Monte Carlo method and allows the variation of all arameters simultaneously. It has been roven to be more efficient than random samling since it estimates the mean value of the function in uestion more accurately [1]. The first ste in LHS is to determine the robability distribution functions that describe the state arameters in the system. If we have an estimate for a secific arameter, we can roduce a normal distribution about that estimate, or if we have no information about a secific arameter, we can use a uniform distribution. The next ste outlined by Blower and Dowlatabadi is to calculate the number of simulations or calculations that need to be erfomed in order to samle the entire arameter sace without relacement. We are then left to divide each of the arameter ranges into eui-robable intervals, samle from these intervals without relacement and erform the simulations or calculations. Once this is done, the resulting data can be lotted and analyzed using uncertainty analysis [1]. This rocess can also be visualized in Figure 2. We can use LHS to better understand what kind of vaccine arameters we would need in order to see a disease-free euilibrium in our system. These arameter values can then be used as guidelines for vaccine develoment in the context of this model Distributions Chosen Although estimates were found for a majority of the arameters examined, I chose to define uniform distributions for all arameters when erforming LHS. Samling from uniform distributions across the entire range of ossible values will give a better sense of the model s dynamics in arameter sace. Uniform distributions also facilitate the division of distributions into eui-robably intervals. Given more time I would like to see how the results of LHS differ when secific distributions are defined for arameters with known means and standard deviations. Uniform distributions were defined for all arameters within the ranges described in Table Difficulties Imlementing the LHS algorithm for general models roved to be relatively straightforward but I exerienced difficulty erforming the algorithm on the secific model described in Section The code, which was develoed for a general model, runs through the LHS algorithm and roduces lots to show how the infected roortion of the oulation varies with various arameter values. The lots (Figure 3) show no atterns or trends between the arameter values and the infected oulation roortions. Even when interactions between arameter values are taken into account, no clear trends were observed. The analysis of the LHS is therefore difficult to do. Blower and Dowlatabadi recommend using a artial rank correlation analysis to understand the results. Due to time constraints I was not able to erform this uncertainty analysis. Because this method did not elicit clear results, I chose to erform a simler sensitivity analysis. 4.2 Screening In order to get a clear icture about how vaccine arameters were affecting the endemic euilibria I chose to erform a screening analysis. This analysis was erformed by varying one arameter and keeing all other arameters constant. This rocedure was erformed while varying,, γ, and θ individually. Four values 5

6 Figure 2: A ictoral examle of the Latin Hyercube samling method [1]. A distribution for each arameter is generated. For arameter A, the mean and standard deviation are known, so a normal distribution is generated. A range of values is known for arameter B, so a uniform distribution is chosen. After slitting each distribution into intervals, an interval is chosen at random and a random value from this interval is also chosen. The randomly chosen values for both arameters are used to calculate a solution and the rocess is reeated until all intervals have been samled without relacement. W Y W Y a b Figure 3: Infected oulation roortions for varying values in LHS. (a) Dislays the 500 interval results for the LHS imlementation. As can be seen, there seems to be no relationshi between infected euilibrium states and the roortion of the oulation that is vaccinated. (b) Simulation oints with a value of less than 0.5 are shown in red, while those with values greater than or eual to 0.5 are shown in blue. As can be seen, there is no correlation between the arameter values and the roortion of the oulation that is infected with HIV at any one time. 6

