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The authors have declared that no competing interests exist.

Conceived and designed the experiments: LX CS. Performed the experiments: LX CS. Analyzed the data: LX LWC HMS CS. Contributed reagents/materials/analysis tools: LX LWC HMS CS. Wrote the paper: LX LWC HMS CS.

Rift Valley fever is a vector-borne zoonotic disease which causes high morbidity and mortality in livestock. In the event Rift Valley fever virus is introduced to the United States or other non-endemic areas, understanding the potential patterns of spread and the areas at risk based on disease vectors and hosts will be vital for developing mitigation strategies. Presented here is a general network-based mathematical model of Rift Valley fever. Given a lack of empirical data on disease vector species and their vector competence, this discrete time epidemic model uses stochastic parameters following several PERT distributions to model the dynamic interactions between hosts and likely North American mosquito vectors in dispersed geographic areas. Spatial effects and climate factors are also addressed in the model. The model is applied to a large directed asymmetric network of 3,621 nodes based on actual farms to examine a hypothetical introduction to some counties of Texas, an important ranching area in the United States of America. The nodes of the networks represent livestock farms, livestock markets, and feedlots, and the links represent cattle movements and mosquito diffusion between different nodes. Cattle and mosquito (

Rift Valley fever (RVF) was first identified in Egypt in 1931

Animal movements, typically motivated by livestock trading and marketing may accelerate the transmission of zoonotic diseases among animal holdings which may cover a vast area

Humans can acquire the infection from the bites of infected mosquitoes or directly from contact with the bodily fluids of infected animals

Epidemiological modeling plays an important role in planning, implementing, and evaluating detection, control, and prevention programs

Proposed here is a deterministic network-based RVF virus transmission model with stochastic parameters. Two competent vector populations:

(A) A hypothetical mosquito diffusion network demonstrating how mosquito move to farms that are smaller than 2 km away. (B) Livestock move bidirectionally between livestock farms and livestock markets but only move unidirectionally to feedlots as demonstrated in the livestock movement network.

The compartmental models are adapted to represent the status of each population during a simulated RVF virus transmission. The models are built based on the principle of the RVF virus transmission flow diagram illustrated in

Para-meter | Description | Range | Assumed most possible value | Units | Source |

contact rate: |
0.1392 | ||||

contact rate: livestock to |
0.1225 | ||||

contact rate: livestock to |
0.16 | ||||

contact rate: |
0.04 | ||||

contact rate: |
0.0015 | Assume | |||

contact rate: livestock to humans | 0.00006 | Assume | |||

contact rate: |
0.000525 | Assume | |||

recovery period in livestock | 3.5 | ||||

recovery period in humans | 5.5 | ||||

longevity of |
31.5 | days | |||

longevity of livestock | 1980 | days | |||

longevity of |
31.5 | days | |||

birth rate of |
weather dependent | ||||

birth rate of livestock | |||||

birth rate of |
weather dependent | ||||

incubation period in |
6 | days | |||

incubation period in livestock | 4 | days | |||

incubation period in |
6 | days | |||

incubation period in humans | 4 | days | |||

mortality rate in livestock | 0.0375 | ||||

transovarial transmission rate in |
0.05 | ||||

development period of |
weather dependent | days | |||

development period of |
weather dependent | days | |||

carrying capacity of |
Assume | ||||

carrying capacity of livestock | Assume | ||||

carrying capacity of |
Assume | ||||

reduction in |
Assume |

There are

There are

The daily number of newborn livestock in location

The number of humans in each node is constant because birth, death, mortality, and mobility of humans are not considered. The number of humans infected by

As a case study, various RVF virus introduction scenarios were tested using the model to determine the hypothetical model outcomes (number of livestock cases and timing of the epidemic). Although the model accounts for their exact locations when simulating RVF virus spread, we do not report any of this information or even discuss ranches in areas smaller than county level. The exact farms and counties are very well masked from the results. Texas cattle ranches were selected because they have large cattle concentrations and we have aggregate survey data on cattle movements in these areas

Range | Source | ||

farm | market | ||

market | farm | ||

farm | feedlot | ||

market | feedlot | ||

feedlot | farm | ||

feedlot | market |

For mosquito diffusion, if the distance between two farms is smaller than an assumed radius, two kilometers, then there is a link between the nodes in the network. The diffusion rates of

Vector competence varies within and between mosquito species

The egg laying rates of

(A) The egg laying rates of

Parameter | Description | Value | Source |

parameter in |
|||

parameter in |
|||

parameter in |
|||

parameter in |
|||

parameter in |
|||

parameter in |
|||

parameter in |
|||

parameter in |
|||

minimum constant fecundity rate | |||

maximum daily egg laying rate | |||

the mean of the daily egg laying rate | |||

variance of function |

Presented is a discrete time compartmental mathematical model based on a network approach. Rift Valley fever is transmitted by several species of mosquito vectors that have varying levels of vector competence; therefore, each genus and species combination requires modeling the vector competence, movement, and life stage development patterns which is too complicated while considering only a single species or genus is not accurate. Consequently, the species are loosely grouped as their genera and the parameters are allowed to vary following PERT distributions. The distribution captures uncertainties on inherent variability between species, as well as variability among individual mosquitoes. The mosquito parameters are functions of climate factors to reflect the impact of climate and season on mosquito dynamics. Only

Different networks are developed for mosquito diffusion and livestock movement considering heterogeneity in both. In the cattle movement network, different types of nodes distinguish between sources, sinks, and transitions.

