AIDS Epidemiology. Min Shim Math 164: Scientific Computing. April 30, 2004

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1 AIDS Epideiology Min Shi Math 64: Scientiic Coputing April 30, 004 Abstract Thopson s AIDS epideic odel, which is orulated in his article AIDS: The Misanageent o an Epideic, published in Great Britain, 989, does not relect eale inective population that has been growing. While the ajority o AIDS patients were assued to be ale hoosexuals, Center or Disease Control and Prevention reports that the percentage o inected eales through heterosexual contacts has been increasing. The odel is iproved by adding new ters or eale population to Thopson s existing odel. With this iproved odel, public education or susceptible and inective populations is suggested as an approach to stop the AIDS epideic.

2 Introduction AIDS started getting attentions recently and it quickly has becoe a death sentence and a ear to a lot o people ainly because there is no cure. Since there is no cure available or AIDS, it is essential to prevent AIDS ro spreading. Many studies have been done to odel the progression o AIDS and how dierent possibilities and situations aect the spread. For exaple, L. Billard and Zhen Zhao discuss AIDS odels that can be described as ulti-stage processes such as three-stage or our-stage odels. 9 Dierent approaches and ethods are used to odel the AIDS epideic ore accurately. Why is it iportant to odel the AIDS epideic? Thopson states that, It is this very low probability o transission o AIDS which distinguishes its epideiological properties ro custoary venereal diseases and gives the key to the ragility o the current AIDS epideic. 3 The increasing nuber o AIDS incidents, regardless o its low probability o transission, indicates that its progression highly depends on people s behavior and is sensitive to other paraeters such as the nuber o sexual contacts or raction o the highly active population. In other words, by changing those paraeters, the AIDS epideic can be stopped. However, when policies are ade to prevent the AIDS epideic, it is crucial to know how each policy will aect the progression o AIDS since it is possible that such policy can backire or have unexpected results that could be oreseen by the odel. By having a atheatical odel o AIDS it is possible to predict and help to ake decisions on what can be done to keep AIDS ro spreading urther. In 997 there were 64,086 cuulative AIDS cases, and in 000 there were 774,467 cases in the United States. Although the nuber o new AIDS cases reported to the CDC declined percent ro 996 to 997, there has been a continuous increase in the nuber o people with AIDS. The late 990 s and early 000 s were ties when AIDS was spotlighted and prevention capaigns becae ore prevalent and public. While these capaigns enlightened the public and ight have changed people s behaviors, at the sae tie, it resulted in placing disrepute on AIDS patients. The stiga shared against AIDS victis leads to shae, depression, withdrawal, and even isolation. The ear o discriination prevents people ro seeking treatent and even sharing their HIV status openly. It is possible to say that new AIDS patients avoid reporting their status because o this negative view and attitude o society toward AIDS. As a result, the nuber o new AIDS cases reported decreased while cuulative cases increased. Thopson attepts to odel the AIDS epideic in AIDS: The Misanageent o an Epideic. His odel, which will be discussed in ore detail in later sections, akes ew assuptions including the exclusion o heterosexual and bisexual populations ro the AIDS inective population. 3 However, as o Deceber 3, 000, percent o inections occurred aong eales and 75 percent o these woen were inected through heterosexual sex. The percentage o inection aong eales is increasing every year. 4 The increasing nubers o newly inected eales through heterosexual contacts indicates that Thopson s odel can be revised to odel the current AIDS epideic ore precisely by including heterosexual populations. Questions The goal o the odel is to answer two kinds o questions: Qualitative: Will the addition o eale population in Thopson s odel actually iprove his odel? Also, what can be done to stop the AIDS epideic?