7 Poulation Size 100 Poulation Size Time years a b Time years Figure 4: Both lots above show the dynamics of a 100 erson oulation over 400 years with the following arameter values: Λ = 1, µ = 0.02, = 0.5, = 0.95, θ = 0.4, s = 0.45, σ = 0.04,β = 0.45, and γ = 0.1. The blue curve corresonds to X(t), urle to V(t), yellow to Y(t), and green to W(t). (a) Initial conditions are used that reresent the HIV revalence in the current U.S. oulation: X(0) = 99.4, V (0) = 0, Y (0) = 0.6, and W (0) = 0. The R value for these arameter choices and initial conditions is (b) Initial conditions are used that have even roortions of eole in each subgrou: X(0) = 25, V (0) = 25, Y (0) = 25, and W (0) = 25. The R value for these arameter choices and initial conditions is Since more interesting results can be seen in endemic euilibria states for the evenly slit oulation, the rest of the analyses will focus on these initial conditions. (one low, two intermediate, one high) were chosen for each arameter. The dynamics of the oulation with the chosen arameters were then observed. After running a few simulations (Figure 4) I chose to focus on a oulation with initial conditions defined by X(0) = 25, V (0) = 25, Y (0) = 25, and W (0) = 25. Initial conditions that resembled current U.S. oulation roortions roduced uninteresting endemic euilibria and will therefore not be analyzed Results The lots in Figure 5 show how the roortion of the oulation of interest changes with varying arameter values. It should be ket in mind that this analysis is secific to the initial conditions chosen, and other initial conditions may roduce different sensitivities. The vaccinated suscetible oulation, which would ideally be as high as ossible, is most sensitive to changes in and γ and least sensitive to changes in and θ. We can maximize the vaccinated oulation by having low γ and and high θ and. This correonds to having low waning immunity, high effectiveness, little modification in behavior and high vaccination rates. The general non-vaccinated infected oulation is again most sensitive to and γ but we see that trends tend to be reversed. High vaccination rates result in small suscetible oulations while high γ values result in high suscetible numbers. These trends make sense, since as waning immunity increases more vaccinated individuals enter the suscetible oulation. Also as vaccination rates increase more individuals are exected to enter the oulation as vaccinated and not suscetible individuals. The vaccinated infected oulation has a lower revalence rate in the oulation then the non-vaccinated infected oulation. We see that all arameters seem to have an effect on the revalence of this grou in the oulation. We observe that increasing and results in an increase in the oulation roortion of vaccinated suscetible individuals. Increasing γ and θ results in the decrease in the number of infected vaccinated individuals. Although none of these results are articularly surrising, it is interesting to note that and θ have a larger effect on the vaccinated infected oulation then on the non-vaccinated infected oulation. When we look at the total infected oulation we see that all arameters affect the endemic euilibria. We notice that to get a small endemic euilibria we want small γ and and large and θ. We notice also that the sensitivity to θ and γ seems to not be linear but looks exonential for both decay and increase. 7

8 V 0.35 a Y b. W c. W Y d Figure 5: These lots describe how changing vaccine arameter values affects various oulations of interest. (a) Vaccinated suscetible individuals are affected most by and γ. (b) on-vaccinated infected individuals are most sensitive to and γ. (c) Vaccinated infected individuals are affected by changes in all arameters. (d) The total infected oulation roortion seems to be affected by all arameter choices. 8

