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idalab seminar #9 When you're sick, please stay at home

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Scientists have been studying spreading mechanisms of epidemic diseases for almost a century and obtained concise results for model systems where everybody is always in contact with everybody else. As the reality of such systems can be doubted, we give a broad overview on how exploring the temporal, social and spatial nature of human contacts yields better descriptions and surprising effects, such as accelerated spreading on social networks, smart vaccination strategies, and ways to predict when a pandemic disease will hit your home town. We show how modern studies record actual face-to-face contact networks and introduce a new model for the systematic search for outbreak mitigation strategies, yielding a simple result: Even the most infectious disease can be killed if you'd just stay at home.

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idalab seminar #9 When you're sick, please stay at home

  1. 1. © 2018 | idalab GmbH | Potsdamer Straße 68 | 10785 Berlin | idalab.de page 1 | confidential Agency for Data Science Machine learning & AI Mathematical modelling Data strategy Benjamin F. Maier When you’re sick, please stay at home Making sense of spreading phenomena using human mobility and contact data idalab seminar #9 | April 27th 2018
  2. 2. When you're sick, please stay at home Making sense of spreading phenomena using human mobility and contact data Benjamin F. Maier // 27.04.2018 // idalab
  3. 3. Black plague https://bit.ly/2vQDLpq • imagine you're a "doctor" in middle- age Europe working for the King • a traveller tells you a strange disease just popped up • people get high fevers, black patches of skin and then die
  4. 4. The King asks When shall the plague be-eth here?
  5. 5. The King asks
  6. 6. The King asks How many will-eth pass away?
  7. 7. The King asks
  8. 8. The King asks How shall us prepare-th?
  9. 9. The King asks Ooh I know that one!
  10. 10. The King is satisfied
  11. 11. The King is satisfied
  12. 12. And just like that • 25% of Europe's population dies • Hamburg loses 70% of its population • society crumbles (crime and violence soars) • 200-300 years living of constant fear of death copyright W. L. Langer, Scientific American, 1964.
  13. 13. Modern Epidemiology fortunately we got smarter but the questions stay the same where? when? how many? what can we do? how can modeling and statistical physics help?
  14. 14. 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  15. 15. Are we lacking chemistry? three kinds of chemicals R<latexit sha1_base64="aunByyIhdEN7vj54N2KmKwtqfTc=">AAAB/3icbVDLSsNAFL3xWeur6tJNsAiuSiKCLgtuBDet2Ae0oUymN+3QySTMTNQSunDtVr/Bnbj1U/wE/8JpmoVtPTBwOOdezp3jx5wp7Tjf1srq2vrGZmGruL2zu7dfOjhsqiiRFBs04pFs+0QhZwIbmmmO7VgiCX2OLX90PfVbDygVi8S9HsfohWQgWMAo0Uaq3/VKZafiZLCXiZuTMuSo9Uo/3X5EkxCFppwo1XGdWHspkZpRjpNiN1EYEzoiA+wYKkiIykuzQyf2qVH6dhBJ84S2M/XvRkpCpcahbyZDoodq0ZuK/3mdRAdXXspEnGgUdBYUJNzWkT39td1nEqnmY0MIlczcatMhkYRq081cisBH/ZTlFE037mITy6R5XnGdilu/KFdv85YKcAwncAYuXEIVbqAGDaCA8AKv8GY9W+/Wh/U5G12x8p0jmIP19QsKbZbA</latexit><latexit