SIS-models If you click on !!! (the menu bar at the top) maple will execute all commands. You can also enter each command as you go along reading this document. You may enter your own commands at any time and place in this document.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkMvJStleGVjdXRhYmxlR0Y0Ri8= QypJKHJlc3RhcnRHJSpwcm90ZWN0ZWRHISIiLUkld2l0aEc2JEYkSShfc3lzbGliRzYiNiNJLkxpbmVhckFsZ2VicmFHRihGJS1GJzYjSSZwbG90c0dGKEYlLUYnNiNJL1ZlY3RvckNhbGN1bHVzR0YoRiU= QyY+SShpdGVyYXRlRzYiZio2JUkiRkdGJUkjUDBHRiVJInRHRiVGJTYjSSlyZW1lbWJlckdGJUYlQCUxOSYiIiFPOSVPLTkkNiMtRiQ2JUY1RjIsJkYvIiIiISIiRjo2I1FAY29tcHV0ZX50aGV+dHRofml0ZXJhdGV+Rl50KFAwKUYlRiVGJUY7Pkkmb3JiaXRHRiVmKjYlRihGKUkiVEdGJTYjSSJpR0YlRiVGJS1JJHNlcUclKnByb3RlY3RlZEc2JC1GJDYlRjVGMjgkL0ZLO0YwRi82I1FQdGhlfm9yYml0fm9mfnRoZX5wb2ludH5QMH51bmRlcn5Ufml0ZXJhdGVzfm9mfkZGJUYlRiVGOw== QyQ+SSp0aW1lcGxvdDJHNiJmKjYlSSJGR0YlSSNQMEdGJUkiVEdGJTYjSSRvcmJHRiVGJUYlQyQ+OCQtSSZvcmJpdEdGJTYlOSQ5JTkmLUklcGxvdEdGJTYpNyY3Iy1JJHNlcUclKnByb3RlY3RlZEc2JDckSSJuR0YlJiZGLzYjRkA2IyIiIi9GQDtGRUY1NyMtRjw2JDckRkAmRkI2IyIiI0ZGRjpGSEZGL0kmc3R5bGVHRiU3JkkmcG9pbnRHRiVGUkklbGluZUdGJUZTL0kmY29sb3JHRiU3JkkmYmxhY2tHRiVGV0klYmx1ZUdGJUkkcmVkR0YlL0kqdGhpY2tuZXNzR0YlRk4vSStzeW1ib2xzaXplR0YlIiM5L0knbGFiZWxzR0YlNyRRJXRpbWVGJVErUG9wdWxhdGlvbkYlNiNRRFRpbWVwbG90fm9mfnRoZX50d29+Y29tcG9uZW50c35pbn5GRiVGJUYlISIi 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 QyQ+SSpwaGFzZWRpYTJHNiJmKjYlSSJGR0YlSSNQMEdGJUkiVEdGJTYjSSRvcmJHRiVGJUYlQyQ+OCQtSSZvcmJpdEdGJTYlOSQ5JTkmLUkmYXJyb3dHRiU2JzwjLUkkc2VxRyUqcHJvdGVjdGVkRzYkNyQmRi82I0kiaUdGJSZGLzYjLCZGQSIiIkZFRkUvRkE7RkUsJkY1RkUhIiJGRS9JK2RpZmZlcmVuY2VHRiVJJXRydWVHRjwvSSZzaGFwZUdGJUY3L0kmY29sb3JHRiVJJHJlZEdGJS9JJ2xhYmVsc0dGJTckSSRQXzFHRiVJJFBfMkdGJTYjUT1UaGV+b3JiaXR+aW5+dGhlfnBoYXNlfnNwYWNlRiVGJUYlRkk= QyQ+SSRTSVNHNiJmKjYkSSZhbHBoYUdGJUkmZ2FtbWFHJSpwcm90ZWN0ZWRHRiVGJUYlT2YqNiNJIlBHRiVGJTYkSSlvcGVyYXRvckdGJUkmYXJyb3dHRiVGJS1JJDwsPkdGJTYlLCgmOSQ2IyIiIkY5KihUJEY5RjZGOSZGNzYjIiIjRjkhIiIqJlQmRjlGPEY5RjksKEY8RjlGOkY5RkBGPyIiIUYlRiU2JkYoRjdGKTklNiNRKlNJU35tb2RlbEYlRiVGJUY/ QyQ+SSZwYXJhbUc2ImYqNiNJIkZHRiU2JEkiYUdGJUkiYkdGJUYlRiVDJT44JSYtOSQ2Iy1JJDwsPkdGJTYkIiIhIiIiNiNGNz44JCwoRjdGNyYtRjE2Iy1GNDYkRjdGN0Y4ISIiRi5GN083JEY6Ri42I1FHY29tcHV0ZX50aGV+cGFyYW1hdGVyc35vZn5hbn5TSVN+bW9kZWxGJUYlRiVGQQ== QyQtSSZwYXJhbUc2IjYjLUkkU0lTR0YlNiRJJmFscGhhR0YlSSZnYW1tYUclKnByb3RlY3RlZEciIiI= General theory QyQtSSZlcXVpMkc2IjYjZio2I0kiUEdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLUkkPCw+R0YlNiQmLS1JJFNJU0dGJTYkSSZhbHBoYUdGJUkmZ2FtbWFHJSpwcm90ZWN0ZWRHNiM5JDYjIiIiJkYxNiMiIiNGJUYlRiVGOw== In a SIS-model, I = P[2] follows logistic growth because P[1]+P[2]=N is constant QyQtSSVzdWJzRyUqcHJvdGVjdGVkRzYkLyZJIlBHNiI2IyIiIi1JIitHNiRGJS9JK21vZHVsZW5hbWVHRipJL1ZlY3RvckNhbGN1bHVzRzYkRiVJKF9zeXNsaWJHRio2JEkiTkdGKi1JIi1HRi82IyZGKTYjIiIjJi0tSSRTSVNHRio2JEkmYWxwaGFHRipJJmdhbW1hR0YlNiNGKUY7Riw= QyQvSSIlRzYiLUkiK0c2JCUqcHJvdGVjdGVkRy9JK21vZHVsZW5hbWVHRiVJL1ZlY3RvckNhbGN1bHVzRzYkRilJKF9zeXNsaWJHRiU2JCZJIlBHRiU2IyIiIy1JIipHRig2JC1GNTYkSSZhbHBoYUdGJUYwLUYnNiQtRic2JEkiTkdGJS1JIi1HRig2Iy1GNTYkSSZnYW1tYUdGKSokRjkhIiItRkA2I0YwIiIi QyQvLUkiK0c2JCUqcHJvdGVjdGVkRy9JK21vZHVsZW5hbWVHNiJJL1ZlY3RvckNhbGN1bHVzRzYkRidJKF9zeXNsaWJHRio2JCZJIlBHRio2IyIiIy1JIipHRiY2JC1GNDYkSSZhbHBoYUdGKkYvLUYlNiQtRiU2JEkiTkdGKi1JIi1HRiY2Iy1GNDYkSSZnYW1tYUdGJyokRjghIiItRj82I0YvLUYlNiRGLy1GNDYkLUY0NiRGOEY7LUYvNiMtRiU2JCIiIi1GPzYjLUY0NiRGLyokRjtGRUZS This is logistic growth with carrying capacity N-\316\263/\316\261 or N-\317\201 where \317\201=\316\263/\316\261 is the relative removal rate. Example 1 QyQ+SSJGRzYiLUkkU0lTR0YlNiQkIiIiISIjJCIjREYrISIi Relative removal rate and contact number