Why are Testing Rates so Low in Sub-Saharan Africa? Misconceptions and Strategic Behaviors

Appendix A Proofs

Corollary 2.2.The propensity Θ_A is an increasing function of the expected transmission rate per act β. Therefore, if the test results of the individual’s partner are expected to be unobservable, then the individual’s overestimation of the HIV transmission rate may promote his decision to be tested for HIV.

Proof The derivative of Θ_A with respect to β is given by:

We conclude that is strictly positive if and    □

Proposition 2.3.If j is not expected to test or if i and j have never engaged in unprotected sex (n=0), then the individual i tests for HIV at t₁ if A≥0.

On the contrary, if j is expected to test at t₁ and if i and j have had n>0 unprotected encounters before t₁, then the incentive for i to be tested at t₁ is reduced by opportunism. In this case, i tests at t₁ if A–Ω≥0, where the opportunism Ω is defined as follows:

If i does not undergo HIV testing at t₁, he will undergo HIV testing at t₂ if he observes that his partner is HIV-positive and if Ω≥0.

Proof If i believes that j will not undergo HIV testing at t₁, then the situation is similar to proposition (2.1), and i will therefore chooses to be tested for HIV if A≥0. Similarly, if n=0, then the test results for j will not be informative to i, and i will again choose to be tested for HIV if A≥0.

If i thinks that j will choose to be tested for HIV, then i has three possible courses of action. First, he may be tested for HIV at t₁ because this testing would allow him to obtain access to HIV-related health care benefits if he is infected. In this situation, if he tests negative and observes that his partner is infected, recall the assumption that i will choose to engage in N unprotected encounters with a new partner whose expected probability of being infected is p_ij. Therefore, if i tests at t₁, his expected lifetime utility may be stated as follows:

Second, i may not undergo HIV testing at t₁ but may instead undergo this testing at t₂ if j is observed to be HIV-positive. In this case, i tests and benefits from health care only if both i and j are infected at the time of the testing decision. If i tests negative at t₂, it is assumed that he finds a new partner for N unprotected sexual encounters. Because i does not undergo testing at t₁, if i is infected but j is uninfected at this time, i is therefore excluded from receiving H–C, the net benefit of health care from being tested for HIV. His expected lifetime utility may then be stated as follows:

Finally, if the testing cost is high compared with its expected benefits, i tests at neither t₁ nor t₂. In this case, he does not obtain any benefits from HIV-related health care if he is infected. Again, it is assumed that he finds a new sexual partner if he observes that j is infected. His expected lifetime utility may then be expressed as follows:

Let us compare these three alternative courses of action. First, i prefers to be tested for HIV at t₁ instead of t₂ if the following condition holds:

where opportunism Ω is defined as follows: Ω=[p_iip_ij+p_ii(1–p_ij)P_n+(1–p_ii)p_ijP_n]H–[p_ij+p_ii(1–p_ij)P_n]C. Secondly, if i does not want to be tested for HIV at t₁ because A–Ω<0, he nevertheless undergoes this testing at t₂ if Finally, if both A–Ω<0 and Ω<0 are satisfied, i tests at neither t₁ nor t₂.   □

Corollary 2.4.The propensity Θ_A is an increasing function of β, the expected per-act transmission rate. Therefore, if the HIV test results of an individual’s partner are expected to be unobservable or if this partner is not expected to be tested for HIV, the overestimation of the transmission rate may foster testing.

By contrast, the propensity Θ_A–_Ω to test at t₁ is an inverted U-shaped function of β. As a consequence, if j is expected to be tested for HIV at t₁ and if the test results are expected to be observable, the reduction of the expected transmission rate to an intermediate level β* would encourage i to be tested at t₁ if there exists a β^*<1 such that Θ_A–_Ω(β^*)≥0.

Proof The first part of this proof is similar to the proof of corollary 2.2.