sha1_base64="aunByyIhdEN7vj54N2KmKwtqfTc=">AAAB/3icbVDLSsNAFL3xWeur6tJNsAiuSiKCLgtuBDet2Ae0oUymN+3QySTMTNQSunDtVr/Bnbj1U/wE/8JpmoVtPTBwOOdezp3jx5wp7Tjf1srq2vrGZmGruL2zu7dfOjhsqiiRFBs04pFs+0QhZwIbmmmO7VgiCX2OLX90PfVbDygVi8S9HsfohWQgWMAo0Uaq3/VKZafiZLCXiZuTMuSo9Uo/3X5EkxCFppwo1XGdWHspkZpRjpNiN1EYEzoiA+wYKkiIykuzQyf2qVH6dhBJ84S2M/XvRkpCpcahbyZDoodq0ZuK/3mdRAdXXspEnGgUdBYUJNzWkT39td1nEqnmY0MIlczcatMhkYRq081cisBH/ZTlFE037mITy6R5XnGdilu/KFdv85YKcAwncAYuXEIVbqAGDaCA8AKv8GY9W+/Wh/U5G12x8p0jmIP19QsKbZbA</latexit><latexit sha1_base64="aunByyIhdEN7vj54N2KmKwtqfTc=">AAAB/3icbVDLSsNAFL3xWeur6tJNsAiuSiKCLgtuBDet2Ae0oUymN+3QySTMTNQSunDtVr/Bnbj1U/wE/8JpmoVtPTBwOOdezp3jx5wp7Tjf1srq2vrGZmGruL2zu7dfOjhsqiiRFBs04pFs+0QhZwIbmmmO7VgiCX2OLX90PfVbDygVi8S9HsfohWQgWMAo0Uaq3/VKZafiZLCXiZuTMuSo9Uo/3X5EkxCFppwo1XGdWHspkZpRjpNiN1EYEzoiA+wYKkiIykuzQyf2qVH6dhBJ84S2M/XvRkpCpcahbyZDoodq0ZuK/3mdRAdXXspEnGgUdBYUJNzWkT39td1nEqnmY0MIlczcatMhkYRq081cisBH/ZTlFE037mITy6R5XnGdilu/KFdv85YKcAwncAYuXEIVbqAGDaCA8AKv8GY9W+/Wh/U5G12x8p0jmIP19QsKbZbA</latexit><latexit sha1_base64="aunByyIhdEN7vj54N2KmKwtqfTc=">AAAB/3icbVDLSsNAFL3xWeur6tJNsAiuSiKCLgtuBDet2Ae0oUymN+3QySTMTNQSunDtVr/Bnbj1U/wE/8JpmoVtPTBwOOdezp3jx5wp7Tjf1srq2vrGZmGruL2zu7dfOjhsqiiRFBs04pFs+0QhZwIbmmmO7VgiCX2OLX90PfVbDygVi8S9HsfohWQgWMAo0Uaq3/VKZafiZLCXiZuTMuSo9Uo/3X5EkxCFppwo1XGdWHspkZpRjpNiN1EYEzoiA+wYKkiIykuzQyf2qVH6dhBJ84S2M/XvRkpCpcahbyZDoodq0ZuK/3mdRAdXXspEnGgUdBYUJNzWkT39td1nEqnmY0MIlczcatMhkYRq081cisBH/ZTlFE037mITy6R5XnGdilu/KFdv85YKcAwncAYuXEIVbqAGDaCA8AKv8GY9W+/Wh/U5G12x8p0jmIP19QsKbZbA</latexit> S<latexit sha1_base64="KrCMcC1AG+T5ZWYfQEih7JdvhQw=">AAAB/3icbVDLSsNAFL3xWeur6tJNsAiuSiKCLotuXLZoH9CGMpnetEMnkzAzUUvowrVb/QZ34tZP8RP8C6dpFrb1wMDhnHs5d44fc6a043xbK6tr6xubha3i9s7u3n7p4LCpokRSbNCIR7LtE4WcCWxopjm2Y4kk9Dm2/NHN1G89oFQsEvd6HKMXkoFgAaNEG6l+1yuVnYqTwV4mbk7KkKPWK/10+xFNQhSacqJUx3Vi7aVEakY5TordRGFM6IgMsGOoICEqL80OndinRunbQSTNE9rO1L8bKQmVGoe+mQyJHqpFbyr+53USHVx5KRNxolHQWVCQcFtH9vTXdp9JpJqPDSFUMnOrTYdEEqpNN3MpAh/1U5ZTNN24i00sk+Z5xXUqbv2iXL3OWyrAMZzAGbhwCVW4hRo0gALCC7zCm/VsvVsf1udsdMXKd45gDtbXLwlTlrg=</latexit><latexit sha1_base64="KrCMcC1AG+T5ZWYfQEih7JdvhQw=">AAAB/3icbVDLSsNAFL3xWeur6tJNsAiuSiKCLotuXLZoH9CGMpnetEMnkzAzUUvowrVb/QZ34tZP8RP8C6dpFrb1wMDhnHs5d44fc6a043xbK6tr6xubha3i9s7u3n7p4LCpokRSbNCIR7LtE4WcCWxopjm2Y4kk9Dm2/NHN1G89oFQsEvd6HKMXkoFgAaNEG6l+1yuVnYqTwV4mbk7KkKPWK/10+xFNQhSacqJUx3Vi7aVEakY5TordRGFM6IgMsGOoICEqL80OndinRunbQSTNE9rO1L8bKQmVGoe+mQyJHqpFbyr+53USHVx5KRNxolHQWVCQcFtH9vTXdp9JpJqPDSFUMnOrTYdEEqpNN3MpAh/1U5ZTNN24i00sk+Z5xXUqbv2iXL3OWyrAMZzAGbhwCVW4hRo0gALCC7zCm/VsvVsf1udsdMXKd45gDtbXLwlTlrg=</latexit><latexit sha1_base64="KrCMcC1AG+T5ZWYfQEih7JdvhQw=">AAAB/3icbVDLSsNAFL3xWeur6tJNsAiuSiKCLotuXLZoH9CGMpnetEMnkzAzUUvowrVb/QZ34tZP8RP8C6dpFrb1wMDhnHs5d44fc6a043xbK6tr6xubha3i9s7u3n7p4LCpokRSbNCIR7LtE4WcCWxopjm2Y4kk9Dm2/NHN1G89oFQsEvd6HKMXkoFgAaNEG6l+1yuVnYqTwV4mbk7KkKPWK/10+xFNQhSacqJUx3Vi7aVEakY5TordRGFM6IgMsGOoICEqL80OndinRunbQSTNE9rO1L8bKQmVGoe+mQyJHqpFbyr+53USHVx5KRNxolHQWVCQcFtH9vTXdp9JpJqPDSFUMnOrTYdEEqpNN3MpAh/1U5ZTNN24i00sk+Z5xXUqbv2iXL3OWyrAMZzAGbhwCVW4hRo0gALCC7zCm/VsvVsf1udsdMXKd45gDtbXLwlTlrg=</latexit><latexit