# Exact Markovian SIR and SIS epidemics on networks and an upper bound for the epidemic threshold

@article{Mieghem2014ExactMS, title={Exact Markovian SIR and SIS epidemics on networks and an upper bound for the epidemic threshold}, author={Piet Van Mieghem}, journal={arXiv: Dynamical Systems}, year={2014} }

Exploiting the power of the expectation operator and indicator (or Bernoulli) random variables, we present the exact governing equations for both the SIR and SIS epidemic models on \emph{networks}. Although SIR and SIS are basic epidemic models, deductions from their exact stochastic equations \textbf{without} making approximations (such as the common mean-field approximation) are scarce. An exact analytic solution of the governing equations is highly unlikely to be found (for any network) due…

## 39 Citations

### Approximate formula and bounds for the time-varying susceptible-infected-susceptible prevalence in networks.

- MathematicsPhysical review. E
- 2016

A formula for the epidemic threshold in the cycle (or ring graph) is presented and the challenge of finding tight upper bounds for the SIS (and SIR) epidemic threshold for all graphs is revisited.

### Accuracy criterion for the mean-field approximation in susceptible-infected-susceptible epidemics on networks.

- Mathematics
- 2015

Mean-field approximations (MFAs) are frequently used in physics. When a process (such as an epidemic or a synchronization) on a network is approximated by MFA, a major hurdle is the determination of…

### Epidemics on networks with heterogeneous population and stochastic infection rates.

- MathematicsMathematical biosciences
- 2016

### Power-law decay in epidemics is likely due to interactions with the time-variant contact graph

- Mathematics
- 2020

Reported COVID-19 data from o¢ cial health agencies in several countries suggest that the prevalence, the average fraction of infected people, decays over time with power-law tails. Moreover, the…

### Heterogeneous SIS model for directed networks and optimal immunization

- Mathematics, Computer ScienceArXiv
- 2016

This work investigates the influence of a contact network structure over the spread of epidemics in an heterogeneous population using a first-order mean-field approximation and proves that the positive steady-state of the original system, that appears above the threshold, can be computed by this lower-dimensional system.

### Heterogeneous SIS model for directed networks and optimal curing policy

- Mathematics, Computer Science
- 2016

This work investigates the influence of the contact network structur e on the spread of epidemics over an heterogeneous population and proves that the epide mic threshold can be computed using a lower-dimensional dynamical system.

### Universality of the SIS prevalence in networks

- MathematicsArXiv
- 2016

A new analysis of the prevalence, the expected number of infected nodes in a network, is presented and physically interpreted and leads to a universal, analytic curve, that can bound the time-varying prevalence in any finite time interval.

### An Epidemic Perspective on the Cut Size in Networks

- Mathematics
- 2018

An epidemic spreads over the network via infectious links between healthy and infected nodes. The rate of increase in the number of infected nodes depends on the number of infectious links, called…

### Time to Extinction for the SIS Epidemic Model: New Bounds on the Tail Probabilities

- MathematicsIEEE Transactions on Network Science and Engineering
- 2019

It is proved that, for an effective infection rate above a threshold depending on the topology of the network, the time to extinction grows exponentially in the size of the population, with probability converging to 1 as the number of population grows large.

### Edge Deletion Algorithms for Minimizing Spread in SIR Epidemic Models

- MathematicsSIAM J. Control. Optim.
- 2022

Under moderate assumptions on the reproduction number, it is proved that the infection numbers are bounded by supermodular functions in the D-Sir model and the IC-SIR model for large classes of random networks.

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