Sexually Transmitted Diseases VMED 5180 September 27, 2016

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1 Sexually Transitted Diseases VMED 518 Septeber 27, 216 Introduction Two sexually-transitted disease (STD) odels are presented below. The irst is a susceptibleinectious-susceptible (SIS) odel (Figure 1) that assues no iune state. The second is a susceptible-inectious-iune (reoved/resistant) (SIR) odel (Figure 2). In both cases, the transition rates ro susceptible to inectious (Sy) and leaving the inectious state () are identical. The lower case y represents the proportion /. The clear dierence is that the SIR odel has an additional state (iune), which or the purposes o this exaple is assued to be lielong, while no such state exists or the SIS odel. The result will be ore requent epideics, cycles, endeicity, etc. in the SIS vs. the SIR odel. The odel selection, as usual, is disease dependent. Susceptible () Sy Inectious () Figure 1. Basic STD-SIS odel. Susceptible () Sy Inectious () Iune (Z) Figure 2. Basic STD-SIR odel. In contrast to other odels that we have explored, e.g. Reed-Frost that assues discrete tie periods, the STD odels presented here are built on the assuption that tie is continuous. As such, transition rates will be expressed as dierential equations. While dierential equations assue an instantaneous rate change that is diicult/ipossible to odel in a spreadsheet, i a 1

2 suiciently sall tie interval is assued, dierence equations ay be used to very closely iic the dierential equations, as t approaches. Equations The transition rates (dierential equations) or this (SIS) odel that assues rando ixing (STD1) are given as d S d S where S = the average sexual partner rate = the probability o an eective contact, = the ortality rate, and = loss o inectiousness rate = 1 D, where D = duration o inectiousness. Interpretation o the equations or the SIS odel ay be ade as ollows: In the irst equation (d/), the change in susceptibles consists o 3 coponents: the individual probability o each susceptible () becoing a case, the inectious individuals that lose inectiousness and return to the susceptible state, and the susceptible individuals that die due to the noral ortality rate. The rate o inection depends on the nuber o sexual contacts (S) each susceptible has, the probability o such a contact transitting disease (), and the proportion o the population that is inectious (/). In the second equation (d/), the change in inectious individuals consists o the sae 3 coponents: the individual probability o each susceptible () becoing a case, the inectious individuals that lose inectiousness and return to the susceptible state, and the susceptible individuals that die due to the noral ortality rate. I we add an iune state to the odel, we ust odiy the equations. The dierential equations or the (SIR) odel are given as 2

3 d S d S dz Z otice that the equation or d/ no longer has the addition o the individuals that lose inectiousness. These individuals are now in the dz/ equation. That s the only change. ote that the ortality rate or all 3 states is the sae. This is eant to be a disease-independent ortality rate. In 193, Ronald Fisher coined the ter net reproductive value (R) or a parasite. It relected the nuber o ospring that a parasite could produce. This ter has been used widely, and ore recently, it has been applied to evaluate the progression o an epideic. Anderson and May (p. 17, 1991) deine R as the basic reproductive rate being the average nuber o secondary inections produced when one inected individual is introduced into a host population where everyone is susceptible. At equilibriu, the eective reproductive rate is R = 1, and we can write the relationship between the basic reproductive rate and the raction o susceptibles in the population as R = Rx * = 1. This relationship ay be used to identiy the critical proportion o susceptibles (x*) in a population below which an epideic will not occur (R < 1). Conversely, R > 1 relects a situation where an epideic will begin to take o. For directly transitted diseases the equation or the basic reproductive rate is: R D and thus, in its generic or, is a unction o the probability o an eective contact, the susceptible population, and the duration o inectiousness. The basic reproductive nuber (R) ay be calculated or the siple SIS odel. Basically, we are exaining the net change in inection, given a single case is introduced in an otherwise susceptible population. Applying the assuptions that = 1 and / = 1, the basic reproductive nuber calculation is sipliied as: R S S ( ). I the STD involves an iune state and the rate o loss o inectiousness () is uch larger than the ortality rate (), R ay be calculated as: S R. 3

4 I the ortality rate is relatively sall copared with the loss o inectiousness rate, R ay be approxiated as: S R It should be noted in R equations that population density is not included in the equations, which in essence states that the sexual activity is not population size dependent and that there is no population density threshold as is assued with other diseases. An extension o the SIS odel includes the incorporation o nonrando ixing, i.e. 2 separate subgroups (ale and eale in this exaple). The low diagra or this odel is presented below in Figure 3. Susceptible ales S y ) Inectious ales ( ) ( ) Susceptible eales S y ) Inectious eales ( ) ( ) Figure 3. SIS-STD odel assuing rando ixing between 2-groups. The new dierential equations or this odel are shown or the ale subgroup as, d = S + - d = S - (. Substituting or and or in the above equations will result in the dierential equations necessary or the eale subgroup. 4

