Frequent implementation of interventions may increase HIV infections among MSM in China

Intervention measures among men who have sex with men (MSM) are usually designed to reduce the frequency of high risk behaviors (within-community level), but unfortunately may change the contact network and consequently increase the opportunity for them to have sex with new partners (between-community level). A multi-community periodic model on complex network is proposed to study the two-side effects of interventions on HIV transmission among MSM in China, in which the wanning process of the impacts of interventions are modelled. The basic reproduction number for the multi-community periodic system is defined and calculated numerically. Based on the number of annual reported HIV/AIDS cases among MSM in China, the unknown parameters are estimated by using MCMC method and the basic reproduction number is estimated as 3.56 (95%CI [3.556, 3.568]). Our results show that strong randomness of the community-connection networks leads to more new infections and more HIV/AIDS cases. Moreover, main conclusion indicates that implementation of interventions may induce more new infections, depending on relative level of between- and within-community impacts, and the frequency of implementation of interventions. The findings can help to guide the policy maker to choose the appropriate intervention measures, and to implement the interventions with proper frequency.


The basic reproduction number
In the following, we define the basic reproduction number of (2)-(3) by using the theory proposed by Wang and Zhao. 1 System (2)-(3) is equivalent to the following system d dt It is obvious that conditions (A1)-(A5) in reference 1 x(t)) respectively. Then we havẽ and it is easy to obtain that r(ΦM(ω)) < 1, where Φ M (t) is the mondromy matrix of the linear T l − period system dy dt =M(t)y and r(ΦM(ω)) is the spectral radius of ΦM(ω). Thus, the condition (A6) in reference 1 also holds.
Let ) and V(t, x), respectively. Then, we have Let Y(t, x) be a 3n × 3n matrix solution of the following system.
where I is a 3n × 3n identity matrix. Therefore, the condition (A7) in reference 1 holds.
Define ψ(t) as the initial periodic distribution of infected individuals with periodic T l .
Then, the distribution of infected individuals infected at time s and are still infected individuals at time t can be given by Y(t, s)F(s)ψ(t). Let C T l be the ordered Banach space of all T l − periodic functions from R to R 3n , which is equipped with the maximum norm ∥ · ∥ and the positive cone C + T l := {ψ ∈ C T l : ψ(t) ≥ 0, ∀t ∈ R}. A linear operator L : C T l → C T l is defined as follows.
Then, we can define the basic reproduction number as R 0 := ρ(L).
From Wang and Zhao, 1 we have the following Lemma.
On the basis of this Lemma, we can calculate the basic reproduction number numerically by finding the positive solution λ 0 of r(W(T l , λ)) = 1.  Table 1 in the main text.