9 5 Conclusions The sensitivity analysis I have erformed verified intuitive arameter choices that would roduce low endemic euilibria values. We see that our screening sensitivity analysis allows us to conclude that in order to minimize the infected oulation we would want a vaccine to be develoed with a low waning immunity rate (γ 0.5) and a high effectiveness ( 0.5). We would also like to see a large roortion of the oulation ( 0.5) vaccinated and no large difference in risky behavior between vaccinated and non-vaccinated individuals (θ 0.5). The screening sensitivity analysis therefore allowed us to make conclusions about the neccessary magnitudes of the vaccine arameters that roduce a low endemic euilibrium. It should be emhasized that these results are only valid for the chosen initial conditions. Throughout this rocess, I learned that it is often beneficial to begin with the simlest rocedure ossible and move on to more comlex analyses once the dynamics of the model are better understood. Beginning with a general screening before imlementing LHS may have narrowed down my focus to a secific subset of the arameter sace. Although the general algorithm develoed and imlemented for LHS works on general models, including the model considered in this aer, the analysis of the results reuires more comlex and advanced methods than those I ossess. 5.1 Future Work Future attemts to finish this roject should aim to better understand the results from LHS in the context of vaccine develoment. Efforts should be made to analyze the results using artial rank correlation analysis. The broad ranges found from the screening method could then be used as starting distributions for the LHS method, and this may narrow the scoe of the LHS analysis. Work should also be done to better understand the dynamics of the model beginning with initial conditions that reresent the United States oulation. Ultimately I would like to better understand how the introduction of a vaccine in the United States would affect the eidemic and the future of HIV in this country. 6 Acknowledgements I would like to thank Prof. Levy and Prof. Yong for their oeness in making this a joint roject for their two classes. I would also like to thank Prof. Yong for his seemingly endless technical suort on the roject and the coding. I also areciate Prof. Levy s enthusiasm about the roject and for contacting Dr. Karen Yokly for information on the statistical analysis. I would like to thank both Dr. Gumel and Dr. Elbasha for their resonses to my s and their hel in understanding the model. Finally I d like to thank Prof. Jacobsen and Prof. de Pillis for assisting me in finding and understanding the model. References [1] S. M. Blower and H. Dowlatabadi. Sensitivity and uncertainty analysis of comlex models of disease transmission: An model, as an examle. International Statistical Review/Revue Internationale de Statistiue, 62(2): , The article outlines the rocess of Latin Hyercube Samling in an HIV model with 25 arameters. It exlains the basic methodology and also exlains why this method is relevant to comutationally intensive roblems, like the one I am working on. [2] Dale Hanson Bourke. The Sketic s Guide to the Global AIDS Crisis. Authentic Publishing, Colorado Srings, This book rovides a brief and uick udate on the AIDS crisis around the globe. It answers main uestions on transmission and revention and as such was used as background information for the aer. 9

10 [3] E. H. Elbasha and A. B. Gumel. Theoretical assessment of ublic health imact of imerfect rohylactic hiv-1 vaccines with theraeutic benefits. Bulletin of mathematical biology, 68(3): , Ar LR: ; PUBM: Print-Electronic; DEP: ; JID: ; 0 (AIDS Vaccines); 2005/01/04 [received]; 2005/04/20 [acceted]; 2006/04/07 [aheadofrint]; ublish. This aer rovides the theoretical framework for the HIV model I will be considering for my aer. The secific stes taken to classify the disease free euilibrium state are given, which heled me in my roduction of the Jacobian for the generalized steady state values. This aer also rovides detailed descritions of the arameters, ranges for these arameters, and also the descritions of state variables. [4] UAIDS: Joint United ations Program on HIV/AIDS. Aids eidemic udate. Technical reort, WHO: World Health Organization, This technical reort gives an udate on the HIV eidemic, and rovides statistics and facts for various regions of the world. This reort also rovides background on major risk factors and revention strategies used across the world. Each country is given an in deth rofile and from these rofiles, the U.S. arameter estimates were made. [5] A. Saltelli, K. Chan, and E. M. Scott, editors. Sensitivity Analysis. Wiley Series in Probability and Statistics. John Wiley Sons. Ltd, West Sussex, This book rovides a great reference on the rocess of running a sensitivity analysis. Besides roviding a background on the different rocesses, it also contains historical information on all the methods. It gives secific examles from industry as to how the sensitivity analyses are used in real life. [6] O. Sharomi, C.. Podder, A. B. Gumel, E. H. Elbasha, and J. Watmough. Role of incidence function in vaccine-induced backward bifurcation in some HIV models. Mathematical Biosciences, 210(2): , Dec PUBM: Print-Electronic; DEP: ; JID: ; 0 (AIDS Vaccines); 2007/02/09 [received]; 2007/05/20 [revised]; 2007/05/22 [acceted]; 2007/07/04 [aheadofrint]; ublish. This aer exlores the HIV model in uestion in great deth and exands uon the model by exloring various incidence functions and their effect on the model dynamics. This aer also rovides a good summary of the differential euations and the dynamics inherent in the system. 10