QyQ+SSRyaG9HNiItSSIqRzYkJSpwcm90ZWN0ZWRHL0krbW9kdWxlbmFtZUdGJUkvVmVjdG9yQ2FsY3VsdXNHNiRGKUkoX3N5c2xpYkdGJTYkJi1JJnBhcmFtR0YlNiNJIkZHRiU2IyIiIyokJkYxNiMiIiIhIiJGOg== QyQ+SSZzaWdtYUc2Ii1JIipHNiQlKnByb3RlY3RlZEcvSSttb2R1bGVuYW1lR0YlSS9WZWN0b3JDYWxjdWx1c0c2JEYpSShfc3lzbGliR0YlNiQiJCsiKiRJJHJob0dGJSEiIiIiIg== QyQtSShkaXNwbGF5RzYiNiM3JC1JKnRpbWVwbG90MkdGJTYlSSJGR0YlLUkkPCw+R0YlNiQiIyYqIiImIiNELUklcGxvdEdGJTYlLUkiK0c2JCUqcHJvdGVjdGVkRy9JK21vZHVsZW5hbWVHRiVJL1ZlY3RvckNhbGN1bHVzRzYkRjhJKF9zeXNsaWJHRiU2JCIkKyItSSItR0Y3NiNJJHJob0dGJTsiIiFGMS9JJmNvbG9yR0YlSShtYWdlbnRhR0YlIiIi Example 2 QyQ+SSJGRzYiLUkkU0lTR0YlNiQkIiImISIkJCIjRCEiIyEiIg== Relative removal rate and contact number QyQ+SSRyaG9HNiItSSIqRzYkJSpwcm90ZWN0ZWRHL0krbW9kdWxlbmFtZUdGJUkvVmVjdG9yQ2FsY3VsdXNHNiRGKUkoX3N5c2xpYkdGJTYkJi1JJnBhcmFtR0YlNiNJIkZHRiU2IyIiIyokJkYxNiMiIiIhIiJGOg== QyQ+SSZzaWdtYUc2Ii1JIipHNiQlKnByb3RlY3RlZEcvSSttb2R1bGVuYW1lR0YlSS9WZWN0b3JDYWxjdWx1c0c2JEYpSShfc3lzbGliR0YlNiQiJCsiKiRJJHJob0dGJSEiIiIiIg== QyQtSShkaXNwbGF5RzYiNiM3JC1JKnRpbWVwbG90MkdGJTYlSSJGR0YlLUkkPCw+R0YlNiUiIyYqIiImIiIhIiNELUklcGxvdEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYlLUkiK0c2JEY2L0krbW9kdWxlbmFtZUdGJUkvVmVjdG9yQ2FsY3VsdXNHRjU2JCIkKyItSSItR0Y7NiNJJHJob0dGJTtGMUYyL0kmY29sb3JHRiVJKG1hZ2VudGFHRiUiIiI= Example 3 QyQ+SSJGRzYiLUkkU0lTR0YlNiQkIiIkISIkJCIjRCEiIyEiIg== Relative removal rate and contact number QyQ+SSRyaG9HNiItSSIqRzYkJSpwcm90ZWN0ZWRHL0krbW9kdWxlbmFtZUdGJUkvVmVjdG9yQ2FsY3VsdXNHNiRGKUkoX3N5c2xpYkdGJTYkJi1JJnBhcmFtR0YlNiNJIkZHRiU2IyIiIyokJkYxNiMiIiIhIiJGOg== QyQ+SSZzaWdtYUc2Ii1JIipHNiQlKnByb3RlY3RlZEcvSSttb2R1bGVuYW1lR0YlSS9WZWN0b3JDYWxjdWx1c0c2JEYpSShfc3lzbGliR0YlNiQiJCsiKiRJJHJob0dGJSEiIiIiIg== QyQtSShkaXNwbGF5RzYiNiM3JC1JKnRpbWVwbG90MkdGJTYlSSJGR0YlLUkkPCw+R0YlNiUiIyYqIiImIiIhIiN2LUklcGxvdEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYlLUkiK0c2JEY2L0krbW9kdWxlbmFtZUdGJUkvVmVjdG9yQ2FsY3VsdXNHRjU2JCIkKyItSSItR0Y7NiNJJHJob0dGJTtGMUYyL0kmY29sb3JHRiVJKG1hZ2VudGFHRiUiIiI= Differentiated SIS model