sha1_base64="KrCMcC1AG+T5ZWYfQEih7JdvhQw=">AAAB/3icbVDLSsNAFL3xWeur6tJNsAiuSiKCLotuXLZoH9CGMpnetEMnkzAzUUvowrVb/QZ34tZP8RP8C6dpFrb1wMDhnHs5d44fc6a043xbK6tr6xubha3i9s7u3n7p4LCpokRSbNCIR7LtE4WcCWxopjm2Y4kk9Dm2/NHN1G89oFQsEvd6HKMXkoFgAaNEG6l+1yuVnYqTwV4mbk7KkKPWK/10+xFNQhSacqJUx3Vi7aVEakY5TordRGFM6IgMsGOoICEqL80OndinRunbQSTNE9rO1L8bKQmVGoe+mQyJHqpFbyr+53USHVx5KRNxolHQWVCQcFtH9vTXdp9JpJqPDSFUMnOrTYdEEqpNN3MpAh/1U5ZTNN24i00sk+Z5xXUqbv2iXL3OWyrAMZzAGbhwCVW4hRo0gALCC7zCm/VsvVsf1udsdMXKd45gDtbXLwlTlrg=</latexit> I<latexit sha1_base64="h96kL/iU2TRh3mE1gzyXOyYaoiY=">AAAB/3icbVDLSsNAFJ34rPVVdekmWARXJRFBl0U3umvBPqANZTK9aYdOJmHmRi2hC9du9RvciVs/xU/wL5ymWdjWAwOHc+7l3Dl+LLhGx/m2VlbX1jc2C1vF7Z3dvf3SwWFTR4li0GCRiFTbpxoEl9BAjgLasQIa+gJa/uhm6rceQGkeyXscx+CFdCB5wBlFI9XveqWyU3Ey2MvEzUmZ5Kj1Sj/dfsSSECQyQbXuuE6MXkoVciZgUuwmGmLKRnQAHUMlDUF7aXboxD41St8OImWeRDtT/26kNNR6HPpmMqQ41IveVPzP6yQYXHkpl3GCINksKEiEjZE9/bXd5woYirEhlClubrXZkCrK0HQzlyLhEZ+ynKLpxl1sYpk0zyuuU3HrF+Xqdd5SgRyTE3JGXHJJquSW1EiDMALkhbySN+vZerc+rM/Z6IqV7xyROVhfv/k2lq4=</latexit><latexit sha1_base64="h96kL/iU2TRh3mE1gzyXOyYaoiY=">AAAB/3icbVDLSsNAFJ34rPVVdekmWARXJRFBl0U3umvBPqANZTK9aYdOJmHmRi2hC9du9RvciVs/xU/wL5ymWdjWAwOHc+7l3Dl+LLhGx/m2VlbX1jc2C1vF7Z3dvf3SwWFTR4li0GCRiFTbpxoEl9BAjgLasQIa+gJa/uhm6rceQGkeyXscx+CFdCB5wBlFI9XveqWyU3Ey2MvEzUmZ5Kj1Sj/dfsSSECQyQbXuuE6MXkoVciZgUuwmGmLKRnQAHUMlDUF7aXboxD41St8OImWeRDtT/26kNNR6HPpmMqQ41IveVPzP6yQYXHkpl3GCINksKEiEjZE9/bXd5woYirEhlClubrXZkCrK0HQzlyLhEZ+ynKLpxl1sYpk0zyuuU3HrF+Xqdd5SgRyTE3JGXHJJquSW1EiDMALkhbySN+vZerc+rM/Z6IqV7xyROVhfv/k2lq4=</latexit><latexit sha1_base64="h96kL/iU2TRh3mE1gzyXOyYaoiY=">AAAB/3icbVDLSsNAFJ34rPVVdekmWARXJRFBl0U3umvBPqANZTK9aYdOJmHmRi2hC9du9RvciVs/xU/wL5ymWdjWAwOHc+7l3Dl+LLhGx/m2VlbX1jc2C1vF7Z3dvf3SwWFTR4li0GCRiFTbpxoEl9BAjgLasQIa+gJa/uhm6rceQGkeyXscx+CFdCB5wBlFI9XveqWyU3Ey2MvEzUmZ5Kj1Sj/dfsSSECQyQbXuuE6MXkoVciZgUuwmGmLKRnQAHUMlDUF7aXboxD41St8OImWeRDtT/26kNNR6HPpmMqQ41IveVPzP6yQYXHkpl3GCINksKEiEjZE9/bXd5woYirEhlClubrXZkCrK0HQzlyLhEZ+ynKLpxl1sYpk0zyuuU3HrF+Xqdd5SgRyTE3JGXHJJquSW1EiDMALkhbySN+vZerc+rM/Z6IqV7xyROVhfv/k2lq4=</latexit><latexit sha1_base64="h96kL/iU2TRh3mE1gzyXOyYaoiY=">AAAB/3icbVDLSsNAFJ34rPVVdekmWARXJRFBl0U3umvBPqANZTK9aYdOJmHmRi2hC9du9RvciVs/xU/wL5ymWdjWAwOHc+7l3Dl+LLhGx/m2VlbX1jc2C1vF7Z3dvf3SwWFTR4li0GCRiFTbpxoEl9BAjgLasQIa+gJa/uhm6rceQGkeyXscx+CFdCB5wBlFI9XveqWyU3Ey2MvEzUmZ5Kj1Sj/dfsSSECQyQbXuuE6MXkoVciZgUuwmGmLKRnQAHUMlDUF7aXboxD41St8OImWeRDtT/26kNNR6HPpmMqQ41IveVPzP6yQYXHkpl3GCINksKEiEjZE9/bXd5woYirEhlClubrXZkCrK0HQzlyLhEZ+ynKLpxl1sYpk0zyuuU3HrF+Xqdd5SgRyTE3JGXHJJquSW1EiDMALkhbySN+vZerc+rM/Z6IqV7xyROVhfv/k2lq4=</latexit> dS dt = −⌘SI dI dt = +⌘SI − ⇢I dR dt = +⇢I <latexit sha1_base64="S+teTb8/z3+oE0UNaWjndjViPD8=">AAACjHicbVDdahQxGM2OP62r1q3eCL0JLoogXWakUEEKBaFYvKmt2xY2w5LJfLMbmp8h+aZ2GeblfIu+gbf6BGa307JtPRA4Oed8fMnJSiU9xvFlJ3rw8NHjldUn3afPnq+96K2/PPa2cgKGwirrTjPuQUkDQ5So4LR0wHWm4CQ7+zL3T87BeWnND5yVkGo+MbKQgmOQxj3GCsdFnR81dY4NfbdDNxkgp0f7jHVbb//G+9B6dJMyN7V0KXS4FLq2xr1+PIgXoPdJ0pI+aXEwXu+8ZrkVlQaDQnHvR0lcYlpzh1IoaLqs8lByccYnMArUcA0+rRc1NPRtUHJaWBeOQbpQlydqrr2f6SwkNcepv+vNxf95owqLT2ktTVkhGHG1qKgURUvnndJcOhCoZoFw4WR4KxVTHmrB0PytLR7lRfiFgZ/Cas1NXjO4KM+bUZLWTEGBTHEzUUD7CXNyMkXmFvdm3mVyt7n75PjjIIkHyfet/u63ttVVskHekPckIdtkl3wlB2RIBPlFfpM/5G+0Fm1Fn6Odq2jUaWdekVuI9v4Bl37FlA==</latexit><latexit