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Revista de Saúde Pública
Print version ISSN 0034-8910On-line version ISSN 1518-8787
Rev. Saúde Pública vol.33 n.4 São Paulo Aug. 1999
http://dx.doi.org/10.1590/S0034-89101999000400002
A theoretical model of the evolution of virulence in sexually transmitted HIV/AIDS
Modelo teórico da evolucão da virulência do HIV/AIDS transmitido sexualmente
FAB Coutinho, E Massad, RX Menezes and MN Burattini
Departamento de Patologia. Laboratórios de Investigação Médica 01. Faculdade de Medicina da Universidade de São Paulo. São Paulo, SP-Brasil
Keywords Acquired Inmunodeficiency Syndrome. Virulence. Models, statistical. | Abstract Introduction Methods Results Conclusion |
Descritores Síndrome de Imunodeficiência Adquirida.Virulência. Modelos estatísticos. | Resumo Introdução Métodos Resultados Conclusão |
INTRODUCTION
A recent paper by Lipstch and Nowak^{10} investigates the evolution of virulence in sexually transmitted HIV/AIDS. Assuming a population with a constant supply of new susceptibles they conclude that, in the long run, new partner acquisition rates should have no effect on the evolution of pathogen virulence. We summarise their arguments below.
They consider the competition of two different strains of virus. Strain 1, called more virulent is more pathogenic to its hosts and more transmissible during the course of a single partnership. Strain 2 , called less virulent for its ability to remain longer in the host without producing AIDS, is therefore less pathogenic to its host but is also assumed to be less transmissible. The rate of new partner infection is assumed to be independent of the total population density or size.
Let X be the number of susceptibles in the population, Y_{1} and Y_{2} represent the number of hosts infected, respectively, with strain 1 and strain 2. N = X + Y_{1 }+ Y_{2} is the total population minus the individuals with AIDS which are assumed to be too ill. The spread of the two strains can be modelled by the following system of differential equations:
(1)
The force of infection l_{i }(i = 1,2) is assumed to be
(2)
where c is the rate of new partner acquisition, b_{i }(i = 1, 2) is the probability that a host with strain i will infect a single susceptible partner and n_{i} is the rate individuals infected with each strain develop full-blown AIDS (in the present paper this parameter is called virulence).
As shown by Brenmerman and Thieme^{2} one of the pathogen strains will drive the other to extinction. The winning strain will be the one with the greatest reproductive number R_{0}. For strain i, we have
(3)
Equation 3 shows that changing the rate of new partner acquisition c scales R_{0i} equally for all strains. Thus, the main conclusion of Lipstch and Nowak^{10} that, in the long run, partner acquisition should have no effect on the evolution of virulence.
This conclusion depends crucially on b_{i} being independent of c and n. This assumption is, however, contradicted by a number of studies on HIV transmission. In section 2, we summarise the biological studies that show that in fact b, for sexually transmitted HIV, should be a function of both c and n. In section 3 we propose a simple form for this dependence and we examine how R_{0} depends on c and n to conclude that low rates of acquisition of new partners favours a less virulent strain.
Epidemiological evidence for the dependence of b on c and n.
It is an already well established fact that the likelihood of sexually related HIV transmission is influenced, among other things, by the presence of coadjuvant factors, in particular other sexually transmitted diseases (STDs), including chlamydia, gonorrhea, herpes and syphilis. The later, in turn, have incidence rates which are directly dependent on the level of sexual activity. In fact, it has been reported by a number of authors^{4,13,17} that STD's can increase the risk of HIV transmission by a factor of up to nine times. In addition, the relationship between HIV and other STDs has been suggested as a possible explanation for the higher prevalence of heterosexually transmitted HIV observed in Africa as compared to the rates observed in western countries^{1}.
Furthermore, the number of new sexual partners has been directly associated with the risk of HIV infection in a number of studies^{3,8,16}. For instance, in the study by Burcham et al.^{3} it has been shown that the relative risk for HIV infection increases by a factor of 1.02 per new sexual partner. It is, therefore, valid to assume the level of sexual activity as a determining factor of the likelihood of HIV transmission.
As for the influence of the viral load on the natural course and transmissibility of HIV infection, several direct and indirect evidences, mainly related to maternal-fetal transmission, point to a positive relationship between the level of viremia and the speed of disease progression and/or the transmission likelihood^{15,18,19}.
In what follows we consider likelihood of transmission as dependent both on the rate of partner exchange and on the level of virulence of HIV, as defined above.
A simple model for the dependence of b on c and n.
It is reasonable to assume a function for b that is a logistic-like curve for both c and n. This function should assume a zero value when either c or n were zero, and should
(4)
saturates when c and n increase to a finite value. A simple function satisfying the above requirements could be:
where ki are positive constants. Figure 1 shows the shape of the function b(c, n), for k_{1} = 0.0333, k_{2 }= 0.5 and k_{3} = 0.1. The values for the parameters ki were arbitrarily chosen to make the function b(c, n) reproduce accepted epidemiological data.
The basic reproductive ratio, R_{0}, is calculated according to equation 3 replacing b with b(c,n) given by equation 4. Figure 2 shows its shape as a function of n for several values of c.
It should be noted that R_{0} is maximised by certain values of n (n_{max}) and its peaks increase with c and always shift to the right, indicating that, for the assumed b, in the sub-population with a lower level of sexual activity, HIV evolves towards a less virulent state. In Figure 3 is shown n_{max} as a function of c.
These results are in agreement with the findings of Ewald^{5} and Massad et al.^{11,12}
DISCUSSION
The evolution of virulence in host-parasite relationships has been the subject of several publications in the past two decades (see the review by Levin^{9} for details). The paradigm of commensalism as a final end in the evolution of host-parasite interactions has been challenged by some theoretical^{14} and experimental works^{6,7}. In the case of HIV virulence, some authors have been addressing the subject with basically two opposite points of view with regard to the importance of sexual activity level. In a seminal paper, Ewald^{5} concludes that the fraction of the host population with the lowest level of sexual activity ends up infected with a less virulent HIV strain, in the sense that it causes disease (AIDS) after a longer period of time. Attempts to provide a mathematical treatment of Ewald's arguments is provided in Massad et al.^{12}, indicating that the rate of acquisition of new sexual partners may influence the evolution of HIV virulence.
On the other hand, as mentioned above, Lipsitch and Nowak^{10} argue against this, demonstrating that when of b_{i} is independent of c and n, the level of virulence at equilibrium is independent of sexual activity. In this paper we show that when b_{i} is considered as a function of c and n it turns out that the evolution of HIV virulence correlates with the rate of acquisition of new sexual partners in the sense that the greater this rate is, the greater the virulence of the HIV strain selected.
This debate is of extreme importance from the point of view of the epidemiology of HIV/AIDS. For such an infection, for which the only effective control measure is education with changing habits and attitudes towards sex, any conclusion regarding the role of sexual activity on the evolution of virulence can constitute an argument for or against such a measure.
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Correspondence to:
Marcelo N. Burattini
Av. Dr. Arnaldo, 455 01246-903 São Paulo - Brasil
E-mail: mnburatt@usp.br
The publication of this article was supported by FAPESP (Process n. 98/13915-5).
Submitted on 10.10.1998. Reviewed on 13.1.1999. Approved on 13.4.1999.