### 3.1 Network model and assumption

In our network scenario, there are *N* nodes distribute and move in a 2D square according to a random mobility model. The transmission range of each node is *t*. Furthermore, we assume the buffer size in a node is limited. As for the intermittently connected mobile ad hoc networks, since the node distribution is sparse enough, we could ignore the impact of the channel collision. Therefore, we also assume that the communication pair of nodes is perfectly separable, it means that for any communication pair of nodes, they would not interfere with any other simultaneous communication. As for the channel collision avoidance, In [11], host A comes into contact with host B and initiates an anti-entropy session. Anti-entropy session have three steps, in the first step, A transmit it summary vector to B. If B has no idol channel, it will neglect the summary vector. As a result, A cannot receive the request from B. So the anti-entropy session stopped. In such case, the channel collision could be avoided.

### 3.2 Two stages forwarding

In intermittently connected mobile ad hoc networks, there exists multiple source-destination pair nodes. Meanwhile, multiple packets are transmitted via multi-hop transmission mode at the same time. In order to explain the transmission process more clearly, we pick one pair of source-destination nodes to describe the model. We consider a set of *N* nodes and each one with a finite transmission range. The nodes move in a closed area. We define that two nodes meet when they come within transmission range of each other, at which point they can exchange their packets. According to the classical SIR model of epidemic theory, at any time point *t*, in general, there are three categories of nodes in the networks. The first category is the set of susceptible node, it means that these nodes do not receive the packet yet and if they meet the nodes which have already carry the packet, they would be the relay nodes. We define the ratio of these nodes as *S*(*t*). The second category is the set of infective node, it means that these nodes have already store, and carry the packet which needed to be relay to the other node or the destination. We define the ratio of these nodes as *I*(*t*). The third category is the set of recovered node, it means that the node which carry the packet has forwarded the packet to the destination. When the node meets the destination, they would remove the packet from their buffer and never receive the same packet. We define the ratio of these nodes as *R*(*t*). In this article, the number of wireless nodes is a constant N. We have the following Equation (1) to show the relation of *S*(*t*), *I*(*t*), and *R*(*t*). In Equation (1), since the number of nodes is an constant *N*, the summation of *S*(*t*), *I*(*t*), and *R*(*t*) is 1.

S\left(t\right)+I\left(t\right)+R\left(t\right)=1

(1)

Towards the variable rate of the node state, we have the differential equations to denote the foundational SIR model. We assume the initial ratio of *S*(0) = *S*_{0} and *I*(0) = *I*_{0}.

\left\{\begin{array}{c}\frac{dI}{dt}=\lambda S\left(t\right)I\left(t\right)-\mu I\left(t\right)\hfill \\ \frac{dS}{dt}=-\lambda S\left(t\right)I\left(t\right)\hfill \end{array}\right.

(2)

In the above differential equations, *dI*/*dt* is the variable rate of relay nodes which have carried the duplicated packets. *dS/dt* is the variable rate of nodes which do not received the packet yet. *I*_{0} and *S*_{0} are the initial values of received packet nodes and no packet nodes separately. *λ* is the meeting probability between any two nodes in the given field, *μ* is the meeting probability between a certain node which has carried the duplicated packet and the destination. Refer to [17], *λ* is shown by the following equation. *λS*(*t*)*I*(*t*) is the transferring amount from state *S* to state *I*. *μI*(*t*) is the reducing amount in time *I*. Therefore, *λS*(*t*)*I*(*t*) is the changing amount of *I* in the time of *dt*.

\lambda =\frac{2tE\left[{V}^{*}\right]}{{L}^{2}}

(3)

In Equation (3), *t* means the transmission range for each node, *E*[*V**] means the relative velocity expectation of mobile nodes. *L*^{2} means the area of given sensing field. According to the *SIR* model of epidemic theory, the relation between *λ* and *μ* is shown by the following equation. *Ω* is the effective linear meeting rate. The value of *μ* is refer to [11].

For Equation (2), we cancel the *dt*, having the following equation.

\left\{\begin{array}{c}\frac{dI}{dS}=\frac{1}{\Omega S}-1\hfill \\ I{|}_{s={s}_{0}}={I}_{0}\hfill \end{array}\right.

(5)

The integral expression of Equation (5) is shown by Equation (6).

dI=\left(\frac{1}{\Omega S}-1\right)dS\Rightarrow \underset{I\left(0\right)}{\overset{I}{\int}}dI=\underset{S\left(0\right)}{\overset{S}{\int}}\left(\frac{1}{\Omega S}-1\right)dS

(6)

By using the property of integral, we could have the solution of Equation (5). It is shown by Equation (7). By using this equation, we represent the *I* for the latter calculation.

I=\left({S}_{0}+{I}_{0}\right)-S+\frac{1}{\Omega}\text{ln}\frac{S}{{S}_{0}}

(7)

According to the *SIR* model of epidemic theory, we conclude ratio of the un-infective node during the packet forward process and denote the ratio as *S*_{
ω
}, it is shown in the following Equation (8).

{S}_{0}+{I}_{0}-{S}_{\omega}+\frac{1}{\Omega}\text{ln}\frac{{S}_{\omega}}{{S}_{0}}=0

(8)

In the process of epidemic spreading, the packet is forwarded by several relay nodes. Based on the above description, it is easy to conclude that the ratio of infective node is shown in the following Equation (9) and we denote it as *X*. Since *S*_{0} is the initial un-infection ratio, *S*_{
w
}is the un-infection ratio for the whole transferring process. Therefore, *X* is the infection ratio for whole process.