sha1_base64="S+teTb8/z3+oE0UNaWjndjViPD8=">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</latexit><latexit sha1_base64="S+teTb8/z3+oE0UNaWjndjViPD8=">AAACjHicbVDdahQxGM2OP62r1q3eCL0JLoogXWakUEEKBaFYvKmt2xY2w5LJfLMbmp8h+aZ2GeblfIu+gbf6BGa307JtPRA4Oed8fMnJSiU9xvFlJ3rw8NHjldUn3afPnq+96K2/PPa2cgKGwirrTjPuQUkDQ5So4LR0wHWm4CQ7+zL3T87BeWnND5yVkGo+MbKQgmOQxj3GCsdFnR81dY4NfbdDNxkgp0f7jHVbb//G+9B6dJMyN7V0KXS4FLq2xr1+PIgXoPdJ0pI+aXEwXu+8ZrkVlQaDQnHvR0lcYlpzh1IoaLqs8lByccYnMArUcA0+rRc1NPRtUHJaWBeOQbpQlydqrr2f6SwkNcepv+vNxf95owqLT2ktTVkhGHG1qKgURUvnndJcOhCoZoFw4WR4KxVTHmrB0PytLR7lRfiFgZ/Cas1NXjO4KM+bUZLWTEGBTHEzUUD7CXNyMkXmFvdm3mVyt7n75PjjIIkHyfet/u63ttVVskHekPckIdtkl3wlB2RIBPlFfpM/5G+0Fm1Fn6Odq2jUaWdekVuI9v4Bl37FlA==</latexit><latexit sha1_base64="S+teTb8/z3+oE0UNaWjndjViPD8=">AAACjHicbVDdahQxGM2OP62r1q3eCL0JLoogXWakUEEKBaFYvKmt2xY2w5LJfLMbmp8h+aZ2GeblfIu+gbf6BGa307JtPRA4Oed8fMnJSiU9xvFlJ3rw8NHjldUn3afPnq+96K2/PPa2cgKGwirrTjPuQUkDQ5So4LR0wHWm4CQ7+zL3T87BeWnND5yVkGo+MbKQgmOQxj3GCsdFnR81dY4NfbdDNxkgp0f7jHVbb//G+9B6dJMyN7V0KXS4FLq2xr1+PIgXoPdJ0pI+aXEwXu+8ZrkVlQaDQnHvR0lcYlpzh1IoaLqs8lByccYnMArUcA0+rRc1NPRtUHJaWBeOQbpQlydqrr2f6SwkNcepv+vNxf95owqLT2ktTVkhGHG1qKgURUvnndJcOhCoZoFw4WR4KxVTHmrB0PytLR7lRfiFgZ/Cas1NXjO4KM+bUZLWTEGBTHEzUUD7CXNyMkXmFvdm3mVyt7n75PjjIIkHyfet/u63ttVVskHekPckIdtkl3wlB2RIBPlFfpM/5G+0Fm1Fn6Odq2jUaWdekVuI9v4Bl37FlA==</latexit>
  16. 16. Now imagine the following three kinds of chemicals R<latexit sha1_base64="aunByyIhdEN7vj54N2KmKwtqfTc=">AAAB/3icbVDLSsNAFL3xWeur6tJNsAiuSiKCLgtuBDet2Ae0oUymN+3QySTMTNQSunDtVr/Bnbj1U/wE/8JpmoVtPTBwOOdezp3jx5wp7Tjf1srq2vrGZmGruL2zu7dfOjhsqiiRFBs04pFs+0QhZwIbmmmO7VgiCX2OLX90PfVbDygVi8S9HsfohWQgWMAo0Uaq3/VKZafiZLCXiZuTMuSo9Uo/3X5EkxCFppwo1XGdWHspkZpRjpNiN1EYEzoiA+wYKkiIykuzQyf2qVH6dhBJ84S2M/XvRkpCpcahbyZDoodq0ZuK/3mdRAdXXspEnGgUdBYUJNzWkT39td1nEqnmY0MIlczcatMhkYRq081cisBH/ZTlFE037mITy6R5XnGdilu/KFdv85YKcAwncAYuXEIVbqAGDaCA8AKv8GY9W+/Wh/U5G12x8p0jmIP19QsKbZbA</latexit><latexit sha1_base64="aunByyIhdEN7vj54N2KmKwtqfTc=">AAAB/3icbVDLSsNAFL3xWeur6tJNsAiuSiKCLgtuBDet2Ae0oUymN+3QySTMTNQSunDtVr/Bnbj1U/wE/8JpmoVtPTBwOOdezp3jx5wp7Tjf1srq2vrGZmGruL2zu7dfOjhsqiiRFBs04pFs+0QhZwIbmmmO7VgiCX2OLX90PfVbDygVi8S9HsfohWQgWMAo0Uaq3/VKZafiZLCXiZuTMuSo9Uo/3X5EkxCFppwo1XGdWHspkZpRjpNiN1EYEzoiA+wYKkiIykuzQyf2qVH6dhBJ84S2M/XvRkpCpcahbyZDoodq0ZuK/3mdRAdXXspEnGgUdBYUJNzWkT39td1nEqnmY0MIlczcatMhkYRq081cisBH/ZTlFE037mITy6R5XnGdilu/KFdv85YKcAwncAYuXEIVbqAGDaCA8AKv8GY9W+/Wh/U5G12x8p0jmIP19QsKbZbA</latexit><latexit 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sha1_base64="aunByyIhdEN7vj54N2KmKwtqfTc=">AAAB/3icbVDLSsNAFL3xWeur6tJNsAiuSiKCLgtuBDet2Ae0oUymN+3QySTMTNQSunDtVr/Bnbj1U/wE/8JpmoVtPTBwOOdezp3jx5wp7Tjf1srq2vrGZmGruL2zu7dfOjhsqiiRFBs04pFs+0QhZwIbmmmO7VgiCX2OLX90PfVbDygVi8S9HsfohWQgWMAo0Uaq3/VKZafiZLCXiZuTMuSo9Uo/3X5EkxCFppwo1XGdWHspkZpRjpNiN1EYEzoiA+wYKkiIykuzQyf2qVH6dhBJ84S2M/XvRkpCpcahbyZDoodq0ZuK/3mdRAdXXspEnGgUdBYUJNzWkT39td1nEqnmY0MIlczcatMhkYRq081cisBH/ZTlFE037mITy6R5XnGdilu/KFdv85YKcAwncAYuXEIVbqAGDaCA8AKv8GY9W+/Wh/U5G12x8p0jmIP19QsKbZbA</latexit> S<latexit 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  17. 