3 Quantitative: How the values o paraeters aect the progression o AIDS? What values o paraeters, i those paraeters are controllable in reality, will stop the AIDS epideic? Assuptions Because o the coplex and unpredictable eects o AIDS, the ollowing sipliying assuptions are ade: Unior ixing. There are no spatial issues. In other words, population ro urban area and those ro suburban area will be treated equally. Interactions are through sexual contacts and only en will transit ale to ale or to eale. In other words, inective eales will not transit. O inected en, 4 percent was inected through ale-to-ale sexual contacts and 75 percent o woen were inected through ale-to-eale sexual contacts. 4 The reainder o both inected en and woen were inected through ostly drug injections. Fro these statistics, it is valid to disregard the possibility o inection through eale-to-eale or eale-to-ale sexual contacts. Male to eale ration in the US population is :. Fraction o eale inective is 8% o inective population initially. Statistics ro 997 are used as initial conditions and the survey says that 8 percent o inective populations were eales. 5 Total population stays constant. In other words, the nuber o iigrating population is kept the sae as the nuber o eigrating population. With this assuption, one o paraeter value can be estiated. Applying these assuptions, the dynaic approach was taken to iprove Thoson s odel. Forulate the Model Thopson s odel is originally the ollowing: dw d dw kα = kα = d = βw µ γ d = βw µ γ d kα = kα = β µ W p λ µ β µ W pλ µ

4 The represents the susceptible population, the inective population and W the interediate population that correspond to the population who have been inected by AIDS but is not yet able to transit the disease to others. The subscripts and o each population represent sexually inactive and active population respectively. κ represents the nuber o contacts per onth, α the probability o transission by a given contact, λ the iigration rate into the practicing gay counity, µ the rate o eigration ro that counity, γ the rate at which an inective leaves the counity as a result o having the disease, β the rate o transoration ro the W state to state, and p the proportion o the total population that exhibits the greater sexual activity 4. As entioned earlier, Thopson s odel is built under the assuption that inections priarily occur through ale-to-ale sexual contacts in the hoosexual ale population. In a new odel, eales and ales are analyzed separately and added to estiate the total populations. While the ale population can be odeled by Thopson s odel, soe changes were ade to odel the eale population to relect ale-to-eale transission o AIDS. The dierential equations that odel the inective eale population is: p k d p k d W d W d W k dw W k dw µ λ α µ λ α γ µ β γ µ β µ β α µ β α = = = = = = The subscript indicates the eale population while indicates the ale population. The dierent kα value will be used in eale odel while the sae values o other paraeters are used or both ale and eale populations. The kα values o eale and ale populations are not necessarily correlated, so they are treated as two dierent paraeters. Choosing Paraeters The incubation period o AIDS varies greatly aong individuals as well the inectious period. As a result, it is very hard to choose the values o paraeters ro statistics because it varies greatly aong dierent sources. Most paraeters are adopted ro previously researched articles, ostly ro Thopson s, while others are estiated intuitively. One o the reasons why it is so hard to odel AIDS progression realistically is that the HIV virus utates and stays dorant or years, aking the behavior o the virus diicult to predict. Following paraeters are used to estiate and validate the odel by coparison with surveyed values. 3

5 Table. Values o Paraeters used in siulation kα kα β µ λ γ p µ The values o kα, µ, λ, γ and p are chosen ro Thopson s article 4. λ is chosen by assuing that the total nuber o population stays the constant: nubers o iigrating population is equal to nubers o eigrating population. On account o the diiculty o easuring or inding literature data on β and, they were chosen intuitively. It was assued that tie lag ro W to state was years and was extracted ro one o Thopson s trials discussed in the article 4. κα was chosen ro estiates ade in Although this was not the ost recent data, it turned out that this value works or the odel. Nuerical Methods Twelve dierential equations o eale and ale populations, as shown in the previous sections, were solved by using ode45 unction o Matlab. How ode45 was used can be ound in Appendix A. There was no diiculty in applying nuerical ethods since ode unctions in Matlab are airly siple to use. Ode45 uses 4 th or 5 th order Runge-Kutta ethod appropriately. The error at the worst case o the nuerical analysis is 4 th order. By varying step-sizes, the odel can output ore accurate results. Validation o the Model The sae values o paraeters are used or both o Thopson s odel and the new odel and the result is as ollows: Table. Coparison o Estiated Population with AIDS Estiated Population by iproved odel Estiated Population by Thopson s odel Surveyed Population , ,470,343, ,467 00,77,600 7,40, ,575 While the projection in 3 years sees very close to the surveyed value, the projection in 5 years shows a great discrepancy ro the surveyed value. People s behaviors change all the tie. Thereore, paraeters used in the odel are ost likely dynaic while they were assued to be static to sipliy the odel. As the period o the projection becoes longer, there will be greater discrepancies between the estiated and surveyed population since the eect o dynaic paraeters will be ore signiicant. Thopson s odel overestiates the AIDS population alost by a actor o 3 or the projection in 3 years copared to the surveyed value. The result is explained by that Thopson s odel treats a growing percentage o eale inective population with lower probability o transission as a part o ale inective population with higher probability. Consequently, estiated AIDS population is greater than what it is. This result displays the iportance o including heterosexual population in the odel and the iproveent ade by the new odel. 4