To prove the second statement, let us demonstrate that Θ_A–Ω is an inverted U-shaped function of β if and and P_n(0)=p_ii(0)=0. Note that Θ_A–Ω≥0, Θ_A–Ω(β=0)=0 and Θ_A–Ω(β=1)=0. Taking the derivative of Θ_A–Ω with respect to β produces the following expression:

The denominator of the above fraction is strictly positive. The numerator is continuous, positive for β=0, and negative for β=1. We conclude that Θ_A–Ω is a U-shaped function of β with an interior maximum between 0 and 1. Because Θ_A–Ω(β=1)=0, i will never want to be tested for HIV if β≈1, n>0 and E(t_j)=1. In this case, if a reduction of β would encourage i to test at t₁.

Proposition 3.1.An individual i will undergo testing for HIV before the onset of AIDS symptoms if:

This condition is satisfied if the expected probability that i is infected at the time of the decision is high and if H₁, the expected value of health care if HIV/AIDS is asymptomatic, is significantly higher than H₂, the expected value of health care after the onset of AIDS symptoms.

Proof Because we assumed that H₂>C, the individual i always undergoes HIV testing at t₂ if he has begun to experience AIDS symptoms. Two cases should therefore be compared. First, i may be tested for HIV at t₁ and at t₂ if symptoms appear; alternatively, i may only undergo testing at t₂ after the onset of AIDS symptoms. The lifetime expected utility of testing both at t₁ and at t₂ if AIDS symptoms appear is given by the following expression:

Similarly, the lifetime expected utility of testing at only t₂ is given by:

We define ω as the net benefit from testing at t₁ prior to the onset of symptoms, that is, as the difference between the two last equations:

The individual i will undergo testing for HIV before the emergence of symptoms if ω is positive.   □

Appendix B. Household model

In Section 2, it was assumed that each partner separately decides whether to be tested for HIV. This section reveals that the strategic mechanisms that are discussed in Section 2 continue to hold if the hypothesis of individual decision-making is relaxed; in particular, this section examines the results if the decision to undergo HIV testing is reached in cooperation with a committed sexual partner and if the couple’s utility function is maximized.

We begin by extending the notation of Section 2. In this section, the expected probability that i is infected is denoted by p_i and the expected probability that j is infected is denoted by p_j. If only one of the partners tests positive, the other is assumed to have N unprotected encounters with another individual, k, whose expected probability to be infected is denoted by p_k. We utilize A_i and A_j to denote the net benefit of testing at t₁ for i and j if their testing decisions are reached individually and if q=0 (the equivalent of A in Section 2). Similarly, Ω_i and Ω_j are defined to be the equivalents of Ω, the opportunism metric that was introduced in Section 2.¹³

If the decision to undergo HIV testing is reached in combination with a committed sexual partner, then proposition 2.3 is slightly modified by the presence of implicit altruism. The intuition underlying the presence of implicit altruism in the testing decision may be outlined as follows. Without loss of generality, let us assume that j undergoes HIV testing at t₁. Her partner i will benefit from this information in two ways. First, if j is revealed to be HIV-positive, i will learn that he is also likely to be infected with HIV (if n>0). In this case, he may test at t₂, leading to an expected gain of Ω_i. Second, i is provided with the opportunity to change partners and to thereby reduce his probability of becoming infected with HIV in the future. This phenomenon leads to an expected utility gain E_j that is equal to the following expression:

E_j should be regarded as a positive externality from j to i; if i is uninfected and if j tests positive, then i will change partners and reduce his probability of dying from AIDS. Similarly, the externality E_i if i tests for HIV may be defined as follows:

In summary, if the decision to undergo HIV testing is jointly reached with a sexual partner, then testing is valued not only because it provides access to health care but also because it informs an individual’s partner and protects him or her from the virus. This phenomenon is summarized in the following proposition, which is the equivalent of proposition 2.3 for a decision to be tested for HIV that is reached in combination with a sexual partner.

Proposition Appendix B.1.We assume that a decision to test for HIV is reached jointly with a committed sexual partner, that an individual’s test results are disclosed to his or her partner (q=1) and that j is expected to be at greater risk for HIV infection than i (p_j>p_i). The couple must to compare four alternatives. First, both partners will test for HIV at t₁ if opportunism is low and externalities are high, that is, if:

These conditions are satisfied if i and j are likely to be infected and if the test results of the partners are not expected to be informative (if P_n is low).

