Using a previously developed Complex Agent Network 3 , we model the HIV epidemics among MSM in Amsterdam through a sexual contact network in which each node represents an individual and edges represent potentially infectious contacts. Please see Figure 2 in 13 as a schematic illustration of virus spreading in a network. The number of edges emanating from a node is the degree of the node and the degree distribution is the distribution of this quantity across the population 14 . In this study, the degree of a node, drawn from a specific degree distribution, is assumed to be static while the neighboring nodes that this node links to may change over time. There are classes of network geometry such as small-world, Poisson and scale-free networks. Since scale-free networks have been shown to be the representative of a variety of human sexual contact networks 15 16 17 , we perform stochastic simulations of the spreading of HIV in undirected scale-free networks. The scale-free networks are characterized by a highly skewed distribution of contacts so that most of the nodes are weakly connected and a small number of nodes have very high connectivity. In particular, the degree distribution of this class of networks follows a power law distribution, i.e., pk = Ak ^-γ where is a normalization factor. After each node is assigned with a degree drawn from a given power law, these nodes are connected at random according to the so-called configuration model 18 19 with the guarantee that the sum of all degrees is an even number and that there are no loops (nodes connected to themselves) and multiple edges (between two nodes).

The disease status of infected individuals is heterogenous with respects to infection stages, disease durations and infectiousness, etc., so that we can model individuals as agents and individuals' status transitions as internal changes of agents. The HIV infection progression occurs in roughly three stages: primary infection (PI), asymptomatic period (AP) and AIDS. Regarding status transition, for instance, infection reaches from PI to AP with a delay of about 3 months. We assume an irreversible "negative ⇀ PI ⇀ AP ⇀ AIDS" transition order for each infection, excluding the case that an individual with AIDS can with the right treatment go back to the asymptomatic phase, because in general patients are most likely diagnosed and treated before reaching the AIDS stage. During the HIV epidemic, each individual's infection status is related to temporally varying infectiousness due to the corresponding different plasma HIV RNA levels. The transmission probability across each edge depends on the status of the two connected individuals. Therefore, the probability with which a susceptible individual becomes infected is determined by the connectivity (degree) of the individual and the infection status of his partners (neighboring nodes) in the network. For simplicity, we assume no HIV mutations during the course of HIV epidemic. Additionally, individuals who die of AIDS will be replaced by healthy individuals, keeping the total of MSM under study static throughout simulations.

Our model distinguishes between steady and casual relationships as two types of sexual relationships, which brings different transmission probabilities that occur across likewise steady or casual edges. Results from previous models show that the transmission dynamics of HIV differ significantly depending on relationship type, i.e., long steady relationships or short casual partnerships 20 . Moreover, there are strong indications that a substantial proportion of new HIV infections occur within steady partnerships 20 21 . Multiple sexual actions can take place along a steady edge (within a steady relationship) yearly, while only one sexual action take place along a casual edge (within a casual partnership) yearly. Each individual can have at most one steady partner with whom he stays for some time. Casual partnerships can be formed between any two individuals, but men with a steady partner have fewer partners than single men. Even steady partners may be involved in casual partnerships during or between steady relationships, though they may negotiate safety agreements to be monogamous or to have no UAI outside the relationship. Considering two partners with one infected and the other susceptible, the number of sexual acts that they have within a specified time duration, together with the infected one's infectiousness and therapy effect, the susceptible one's susceptibility, and risk behavior involving the two, determines the transmission probability from the infected one to the susceptible one across their partnership edge (see Section 3.3 in 3 for details).

Based on the Amsterdam cohort study (ACS) and literature, we estimate the sociological, virological and medical values and distributions of important parameters, and these are as listed in Table 1. The ACS of HIV infection and AIDS among MSM living in Amsterdam was initiated in 1984 and has involved 2299 men until 2006 22 . Of the 2299 MSM, 571 were HIV-positive at study entry and 192 seroconverted during follow-up until the end of 2006. Consequently, characteristics of MSM's behavior and surveyed yearly HIV incidence over calendar years for the population are reported periodically 23 . How parameters in Table 1 are fed into the model and affect the infection probability between a HIV-positive and a HIV-negative is explicitly explained in Sec. 3.3 and 5.2.2 in 3 . Specifically, the infection probability per sexual action is computed by multiplying a multi-year statistic of transmission probability per action and several coefficients indicating, e.g., personal infectivity reduction due to treatment, personal infectivity increases due to risky behavior, and reduction of transmission probability between two steady partners if they reach a safety agreement on having no contacts outside the relationship.