17. Influenza in a boarding school © Susanne Klimroth a school with N = 763 boys S + I η −→ 2I<latexit sha1_base64="sTBdxuSBgbttBI++0MKD250ojDo=">AAACKXicbVDLSgMxFM34tr6qLt0Ei6AIZUYEXQpuFDeK9gGdUjLpbRuaSYbkjlqGfoMf4tqtfoM7devKvzB9LLR6IHA4517OzYkSKSz6/rs3NT0zOze/sJhbWl5ZXcuvb5StTg2HEtdSm2rELEihoIQCJVQTAyyOJFSi7unAr9yCsUKrG+wlUI9ZW4mW4Ayd1MjvXdN9ek5Di4x3DcgsBGT9LJRatY1od5AZo+/69OC8kS/4RX8I+pcEY1IgY1w28l9hU/M0BoVcMmtrgZ9gPWMGBZfQz4WphcSlsjbUHFUsBlvPhl/q0x2nNGlLG/cU0qH6cyNjsbW9OHKTMcOOnfQG4n9eLcXWcT0TKkkRFB8FtVJJUdNBP7QpDHCUPUcYN8LdSnmHGcbRtfgrRcEd3g9zcq6bYLKJv6R8UAz8YnB1WDi5GLe0QLbINtklATkiJ+SMXJIS4eSBPJFn8uI9eq/em/cxGp3yxjub5Be8z2+QA6eB</latexit><latexit 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  18. 18. Influenza in a boarding school a school with N = 763 boys on the first day, 3 are infected dS dt = −⌘SI dI dt = +⌘SI − ⇢I dR dt = +⇢I <latexit sha1_base64="S+teTb8/z3+oE0UNaWjndjViPD8=">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</latexit><latexit sha1_base64="S+teTb8/z3+oE0UNaWjndjViPD8=">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</latexit><latexit sha1_base64="S+teTb8/z3+oE0UNaWjndjViPD8=">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</latexit><latexit sha1_base64="S+teTb8/z3+oE0UNaWjndjViPD8=">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</latexit> on average, an infected
 will infect 1.66 people / day s/he will need 2.2 days to 
 recover recovery rate ⇢ = (2.2d)−1 <latexit sha1_base64="nYIwO6vXt3SxJEh7buKJgnGwOPU=">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</latexit><latexit 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sha1_base64="nYIwO6vXt3SxJEh7buKJgnGwOPU=">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</latexit> github.com/benmaier/boarding_school_SIR infection rate ⌘ = 1.66d−1 <latexit 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  19. 19. The basic reproduction number • the basic reproduction number is a measure for the
 severity of a disease • looking at fraction of total cases of infected R0 = ⌘ ⇢<latexit sha1_base64="REp4iFnPBgcW1+iLQq0shovMlbo=">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</latexit><latexit 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sha1_base64="REp4iFnPBgcW1+iLQq0shovMlbo=">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</latexit> R0 < 1<latexit 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sha1_base64="nU1oZiIgQUw89BPBq+1nlI5yKx8=">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</latexit> diseasefree epidemic thresholdR0 = 1<latexit 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sha1_base64="0c1bwe2iiHYhi5e7fTnEZ4D9qN0=">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</latexit> R0 > 1<latexit 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sha1_base64="tavznSXnilZTOXdZPlLQf0WqQ0g=">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</latexit><latexit sha1_base64="tavznSXnilZTOXdZPlLQf0WqQ0g=">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</latexit><latexit