6 Results The raction o inective is investigated to see how each paraeter aects the AIDS progression by varying one paraeter at a tie. Figure. Fraction o Inective with varying kα Figure. Fraction o Inective with varying values β values Fro Figure, it is noticeable that increasing kα values results in a rapid increase o raction o inective population. Higher nubers o sexual contacts or a higher probability o transitting AIDS per contact will aect the raction o AIDS inective to be higher. To reduce the nuber o sexual contacts, which is the κ value, the public should be discouraged ro having highly sexually active liestyles. To lower the probability o transitting AIDS per contact, α, educating AIDS inectives to not have sexual intercourse or the public to have sae sex with protection is a possible approach. Catania states that observed increases over tie in condo use across all heterosexual at-risk population segents are consistent with the observed declines trends in HIV and syphilis in the 990s. 7 While condos do not provide 00% protection against AIDS, it still can reduce the probability o transission and should be recoended. The bathhouse phenoenon was highly connected to the AIDS epideic. Bathhouses provided places or ale hoosexuals to eet new partners and to have sexual contacts. Most ale hoosexuals who cae to bathhouses are highly sexually active and it was suggested that closing down bathhouses would reduce the nuber o sexual contacts o this population. However, CDC news reported on October 0, 003 that new epideiological data, cobined with anecdotal evidence and research, show that en who have sex with en MSM are increasingly using Internet chat roos to schedule "real tie" sexual encounters. The Internet's role in this upturn is also apparent ro the data. In the irst hal o 00, percent o the MSM with syphilis reported eeting their partners online; by the irst hal o 003, this percentage increased to 40 percent. While closing bathhouses seeed like a solution to slow down the AIDS epideic, the replaceent o bathhouses with Internet chat roos shows the necessity o solutions that can aect the ways people think and behave. 5

7 As population stays at the W state longer, the sall value o β, the raction o inective grows ore slowly as observed ro Figure. To keep the population at the W state longer, treatent is a reasonable ethod. Although there is no treatent that can cure AIDS copletely, i the HIV positive population can stay at the W state as long as possible, the transission rate o AIDS will be diinished depending on how long population can stay at the W state. However, treatent can have side eects. Allreer.co 8, a health website, states: Antiviral therapy suppresses the replication o the HIV virus in the body. A cobination o several antiretroviral agents, tered Highly Active Anti-Retroviral Therapy HAART, has been highly eective in reducing the nuber o HIV particles in the blood strea as easured by a blood test called the viral load. However, HIV tends to becoe resistant in patients who do not take their edications every day. Also, certain strains o HIV utate easily and ay becoe resistant to HAART especially quickly. Treatents ight provide a short-ter solution to reduce the AIDS epideic but they also can coplicate the atter ore as entioned in the quote above. While eect o treatent can be incorporated in the odel, that situation was not considered in the interest o keeping the odel as siple as possible. Figure 3. Fraction o Inective with varying values Figure 4. Fraction o Inective with varying γ values is the higher ratio o sexual contacts o the ore proiscuous population and an increase in increases the raction o inective alost exponentially, as shown in Figure 3. By keeping low, the raction o inective can also be kept near 0 and stable. Figure 3 is very siilar to Figure since increases kα o the ore proiscuous population. One o possible approaches to keep relatively low is to educate and convince the inective population to reduce the nuber o sexual contacts and use protections at all ties. Another actor that can change the raction o inective drastically is γ, the death rate. γ also represents how long the inective population stays inective, so as γ decreases, the raction o inective increases. In other words, when the inective population stays inective and sexually active or a longer period o tie, the inective population will increase aster. The paraeters that aect the outcoes o the odel signiicantly are greatly correlated to people s sexual liestyles and preerences. However, the 6