The model can be used to predict the HIV epidemic by tuning the parameters related to two important factors: therapy effectiveness and risk behavior. Ref. 3 provides the value for the risk behavior factor in 1985-2005 among people infected and people not infected with HIV. It is strongly believed that risk behavior decreased until 1996 because of fear of the harm of HIV. The introdction of HAART in 1996 was associated with a slow increase in risk behavior. To investigate the counterbalancing impact of therapy effectiveness and risk behavior, we design several scenarios as listed in Table 2 to tentatively predict the HIV epidemic among a fixed number of 2299 individuals which is the total of MSM involved in ACS substudies since 1984. A reference scenario uses default values of parameters (Table 1) which remain unchanged during the course of simulations. Prediction scenarios 1-5 are designed based on various combinations of therapy reduction and risk behavior factors. We perform stochastic simulations of these scenarios from 1985 to 2044 (initialized according to the situation in 1984) with 1-year time steps and compare the results of the prediction scenarios to that of the reference scenario, with the aim to evaluate the significance of the factors. Each simulation result is an average over 30 realizations.

The incidence resulting from the reference scenario (RS in Table 2) simulation (1985-2020) versus the ACS historical incidence data (1985-2006) is shown in Figure 1. The curve of the simulated incidence is fluctuating less intensively compared to that of the ACS historical data. This is most likely due to the fact that the ACS has traces and records in incomprehensive populations every year (e.g., 2 HIV positives out of 511.12 in 1988 and 5 out of 374.54 in 2002, see Table five in 23 ) while we model the HIV epidemic in a population with a fixed size of 2299.

Compared to 1.66 in 2006, the RS incidence does not change noticeably at the time steps following 2006. For example, the incidence increases to 1.69 in 2015 and 1.74 in 2020, and then declining to 1.63 in 2030 and 1.60 in 2040.

To validate our model, a chi-square goodness of fit test and a paired-sample T-test are performed, comparing the simulated incidence of RS and the ACS historical incidence in 1985-2006. The null-hypothesis under consideration is that there is no difference between these two series of data with a significance level of 0.05 (data not shown). Meanwhile, we believe that the resulting validity also applies to the testing for all the prediction scenarios that share identical parameter setting to that of RS in the range of years of 1985-2006 (see Figure 2).

The incidence resulting from prediction scenarios, together with the RS incidence and ACS historical incidence, is shown in Figure 2 and Figure 3. To further compare these data and assess the effect of parameter tuning, we list the simulated incidence (see Figure 3) and the corresponding increase or decline percentage in each specified year in Table 3. Arrows indicate the percentage-wise increase or decrease of simulated incidence resulting from the prediction scenarios, compared to that resulting from the reference scenario.

The P1 incidence is higher in each specified year than the RS incidence though a moderate treatment-induced infectivity reduction factor is adopted in both simulations. We observe an increases of 10.95% in 2030 and of 2.87% in 2040, respectively.

Once the optimistic therapy is adopted in the following P2-P4 simulations, the resulting incidence shows an obvious overall downtrend compared to the RS incidence, since down arrows are predominantly present in the P2, P3 and P4 columns of Table 3. Among incidence resulting from these 3 simulations, the P2 incidence declines to the largest extent, e.g., by 11.24% in 2030 and 28.7% in 2040.

Surprisingly, for the P5 simulation an overall uptrend occurs compared to the RS incidence, since up arrows are predominantly present in the P5 column of Table 3. Although the optimistic treatment-induced infectivity reduction factor is used for all P2-P5 simulations, the P5 incidence shows a totally different upgoing trend compared to the largely downgoing incidence resulting from the P2, P3 and P4 simulations in the long run. For example, in 2035 there is a 5.5%↑ for P5 incidence, while there are 26.3%↓, 10.7%↓ and 9.63%↓ for P2, P3 and P4 incidence, respectively. Additionally, from 2030 onward the P5 incidence approximately remains more or less at the same level as the RS incidence, for instance, 1.64 versus 1.63 in 2030 and 1.58 versus 1.60 in 2040, respectively.

In addition, the yearly fraction of the diagnosed (the number of the diagnosed divided by the number of the infected), inferred from our simulated results, is fluctuating within the range of 0.35-0.56 and finally approaching 0.41 (data not shown).