sha1_base64="tavznSXnilZTOXdZPlLQf0WqQ0g=">AAACM3icbVDLSgMxFM34rPVZ3QhugkVwVSYi6EoEXbhUsSp0hpJJ77TBJDMkd9Qy9Bvc6of4MeJO3PoPprULXwcCJ+fcy733JLmSDsPwJZiYnJqema3MVecXFpeWV2qrly4rrICmyFRmrxPuQEkDTZSo4Dq3wHWi4Cq5ORr6V7dgnczMBfZziDXvGplKwdFLzfN2eMDaK/WwEY5A/xI2JnUyxmm7FqxHnUwUGgwKxZ1rsTDHuOQWpVAwqEaFg5yLG96FlqeGa3BxOdp2QLe80qFpZv0zSEfq946Sa+f6OvGVmmPP/faG4n9eq8B0Py6lyQsEI74GpYWimNHh6bQjLQhUfU+4sNLvSkWPWy7QB/RjikN5768wcCcyrbnplBHc57eDFovLSEGKkeKmq4DWWWRlt4eRHf0HVZ8l+53cX3K502Bhg53t1g+Px6lWyAbZJNuEkT1ySE7IKWkSQSR5II/kKXgOXoO34P2rdCIY96yRHwg+PgF6paoI</latexit> endemic
  20. 20. The basic reproduction number • the basic reproduction number is a measure for the
 severity of a disease R0 = ⌘ ⇢<latexit sha1_base64="REp4iFnPBgcW1+iLQq0shovMlbo=">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</latexit><latexit sha1_base64="REp4iFnPBgcW1+iLQq0shovMlbo=">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</latexit><latexit sha1_base64="REp4iFnPBgcW1+iLQq0shovMlbo=">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</latexit><latexit sha1_base64="REp4iFnPBgcW1+iLQq0shovMlbo=">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</latexit> TABLE 2.1. Some Estimated Basic Reproductive Ratios. Infectious Disease Host Estimated R0 Reference FIV Domestic Cats 1.1–1.5 Smith (2001) Rabies Dogs (Kenya) 2.44 Kitala et al. (2002) Phocine Distemper Seals 2–3 Swinton et al. (1998) Tuberculosis Cattle 2.6 Goodchild and Clifton-Hadley (2001) Influenza Humans 3–4 Murray (1989) Foot-and-Mouth Livestock farms 3.5–4.5 Ferguson et al. (2001b) Disease (UK) Smallpox Humans 3.5–6 Gani and Leach (2001) Rubella Humans (UK) 6–7 Anderson and May (1991) Chickenpox Humans (UK) 10–12 Anderson and May (1991) Measles Humans (UK) 16–18 Anderson and May (1982) Whooping Cough Humans (UK) 16–18 Anderson and May (1982) from Modeling Infectious Diseases in Humans and Animals, Keeling & Rohani
  21. 21. Problems with this • assumption: well-mixed • we live in a petri-dish • we don't move around • every individual is the same • human contact structure is homogeneous
  22. 22. Problems with this • assumption: well-mixed • we live in a petri-dish • we don't move around • every individual is the same • human contact structure is homogeneous HALLO ECHO echo gibts nich mehr
  23. 23. We live in space JUN 1348 DEC 1347 DEC 1348 JUN 1349 DEC 1349 JUN 1350 DEC 1350 Noble, Nature (1974) vol. 250 (5469) pp. 726-728 Fort, J. & Méndez, V., Phys Rev Lett (1999). copyright W. L. Langer, Scientific American, 1964. ∂tI = αIS/N − βI + D∂2 xI ∂tS = −αIS/N + D∂2 xS v ∝ p (R0 − 1)D
  24. 24. 4000 airports 25000 direct connections 3 billion passengers per year 14 billion person km per day 40.000.000 liters of blood per day copyright D. Brockmann
  25. 25. D. Brockmann D. Helbing D. Brockmann & D. Helbing, Science, 2013
  26. 26. Arrival Time D. Brockmann & D. Helbing, Science, 2013Geographical Distance
  27. 27. Arrival Time D. Brockmann & D. Helbing, Science, 2013Effective Distance
  28. 28. Problems with the traditional view • assumption: well-mixed • we live in a petri-dish • we don't move around • every individual is the same • human contact structure is homogeneous HALLO ECHO echo gibts nich mehr
  29. 29. people are nodes contacts are edges Human contacts described as complex networks average number of friends of a person (average degree) hki = 1 N N’ i=1 ki <latexit sha1_base64="4gGLGDdEHdAVYqvN9eANWo+vl0g=">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</latexit><latexit 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sha1_base64="4gGLGDdEHdAVYqvN9eANWo+vl0g=">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</latexit><latexit sha1_base64="4gGLGDdEHdAVYqvN9eANWo+vl0g=">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</latexit> sum degrees of all nodes divide by number of nodes another way of calculating that N : number of nodes m : number of edges<latexit 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sha1_base64="1gj8F9dCqsKlXqfsQusblBNEXSk=">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</latexit><latexit sha1_base64="1gj8F9dCqsKlXqfsQusblBNEXSk=">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</latexit><latexit sha1_base64="1gj8F9dCqsKlXqfsQusblBNEXSk=">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</latexit> i j a b c d N… i=1 ki <latexit sha1_base64="RzOlRfvvEHOhr/OVFZ69SlqiDWs=">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</latexit><latexit sha1_base64="RzOlRfvvEHOhr/OVFZ69SlqiDWs=">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</latexit><latexit sha1_base64="RzOlRfvvEHOhr/OVFZ69SlqiDWs=">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</latexit><latexit sha1_base64="RzOlRfvvEHOhr/OVFZ69SlqiDWs=">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</latexit> = 5 + 1 + 1 ´Ø¨ a + 1 + 1 ´Ø¨ b + 1 + 1 ´Ø¨ c + 1 + 1 ´Ø¨ d + … <latexit sha1_base64="xk23eg1iuc5u8FnH0CbrED5HdWI=">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</latexit><latexit sha1_base64="xk23eg1iuc5u8FnH0CbrED5HdWI=">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</latexit><latexit sha1_base64="xk23eg1iuc5u8FnH0CbrED5HdWI=">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</latexit><latexit sha1_base64="xk23eg1iuc5u8FnH0CbrED5HdWI=">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</latexit> = 2m<latexit sha1_base64="qyN9LAYQEvu2LtVkMpqITd4UARs=">AAACMnicbVDLSgMxFM34tr51I7gJFsFVmRFBN0LRjUsF+4DOUDLpnTaYZIbkjrYM/QW3+iH+jO7ErR9hWrvQ1gOBk3Pu5d574kwKi77/5s3NLywuLa+sltbWNza3tnd26zbNDYcaT2VqmjGzIIWGGgqU0MwMMBVLaMT3VyO/8QDGilTf4SCDSLGuFongDEfSBT1R7e2yX/HHoLMkmJAymeCmvePth52U5wo0csmsbQV+hlHBDAouYVgKcwsZ4/esCy1HNVNgo2K87JAeOaVDk9S4p5GO1d8dBVPWDlTsKhXDnp32RuJ/XivH5DwqhM5yBM1/BiW5pJjS0eW0IwxwlANHGDfC7Up5jxnG0eXzZ4pF0XdXaHjkqVJMd4oQ+tnDsBVERSghwVAy3ZVAy0FoRLeHoRn/hyWXZTCd3Cypn1QCvxLcnparl5NUV8gBOSTHJCBnpEquyQ2pEU565Ik8kxfv1Xv3PrzPn9I5b9KzR/7A+/oGvFWpqA==</latexit><latexit sha1_base64="qyN9LAYQEvu2LtVkMpqITd4UARs=">AAACMnicbVDLSgMxFM34tr51I7gJFsFVmRFBN0LRjUsF+4DOUDLpnTaYZIbkjrYM/QW3+iH+jO7ErR9hWrvQ1gOBk3Pu5d574kwKi77/5s3NLywuLa+sltbWNza3tnd26zbNDYcaT2VqmjGzIIWGGgqU0MwMMBVLaMT3VyO/8QDGilTf4SCDSLGuFongDEfSBT1R7e2yX/HHoLMkmJAymeCmvePth52U5wo0csmsbQV+hlHBDAouYVgKcwsZ4/esCy1HNVNgo2K87JAeOaVDk9S4p5GO1d8dBVPWDlTsKhXDnp32RuJ/XivH5DwqhM5yBM1/BiW5pJjS0eW0IwxwlANHGDfC7Up5jxnG0eXzZ4pF0XdXaHjkqVJMd4oQ+tnDsBVERSghwVAy3ZVAy0FoRLeHoRn/hyWXZTCd3Cypn1QCvxLcnparl5NUV8gBOSTHJCBnpEquyQ2pEU565Ik8kxfv1Xv3PrzPn9I5b9KzR/7A+/oGvFWpqA==</latexit><latexit