5 The ipact o the introduction o 2 sexes in the odel can be seen in the reproductive rate (R): Male R Feale R R = and R = S ( ) S ( ), (note this odel only assues ale eale transission). Thus R deterines i the disease will continue in the ale population and R perors a siilar calculation or the eale population. In the equations above, it is iplicit that the epideic begins in the other (i) subgroup and thus j = 1 and does not appear in the equation. For exaple, in the equation or R, the coplete equation would include in the nuerator; however, since it is 1, it is not included to sipliy the equation. The result is the overall reproductive nuber at any tie is calculated as R R R Assuing a wholly susceptible population, i.e. / = 1, and the ortality rate is very sall, the overall basic reproductive rate is calculated as R = S S. Sexual contact heterogeneity The original odel (Model I) ignores new sexual contact variability and assues that each sex has a hoogeneous ixing rate, with little or no variability, i.e. 2 =. One proble with rando ixing (even between dierent sexes) is that it does not typically occur or sexually transitted diseases. That is, the nuber o sexual contacts ade by a given individual is not adequately represented by the ean, S. This variability is illustrated ro results o a survey conducted in the UK concerning sexual activity in that population (Figures 5 and 6). As can be seen ro the igures, although the ean nuber o reported annual sexual partners ay be approxiately 2, it ranges widely, ro to ore than 1. Siilarly variable results were 5

6 obtained in a 1986 survey o sexual activity aong ale hoosexual/bisexual residents in ondon (Figure 7). Probability distribution o average nuber o sexual partners/yr (UK survey) OW MEDIUM HIGH > 1 uber o sexual partners/yr. Figure 5. Probability distribution o nuber o sexual partners per year categories (source: Anderson and May, 1991). Survey inoration o sexual observations (an eperical observation) slope =3 og ean Figure 6. Relationship between the ean and variance o nuber o sexual contacts per year (source: Anderson and May, 1991) 6

7 Frequenc >13 Partners per onth Figure 7. Sexual activity aong ale hoosexual/bisexual residents in ondon surveyed in 1986 (source: May and Anderson, 1987). For this reason, easures o the ean nuber o contacts as well as its variability need to be considered in the odel. This ay be done by evaluating and including the variance ( 2 ), and coeicient o variation ( 2 /) to the ean, to ore accurately represent the result o variability in sexual contacts. The nuber o sexual contacts in a non-randoly ixing population ay be represented as: where = the ean nuber o contacts. c = + 2 /, on-rando ixing STD odel aong groups In addition to the exaples discussed above another odel ay include the assuption that sexual contacts are ost likely ade aong individuals within a given sexually active group. For instance individuals who have a low nuber o sexual partners are likely to have one another as their sexual partner, whereas individuals who have a high nuber o sexual contacts are ore likely to have such contacts with one another. Matheatically, this ay be expressed as ollows (using 1 dierential equation or the susceptible population only) d (, M H S ( P ) S ( P ) S M ( P )) H M H where, M and H reer to low ediu and high contact group, respectively. Thereore, the rate o change in the low contact group is a unction o the probability that a eber is this group has a sexual contact either within its own group or one o the other two higher contact groups. One interesting product o this relationship is that the resulting epideic ay not take on a traditional shape, with a single peak. Rather it is likely that there ay be ultiple peaks, resulting ro the ultiple within group epideics (see Figure 9). 7

8 Case incidence Tie Figure 9 Hypothetical epideic arising ro STD transission aong 3 contact groups. This apparently unusual epideic pattern ay be easily explained by transission o the disease within and aong 3 separate contact groups, beginning with the high contact group and then oving in to the ediu and low contact groups, respectively, as shown in Figure 1. 8

9 35 3 Case incidence H M Tie Figure 1. Hypothetical epideic arising ro STD transission aong 3 contact groups Reerences 1. Anderson, R.M. and May, R.M., Social heterogeneity and sexually transitted diseases, Ch. 11 In: Inectious Disease o Huans: dynaics and control, Oxord Science Publications, Oxord, England. 2. Fisher, RA, 193. The General Theory o atural Selection, Clarendon, Oxord. 3. May, R.M. and Anderson, R.M., Transission dynaics o HIV inection, ature, 326: