Figure 1.
Flow Diagram of the Compartmental Model Describing Gonorrhea Transmission within a Homogenous Core Group
Not shown are the flows out of each compartment at rate λ. Table 1 summarizes the variables and parameters. A detailed explanation of the model is given in the text.
Table 1.
Variables and Parameters of the Model
Figure 2.
The Process of Resistance Emergence
The bars indicate between-host fitness levels of the different strains. Solid curved arrows show conversion events that occur frequently due to large or expanding source populations.
Dashed arrows show conversion events that occur infrequently due to small source populations.
(A) Without treatment, all resistant strains are less fit than the sensitive strain. Therefore, resistance emergence is not possible.
(B) Treatment of a small fraction of the population reduces fitness of the sensitive strain enough to allow for emergence of the fittest resistant strain. For that to happen, one frequent and two rare conversions need to occur.
(C) Further increase in treatment level allows both the second and third resistant strains to emerge. For the second resistant strain to emerge, one frequent and one rare conversion need to occur. Subsequently, the third resistant strain is rapidly generated and will outcompete all other strains.
(D) Treatment of a large fraction of the population results in all conversion events being frequent and in rapid emergence of resistance.
Figure 3.
Number of Introductions Q = μt Ît per Year
Parameter choices are described in the text and in Table 1. Note that above a certain level of treatment, indicated by f*, the basic reproductive number for the sensitive strain is , which makes the endemic steady state unsustainable and leads to disease extinction. In this study, we only consider the endemic situation with 0 ≤ f < f*.
Figure 4.
Years until Resistance Emerges—Deterministic Model
The black diamonds show the earliest time at which any one of the resistant strains emerges (reaches a level of 5% of total infecteds), obtained from simulations of the full deterministic system (Equation 1). The red dashed, green dash-dotted, and blue solid lines show the analytic approximation (Equation 7) for the time to emergence of the first, second, and third resistant strain. The vertical black lines indicate the level of treatment at which the fitness (R0) of the respective resistant strain is the same as that of the sensitive. For the top panels, fitness levels of the resistant strains are 75%, 85%, and 95% of the sensitive strain in the absence of treatment, resulting in values for the basic reproductive numbers as indicated. For the bottom panels, fitness levels of the resistant strains are 60%, 75%, and 90% of the sensitive strain. The left panels show results for conversion rates μt = μ1 = μ2 = 10−1, the right panels show results for μt = μ1 = μ2 = 10−3. Other parameter choices are given in Table 1.
Figure 5.
Probability That Resistance Emerges within One Year
The black diamonds show the probability that any one of the resistant strains emerges (reaches a level of 5% of total infecteds), obtained from stochastic simulations of the full system (stochastic version of Equation 1). The red dashed, green dash-dotted, and blue solid lines show the analytic result (Equation 9 with τ =1). The vertical black lines indicate the level of treatment at which the fitness of the respective resistant strain is the same as that of the sensitive strain. Parameters are chosen as in Figure 4.
Figure 6.
Years until Resistance Emerges—Stochastic Model
Boxplots show distribution of times to emergence (resistance at a level of 5% of total infecteds) for 5,000 simulations of the stochastic model. The red dashed, green dash-dotted, and blue solid lines show the analytic approximations for the time to emergence of the first, second, and third resistant strain. The vertical black lines indicate the level of treatment at which the fitness of the respective resistant strain is the same as that of the sensitive. For comparison, the black dashed line shows the deterministic result (Equation 7). Parameters are chosen as in Figure 4.
Figure 7.
Years until Emergence Occurs as a Function of Treatment
(A) All conversion probabilities are μi = 10−3. The green dashed line shows a situation with two resistant strains with fitness 60% and 90% that of the sensitive strain. The blue dash-dotted line shows three resistant strains with fitness 60%, 75%, and 90%, and the red solid line shows four resistant strains with fitness 60%, 70%, 80%, and 90%.
(B) Same number of strains and fitness levels as (A) but the product of all conversion probabilities is kept the same. We choose μt = 10−2 for all three cases and μ1 = 10−6 for the two-strain scenario (dashed green line), μ1 = μ2 = 10−3 for the three-strain scenario (dash-dotted blue line), and μ1 = μ2 = μ3 = 10−2 for the four-strain scenario (solid red line).
(C) Three resistant strains with fitness of 60%, 75%, and 90%. Conversion rates are μi = 10−1 (dashed green line), μi = 10−2 (dash-dotted blue line), and μi = 10−3 (solid red line).
(D) Same as (C) but with conversion rates μt = 10−2, μ1 = μ2 = 10−3 (dashed green line), μt = 10−4, μ1 = μ2 = 10−2 (dash-dotted blue line), and μt = μ2 = 10−3, μ1 = 10−2 (solid red line).