X={S}_{0}-{S}_{\omega}

(9)

From the Equation (8), at the beginning of spreading process, we think about that the initial *I*_{0} is small enough and we can omit it. Therefore, we have the following Equation (10).

X+\frac{1}{\Omega}\text{ln}\left(1-\frac{X}{{S}_{0}}\right)\approx 0

(10)

By using the Taylor expansion for above equation, we have the Equation (11). Through Equation (11), we can evaluate the number of infection nodes and the number of un-infection nodes when the packet copies reached the equilibrium point. Furthermore, based on the number of infection to evaluate the time of equilibrium point. The timer of nodes would notify that do not forward the copies anymore and switch to the many to one scheme.

X\left(1-\frac{1}{{S}_{0}\Omega}-\frac{X}{2{S}_{0}^{2}\Omega}\right)\approx 0

(11)

From the above description, we can evaluate the amount number of the un-infected node when the network gets epidemic spreading equilibrium point. When the SIR model is at the equilibrium point, the spreading speed of the packet would not be faster than the scheme of many-to one scheme. Therefore, the forwarding mode transfer to the many-to-one mode which means the nodes which carry the packet only send the packet to the destination instead of conventional epidemic theory. In the above description, we can conclude the amount number of the infective nodes. It is means that we can know which node carry the packet. Meanwhile, we need to know when the infective nodes should change their forwarding mode to many-to-one. In the following section, we will estimate the expected time from the original packet is generated to the time of the network get the epidemic equilibrium point.

In the process of epidemic forwarding, the source node generates packet which is needed to be relayed to the destination. Along with the random moving in the sense field, the source would send the packet to any node that it will meet. Furthermore, when the relay nodes received the original packet, the packet will be copied and stored in the buffer of the relay nodes. Then the relay nodes would send the copy to any node they will meet. This process is based on the epidemic theory. In our forwarding scheme, we introduce the two sub-process forwarding. Therefore, we need to know when the first sub-process should transfer to the second sub process. In this section, we introduce the single absorbing Markov chain to estimate the time of equilibrium point. Nodes stop the epidemic process is based on a timer. Nodes would predict the time when the forwarding scheme would transfer to many to one scheme, when the forwarding scheme transfer to many to one, then the forwarding activity switch to the second stage. For stage transition, the equilibrium point is not the optimal point. The equilibrium point is the infection ratio which is calculated by the encounter probability of nodes. When the packet copies reached this equilibrium point, the spreading of packet would not be faster than the scheme of many to one scheme. We also calculate the time when reached this equilibrium point by using Markov chain. Nodes stop the epidemic process is based on a timer. Nodes would predict the time when the forwarding scheme would transfer to many to one scheme, this special is the timer to stop the epidemic process.

According to the encounter probability, each node calculates the ratio of infection when the packet copies reached the equilibrium point. Then, by using the Markov chain, the node would predict the time when the packet copies reached the equilibrium point. When this time is reached, forwarding node will stop the forwarding process since the packet has been delivered to the destination.

In Figure 1, the numeric which in the circle denotes the number of the infective nodes at current time state. We formulate the numeric as the transition state for the single absorbing Markov chain. The states 1, 2, 3..., *n-* 2, *n-* 1, *n* denote the transient state in the chain. The *D* is the single absorbing state of above Markov chain. From the aspect of epidemic theory, every state means the infective node at time *t*, e.g., the state of *x* means that at time *t*, there are *x* nodes were infected. At the state of *n*, the network gets to the epidemic spreading equilibrium. For the absorbing state *D*, it denotes the destination of the packet. In the first sub-process which is based on epidemic theory, each node which has carried the duplicate packet will distribute the packet copy to other nodes which do not have this packet when they are within the transmission range. Therefore, when there are *x* copies of the packet in the network (*x* nodes are infected), the *x* + 1 copy is generated by the transition rate *λx* (*N*-*x*). At any state *x*, the transition rate of reaching the destination node for any one infective node could be denoted by *λx* (transition rate from *x* to *D*).

In order to separate the whole forwarding process into two sub-process, we need to let the nodes know that when the forwarding mode should transfer from the first sub-process into the second one. Therefore, we need to know the time when the epidemic forwarding gets to the spreading equilibrium point. According to the above Markov chain, we conclude that when there are *i* nodes are infected, the inter-meeting time which infected the next *i* + 1 node could denote by the following Equation (12) which is based on the epidemic SIR model. We cumulate the total expected time which from 0 infection node to *n*_{
th
}infection node. We define this expected time as *T*_{
n
}.

{t}_{i\to i+1}=\frac{1}{\lambda SI}

(12)

Denote the expect time from state 1 to *n* as *T*_{
n
}, we have the following Equation (13) to show the expression of *T*_{
n
}. The parameter *n* is the local information of node and it is the total number of infection nodes.

\begin{array}{ll}\hfill {T}_{n}& =\frac{1}{\left(N-1\right)\lambda}+\frac{1}{2\left(N-2\right)\lambda}+\cdots +\frac{1}{n\left(N-n\right)\lambda}\phantom{\rule{2em}{0ex}}\\ =\frac{1}{\lambda N}\sum _{i=1}^{n}\left(\frac{1}{i}+\frac{1}{N-i}\right)\phantom{\rule{2em}{0ex}}\end{array}

(13)