The model can be used to simulate networks with the number of nodes up to thousands with the easily solvable discrete time model. To use the model in any location, one only needs the initial populations, the movement rates, ranges of the parameters, and climate factors in each location to obtain the epidemic curve.

Sixteen initial conditions shown in

Farm size | Quantity | Infected | |||

Cattle | |||||

Small | Few | Cattle-f-s | |||

Many | Cattle-m-s | ||||

Large | Few | Cattle-f-l | |||

Many | Cattle-m-l |

Initial | source | of | infection | |||

Farm size | Initial infection size | Outcome characteristics | Cattle | |||

Small | Few ( |
average | small | very small | very small | |

very large | very large | large | average | |||

very large | very large | average | very small | |||

very long | very long | long | medium | |||

medium | long | very long | short | |||

Many ( |
very small | large | very large | average | ||

average | small | very small | small | |||

very small | small | average | very small | |||

short | short | short | short | |||

short | very short | very short | very short | |||

Large | Few ( |
very small | very small | very small | small | |

very large | large | average | very large | |||

very small | small | very small | average | |||

long | long | short | very long | |||

very long | medium | short | long | |||

Many ( |
very large | very large | very large | very small | ||

very small | small | small | large | |||

average | large | average | small | |||

short | very short | very short | long | |||

very short | short | short | medium |

The suffix l or s, (which denote large or small farms) were removed from the initial condition labels when comparing results with different initial infections in the same scale of initial location. The impact of the Rift Valley fever epidemic in terms of infected cattle depends on the size of the initial infection.

When the initial condition of the outbreak is assumed to be

When the initial condition of the outbreak is assumed to be Cattle-f (few cattle), the simulations result in a larger cumulative number of infected cattle than the ones obtained in the case of Cattle-m (many cattle).

The total number of infected humans and the total number of farms with at least one infected human remain fewer than one regardless of initial infection conditions. This is likely because the human population of each farm is assumed to be fewer than

The temporal characteristics of Rift Valley fever cases followed the general trend that fewer infected individuals in the initial introduction resulted in a delayed epidemic peak. When the initial condition of the outbreak is assumed to be

The original meta-population model for Rift Valley fever described by

The model can be used to study not only local transmission between hosts and vectors, but also trans-location transmission of RVF virus with the network approach. The roles of mosquitoes and livestock in RVF virus transmission can be studied independently because they have separate networks. One infected farm node can spread the infection to other nodes connected to it; therefore, more nodes can be infected over time. The temporal and spatial evolution of RVF virus and its driving force can be analyzed. The spread of RVF virus is estimated within farms as well as between farms, markets, and feedlots. The goal of the simulation analysis is to provide insights into possible pathways for rapid spread of RVF virus among farms and counties. Using the cattle networks, the impact of cattle movement from trade can be investigated as newborn calves mature to weaning and on to harvest. The cattle farms are the source nodes where the cattle are born and raised for several months before being sold through markets or direct to feedlots, or to other farms as stockers or replacement females. Cattle on an infected farm may become infected and then carry the virus to the livestock market or else transition nodes before being sold to another farm, which may introduce the virus to a new farm. On the other hand, infected cattle movement to feedlots (sink nodes) does not propagate the transmission because there is no further transfer of cattle from the nodes except onto slaughter. Different mitigation strategies can be applied according to each node type (source, sink, and transition) within livestock movement network.

Discrete time modeling is appealing in the way it describes the epidemic process, which is conceptualized as evolving through a set of discrete time epochs instead of continuously

In large populations, with a large scale of epidemic incidence, deterministic models can provide good approximations

Concerning the discussion of simulation results,

Time to peak infection is the time until the maximal number of cases is observed and epidemic duration is the amount of time an epidemic persists.

Cattle can be spreaders of virus because they are frequently bought and sold

There are no human cases (integers) in the simulations regardless of initial starting conditions because of the small constant human population in each node of the study region. In high population areas, there can be a large number of human cases. Humans are often exposed to fewer mosquitoes than cattle, especially in more developed countries, which results in lower probability of being infected by mosquitoes. The probability that humans are infected by cattle is also low in this region because the model does not account for contact with the virus via animal slaughter. Hence, the number of infected humans in each farm produced by simulations is fewer than

In conclusion, the general epidemiological trend of a smaller initial infection observed through various simulations with various initial staring locations is:

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We are grateful to Kenneth J. Linthicum for his suggestions on mosquito models, and we are thankful to Bo Norby, Doyle Fuchs, Bryanna Pockrandt, and Phillip Schumm for their help in producing this work. We gratefully thank two anonymous referees for their valuable comments and suggestions which lead to an improvement of our manuscript.