# Hop-by-Hop Worm Propagation with Carryover Epidemic Model in Mobile Sensor Networks

## Abstract

**:**

## 1. Introduction

## 2. Related Work

## 3. Assumptions

## 4. Hop-By-Hop Worm Propagation in Mobile Sensor Networks

## 5. Analysis

**Lemma 1.**In carryover epidemic model, the cumulative number of infected nodes from the 0th time slot to the τth time slot is ${I}_{\tau}-{q}_{\tau}$.

**Proof.**Recall that ${I}_{t+1}-{I}_{t}+{q}_{t}$ is the infection quota in the $t+1$th time slot ($t\ge 1$) in carryover epidemic model, where ${I}_{1}-{I}_{0}$ is the infection quota in the first time slot and ${q}_{t}$ is the infection quota deficit in the tth time slot. The number of infected nodes in a time slot is the difference between the infection quota and the quota deficit in a time slot. Hence, the cumulative number of infected nodes from the 0th time slot to the τth time slot is given by:

**Lemma 2.**If $j>\tau $ and $\mu \le j\left(\frac{\tau}{j-\tau}\right)ln\frac{j}{\tau}$, then

**Proof.**For any $y>0$, the Markov’s inequality is given by:

**Lemma 3.**If $j<\tau +1$ and $\mu \ge j\left(\frac{\tau +1}{j-\tau -1}\right)ln\left(\frac{j}{\tau +1}\right)$, then

**Proof.**For any $y<0$, the Markov’s inequality is given by:

## 6. Simulation Study

#### 6.1. Simulation Configurations

#### 6.2. Simulation Results

## 7. Conclusions

## Acknowledgments

## Conflicts of Interest

## References

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**MDPI and ACS Style**

Ho, J.-W. Hop-by-Hop Worm Propagation with Carryover Epidemic Model in Mobile Sensor Networks. *Computers* **2015**, *4*, 283-292.
https://doi.org/10.3390/computers4040283

**AMA Style**

Ho J-W. Hop-by-Hop Worm Propagation with Carryover Epidemic Model in Mobile Sensor Networks. *Computers*. 2015; 4(4):283-292.
https://doi.org/10.3390/computers4040283

**Chicago/Turabian Style**

Ho, Jun-Won. 2015. "Hop-by-Hop Worm Propagation with Carryover Epidemic Model in Mobile Sensor Networks" *Computers* 4, no. 4: 283-292.
https://doi.org/10.3390/computers4040283