8 issue o creating a policy to change people s behaviors, especially sexual ones, becoes very diicult. Creating such a policy coes down to a decision o choosing the ajority s good by restraining reedo o the AIDS inective population: that then iplies that AIDS patients are not treated equally. Unlike other diseases, creating policies to prevent the epideic o AIDS becoes a politicized atter because transission o AIDS highly involves one s sexual liestyle, which connects to one s rights and reedo in this society. Lastly, as seen in Figure 5, the proportion o the total population that exhibits the greater sexual activity does not see to aect the inective population as uch. Although changing the p value does not aect the raction o inective population as uch, a slight decrease in raction o inective population is still better than an increase. Furtherore, Figure 5 relects the doinancy o kα and values in the projections o AIDS. Figure 5. Fraction o Inective with varying p values Conclusion The revised odel o AIDS was built by including the heterosexual population to the existing odel o Thopson. The iproveent o the new odel was validated by coparing its estiates to that o Thopson s and the surveyed values. Thopson s odel overestiated the AIDS inective population by alost a actor o 3, and the new odel proved to be ore accurate. As ore inoration on AIDS are publicized and distributed to the public, it can aect people s behaviors, so paraeters used in the odel are ost likely dynaic. However, because o its coplexity, paraeters were assued to be static; this can cause great discrepancies since paraeters can be greater or saller than what were chosen over the period o tie in interest. While not all conditions can be included, it is iportant to consider ajor trends such as increasing nubers o inected eales. Although the odel did not include other current ajor trends such as dierent transission possibilities in dierent age groups, it still can be used to see how each situation and approach can alter the progression o AIDS. AIDS is not a new disease but it was noticed only ew decades ago. One o reasons why it has becoe a big issue is because it ties with one s sexual liestyle. Because o its tie with personal preerences, lot o people eel hesitated to discuss about it; this causes AIDS patients to hide ro public or decide not to do anything about it. The ost iportant approach to lower newly inective population is to create an environent in which AIDS patients will not eel so shaed. When people are ore open and accepting about AIDS, it can be diagnosed in earlier stages or even prevented. 7

9 Publicizing awareness o AIDS or having AIDS education along with sex education in high schools can help adolescents to be ore knowledgeable and aware o AIDS. It ight not show a short ter result, but in a long run, these kinds o education can result in the higher nuber o sae sexual contacts with protection and the lower nuber o sexual contacts in uture generation. When what is entioned above can be achieved, the AIDS epideic can be stopped. 8

10 Reerences. Centers or Disease Control and Prevention CDC. HIV/AIDS Surveillance Report 997; 9No.: -44. United Nations Foundations. 7 Apr < 3. Thopson, J.R. AIDS: The Misanageent o an Epideic. Coputers Math. Applic. Vol.8 989: Centers or Disease Control and Prevention CDC. Morbidity and Mortality Weekly Report 00; 50No.: AIDS.ORG. 7 Apr < 6. Brookeyer, Ron and Gail, Mitchell H. AIDS Epideiology: A Quantitative Approach. New ork: Oxord University Press, Catania, J. A., J. Canchola, D. Binson, M. M. Dolcini, J. P. Paul, L. Fisher, K. H. Choi, L. Pollack, J. Chang, W. L. arber, J. R. Heian and T. Coates 00. "National trends in condo use aong at-risk heterosexuals in the united states." Journal o Acquired Iune Deiciency Syndroes 7: AllReer. 7 Apr < 9