Second, only the partner who is at greater risk of HIV infection, j, will be tested for HIV at t₁, and the remaining partner, i, will be tested for HIV at t₂ if j tests positive for HIV if opportunism is high and if externalities are low (strategic behavior):

These conditions are satisfied if both partners are likely to be infected and if the HIV test results for j is expected to reflect the serostatus of i (in other words, if P_n is high).

Third, only j, the partner who is more at risk for HIV infection, undergoes testing for HIV at t₁, and i refrains from being tested at t₂ if only j is likely to be infected:

These conditions are satisfied if i is unlikely to be infected, if j is likely to be infected and if the expected probability P_n that the virus was transmitted between i and j is low.

Finally, neither i nor j will take an HIV test if both i and j are unlikely to be infected, that is, if:

Proof If both i and j are tested for HIV at t₁, the expected lifetime utility of the couple is given by:

If only i is tested for HIV at t₁, and j is tested for HIV at t₂ if i is HIV-positive, then the expected lifetime utility of the couple may be expressed as follows:

If only j is tested for HIV at t₁, and i is tested for HIV at t₂ if j is HIV-positive, then the expected lifetime utility of the couple is expressed as follows:

If i is tested for HIV at t₁, and j refrains from being tested for HIV at t₂ (even if i has tested positive for HIV), then the expected lifetime utility of the couple is given by:

If j is tested for HIV at t₁, and i refrains from being tested for HIV at t₂ (even if j has tested positive for HIV), then the expected lifetime utility of the couple may be expressed as follows:

If neither member of a couple is tested for HIV infection, then the expected lifetime utility of the couple is given by:

If only one of the members of a couple undergoes HIV testing, the partner who has a higher risk of HIV infection will be tested for HIV. Indeed, is >0 if p_j≥p_i. Similarly, if p_j≥p_i. Therefore, the two cases in which only i undergoes HIV testing are dismissed, and the couple must only compare the remaining four alternatives.

First, both partners will undergo HIV testing at t₁ if the following three conditions are satisfied: and When p_j>p_i, these three conditions reduce to the system of conditions (B.1).

Second, only the partner who is at greater risk of being infected with HIV, j, undergoes HIV testing at t₁ and his partner, i, undergoes testing at t₂ if the following three conditions are satisfied: and

Third, the member of the couple who is at higher risk of HIV infection, j, undergoes HIV testing at t₁ but i is not tested at t₂ if the following three conditions are satisfied: and

Finally, neither i nor j will be tested for HIV if and If p_j>p_i, then these three conditions reduce to the system of conditions (B.4).

Within this modified framework, the relationship between testing and P_n in the hypothesis of couple-based decision making is similar to the relationship that is identified under the hypothesis of individual decision-making. Section 2 revealed that individual i acts strategically and postpones testing if three conditions are satisfied. First, his partner should be expected to test at t₁, and the results of this test should be observable. Second, P_n and opportunism Ω should be high. In other words, the observation of the partner’s test results should be expected to yield significant information about i’s serostatus (serodiscordance should be regarded as unlikely). Third, A–Ω should be sufficiently low to ensure that i will prefer to postpone testing instead of being tested at t₁.

If the decision to test is reached in combination with a sexual partner, similar strategic behavior will emerge if these three conditions are satisfied. To observe this similarity, it must first be noted that the externalities E_i and E_j are decreasing functions of P_n. Therefore, these positive externalities are sharply reduced after several risky sexual encounters, particularly if the transmission rate of HIV is overestimated.

With this in mind, we can demonstrate that the three aforementioned conditions coincide with the system of conditions (B.2) that identifies strategic behavior in the modified framework for couples-based decisions. The first condition of system (B.2) states that j should be willing to undergo HIV testing: A_j represents the expected health benefits from testing, E_j is the externality on i, and Ω_i represents the gain from the information that is transmitted to j. The second condition of system (B.2) states that opportunism Ω_j should be high. This condition is only satisfied if P_n is sufficiently high to ensure that the HIV test results for one partner in a couple are expected to be informative about the serostatuses of both members of the couple. Finally, the third condition of system (B.2) reduces to A_i–Ω_i if P_n is high, as the externality E_i vanishes after several risky sexual encounters between i and j if the transmission rate of HIV is overestimated.