sha1_base64="qyN9LAYQEvu2LtVkMpqITd4UARs=">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</latexit><latexit sha1_base64="qyN9LAYQEvu2LtVkMpqITd4UARs=">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</latexit> = 2m N<latexit sha1_base64="UmE5sDesUs6gdE6pUr2g8nOXRso=">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</latexit><latexit sha1_base64="UmE5sDesUs6gdE6pUr2g8nOXRso=">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</latexit><latexit sha1_base64="UmE5sDesUs6gdE6pUr2g8nOXRso=">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</latexit><latexit sha1_base64="UmE5sDesUs6gdE6pUr2g8nOXRso=">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</latexit> k = 2 k<latexit sha1_base64="ULMenQjodBWsgKFxNEq3ELDKQuM=">AAACM3icbVDLSgMxFM34rPVZXeoiWARXZUYEXRZ04VLBqtAZJJPeaUOTzJDcqZahX+BWP8WPEXfi1n8wHWfh60Dg3HPv5dycOJPCou+/eDOzc/MLi7Wl+vLK6tr6RmPzyqa54dDhqUzNTcwsSKGhgwIl3GQGmIolXMfDk2n/egTGilRf4jiDSLG+FongDJ10MbzdaPotvwT9S4KKNEmF89uGtxP2Up4r0Mgls7Yb+BlGBTMouIRJPcwtZIwPWR+6jmqmwEZFeemE7jmlR5PUuKeRlur3jYIpa8cqdpOK4cD+7k3F/3rdHJPjqBA6yxE0/zJKckkxpdNv054wwFGOHWHcCHcr5QNmGEcXzg8XDXd4X/rUQ8d5qhTTvSKE+2w06QZREUpIMJRM9yXQZhAa0R9gaMp64vIMfqf3l1wdtAK/FVwcNtunVbI1sk12yT4JyBFpkzNyTjqEEyAP5JE8ec/eq/fmvX+NznjVzhb5Ae/jE7xbqyE=</latexit><latexit sha1_base64="ULMenQjodBWsgKFxNEq3ELDKQuM=">AAACM3icbVDLSgMxFM34rPVZXeoiWARXZUYEXRZ04VLBqtAZJJPeaUOTzJDcqZahX+BWP8WPEXfi1n8wHWfh60Dg3HPv5dycOJPCou+/eDOzc/MLi7Wl+vLK6tr6RmPzyqa54dDhqUzNTcwsSKGhgwIl3GQGmIolXMfDk2n/egTGilRf4jiDSLG+FongDJ10MbzdaPotvwT9S4KKNEmF89uGtxP2Up4r0Mgls7Yb+BlGBTMouIRJPcwtZIwPWR+6jmqmwEZFeemE7jmlR5PUuKeRlur3jYIpa8cqdpOK4cD+7k3F/3rdHJPjqBA6yxE0/zJKckkxpdNv054wwFGOHWHcCHcr5QNmGEcXzg8XDXd4X/rUQ8d5qhTTvSKE+2w06QZREUpIMJRM9yXQZhAa0R9gaMp64vIMfqf3l1wdtAK/FVwcNtunVbI1sk12yT4JyBFpkzNyTjqEEyAP5JE8ec/eq/fmvX+NznjVzhb5Ae/jE7xbqyE=</latexit><latexit sha1_base64="ULMenQjodBWsgKFxNEq3ELDKQuM=">AAACM3icbVDLSgMxFM34rPVZXeoiWARXZUYEXRZ04VLBqtAZJJPeaUOTzJDcqZahX+BWP8WPEXfi1n8wHWfh60Dg3HPv5dycOJPCou+/eDOzc/MLi7Wl+vLK6tr6RmPzyqa54dDhqUzNTcwsSKGhgwIl3GQGmIolXMfDk2n/egTGilRf4jiDSLG+FongDJ10MbzdaPotvwT9S4KKNEmF89uGtxP2Up4r0Mgls7Yb+BlGBTMouIRJPcwtZIwPWR+6jmqmwEZFeemE7jmlR5PUuKeRlur3jYIpa8cqdpOK4cD+7k3F/3rdHJPjqBA6yxE0/zJKckkxpdNv054wwFGOHWHcCHcr5QNmGEcXzg8XDXd4X/rUQ8d5qhTTvSKE+2w06QZREUpIMJRM9yXQZhAa0R9gaMp64vIMfqf3l1wdtAK/FVwcNtunVbI1sk12yT4JyBFpkzNyTjqEEyAP5JE8ec/eq/fmvX+NznjVzhb5Ae/jE7xbqyE=</latexit><latexit sha1_base64="ULMenQjodBWsgKFxNEq3ELDKQuM=">AAACM3icbVDLSgMxFM34rPVZXeoiWARXZUYEXRZ04VLBqtAZJJPeaUOTzJDcqZahX+BWP8WPEXfi1n8wHWfh60Dg3HPv5dycOJPCou+/eDOzc/MLi7Wl+vLK6tr6RmPzyqa54dDhqUzNTcwsSKGhgwIl3GQGmIolXMfDk2n/egTGilRf4jiDSLG+FongDJ10MbzdaPotvwT9S4KKNEmF89uGtxP2Up4r0Mgls7Yb+BlGBTMouIRJPcwtZIwPWR+6jmqmwEZFeemE7jmlR5PUuKeRlur3jYIpa8cqdpOK4cD+7k3F/3rdHJPjqBA6yxE0/zJKckkxpdNv054wwFGOHWHcCHcr5QNmGEcXzg8XDXd4X/rUQ8d5qhTTvSKE+2w06QZREUpIMJRM9yXQZhAa0R9gaMp64vIMfqf3l1wdtAK/FVwcNtunVbI1sk12yT4JyBFpkzNyTjqEEyAP5JE8ec/eq/fmvX+NznjVzhb5Ae/jE7xbqyE=</latexit> k = 3 k = 1 k = 2 number of contacts each person has
 is called degree

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