11 Appendix A. Matlab Code unction aids % Predicts epideic o AIDS % W is an interediate individual who has been inected by AIDS but is not % yet able to transit disease: represents tie delay % is an individual who has been inected and able to transit disease % is susceptibles % subscript indicates sexually inactive and indicates active population % F stands or eale population o W,, and and M stands or ale % population % k = average nuber o sexual contacts in a onth % a = probabilityo contacting AIDS through a single contact % lada = iigration rate % u = igration rate % gaa = death rate o AIDS % p = high contact raction % tau = ratio o actives to inactives % b = rate o tie delay ro W to % q = ratio o eale to ale % Total nubers o W, and are calculated by adding F and M % For eale: % dw/ = ka**tau*/tau* - bu*w % dw/ = ka*tau**tau*/tau* - bu*w % d/ = b*w - ugaa* % d/ = b*w - ugaa* % d/ = -ka**tau*/tau* -p*.5*lada - u* % d/ = -ka*tau**tau*/tau* p*.5*lada - u* % For ale: % dw/ = ka**tau*/tau* - bu*w % dw/ = ka*tau**tau*/tau* - bu*w % d/ = b*w - ugaa* % d/ = b*w - ugaa* % d/ = -ka**tau*/tau* -p*.5*lada - u* % d/ = -ka*tau**tau*/tau* p*.5*lada - u* % tie variables tspan = [0::*5]; % initial conditions % eale inective is initially 8% o inective population % data ro 997 is used % it was assued that 90 % o inective population is sexually active % [W, W,,,,, W, W,,,, ] W=5000; W=40000; =round64086*.8*.-5000; =round64086*.8* ; = ; = ; W=round.*W; W=round.*W; =round64086*.8*.-w; =round64086*.8*.9- W; =; =; D = 0; % Calculates cuulative death init = [W, W,,,,, W, W,,,,, D]; % paraeters 0

12 ka =.03; b = /*; u =.0047; lada = u*; tau = 5; gaa = /*3; p =.; ka =.00; % resolution o solution can be changed by changing reine actor options = odeset'reine', ; %coented or loops can be used to generate a plot with varying paraeter %values %or i = :4 [t,u] = ode45@ode, tspan, init, options, ka, b, u, lada, tau, gaa, p, ka; %S3 = u:,3u:,4u:,9u:,0./u:,u:,u:,3u:,4u:,5u:,6u:,7u:,8u:,9u:,0u:, u:,; %ka = ka -.003*i; %A:,i = S3; %end %Total nuber o AIDS inective at the end o tspan T = ulengthu,3ulengthu,4ulengthu,9ulengthu,0 % codes used to plot the result o or loop %plott, A:,, '', t, A:,, 'b',t, A:,3, 'y', t, A:,4,'g'; %legend'ka=.03', 'ka=.00', 'ka=.04', 'ka=.005' %xlabel'tie onths'; ylabel'fraction o Inective'; title'fraction o Inective with varying ka values'; %igure; %plot cuulative death %plott, u:,3; %calculates raction o inective or eale, ale and total respectively %S = u:,3u:,4./u:,u:,u:,3u:,4u:,5u:,6; %S = u:,9u:,0./u:,7u:,8u:,9u:,0u:,u:,; %S3 = u:,3u:,4u:,9u:,0./u:,u:,u:,3u:,4u:,5u:,6u:,7u:,8u:,9u:,0u:, u:,; % plots interediates and inectives igure; plott, u:,u:,7,'bo';hold on; plott, u:,u:,8,'bx';hold on; plott, u:,3u:,9,'g';hold on; plott, u:,4u:,0,'gs'; legend'w','w','','';xlabel'tie onths';ylabel'population';title'nuber o Interediates and Inectives'; % plots susceptibles igure; plott, u:,5u:,,'d';hold on; plott, u:,6u:,,'v'; legend'',''; xlabel'tie onths'; ylabel'population';title'nuber o Susceptibles';

13 unction uprie = ODEt,u,ka,b,u,lada,tau,gaa,p,ka % ODEs % u = W, u = W, u3 =, u4 =, u5 =, u6 = % u7 = W, u8 = W, u9 =, u0 =, u =, u = uprie = [ka*u5*u9tau*u0/u5u3tau*u4u6 - bu*u; ka*tau*u6*u9tau*u0/u5u3tau*u4u6 - bu*u; b*u - ugaa*u3; b*u - ugaa*u4; -ka*u5*u9tau*u0/u5u3tau*u4u6 -p*.5*lada - u*u5; -ka*tau*u6*u9tau*u0/u5u3tau*u4u6 p*.5*lada - u*u6; ka*u*u9tau*u0/uu9tau*u0u - bu*u7; ka*tau*u*u9tau*u0/uu9tau*u0u - bu*u8; b*u7 - ugaa*u9; b*u8 - ugaa*u0; -ka*u*u9tau*u0/uu9tau*u0u -p*.5*lada - u*u; -ka*tau*u*u9tau*u0/uu9tau*u0u p*.5*lada - u*u gaa*u3u4u9u0];