QyQ+SSZTSVN0b0c2ImYqNihJI2FmR0YlSSNnZkdGJUkjTmZHRiVJI2FtR0YlSSNnbUdGJUkjTm1HRiVGJUYlRiVPZio2I0kiUEdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLUkkPCw+R0YlNiQsKCY5JDYjIiIiRjwqKFQkRjwsJlQoRjxGOSEiIkY8JkY6NiMiIiNGPEY8KiZUJkY8RjlGPEZBLChGQkY8KihUKkY8LCZULkY8RkJGQUY8RjlGPEY8KiZULEY8RkJGPEZBRiVGJTYuRihGOkYpOSVGKjkmRis5J0YsOShGLTkpNiNROWRpZmZlcmVudGlhdGVkflNJU35tb2RlbEYlRiVGJUZB General theory QyQ+SSJGRzYiLUkmU0lTdG9HRiU2KCZJJmFscGhhR0YlNiNJImZHRiUmSSZnYW1tYUclKnByb3RlY3RlZEdGKyZJIk5HRiVGKyZGKjYjSSJtR0YlJkYuRjMmRjFGMyIiIg== QyQtSSJGRzYiNiNJIlBHRiUiIiI= The equilibria QyQtSSlzaW1wbGlmeUc2IjYjLUkmc29sdmVHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiU2JDwkLyYtSSJGR0YlNiNJIlBHRiU2IyIiIiZGM0Y0LyZGMDYjIiIjJkYzRjk3JEY2RjtGNQ== QyQtSSZlcXVpMkc2IjYjZio2I0kiUEdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLUkkPCw+R0YlNiQmLUkiRkdGJTYjOSQ2IyIiIiZGMTYjIiIjRiVGJUYlRjY= Example 1 QyQ+SSJGRzYiLUkmU0lTdG9HRiU2KCQiIiMhIiQkIiIkISIjIiQrIiQiIiVGKyQiIiJGLkYvISIi The equilibria are QyQtSSZlcXVpMkc2IjYjSSJGR0YlIiIi QyQtSSp0aW1lcGxvdDJHNiI2JUkiRkdGJS1JJDwsPkdGJTYkIiNJIiNdIiM/IiIi QyQtSSp0aW1lcGxvdDJHNiI2JUkiRkdGJS1JJDwsPkdGJTYkIiNTIiIiIiM/Riw= Notice that at the outset the disease is decreasing in one part of the population and increasing in the other part. QyQ+SSNkZkc2Ii1JKWRpcmZpZWxkR0YlNidJIkZHRiUiIycpIiMpKSIjJyoiIykqISIi QyQ+SSRvcmJHNiItSSRzZXFHJSpwcm90ZWN0ZWRHNiQtSSpwaGFzZWRpYTJHRiU2JUkiRkdGJUkjUDBHRiUiIzUvRi4tSSZ1bmlvbkdGKDYkLUYyNiQtRjI2JDwjLUYnNiQtSSQ8LD5HRiU2JC1JIitHNiRGKC9JK21vZHVsZW5hbWVHRiVJL1ZlY3RvckNhbGN1bHVzRzYkRihJKF9zeXNsaWJHRiU2JCIjJyktSSIqR0ZANiQkIiIjISIiSSJrR0YlIiMnKi9GTjsiIiFGLzwjLUYnNiQtRjw2JCIjKSktRj82JEZPRkhGUDwjLUYnNiQtRjw2JEY+IiMpKkZQPCMtRic2JC1GPDYkRkdGWUZQRk0= QyQtSShkaXNwbGF5RzYiNiNJI2RmR0YlIiIi QyQtSShkaXNwbGF5RzYiNiNJJG9yYkdGJSIiIg== QyQtSShkaXNwbGF5RzYiNiRJI2RmR0YlSSRvcmJHRiUiIiI= QyQtSShkaXNwbGF5RzYiNiMtSSRzZXFHJSpwcm90ZWN0ZWRHNiQtSSpwaGFzZWRpYTJHRiU2JUkiRkdGJUkjUDBHRiUiIzUvRi8tSSZ1bmlvbkdGKTYkLUYzNiQtRjM2JDwjLUYoNiQtSSQ8LD5HRiU2JC1JIitHNiRGKS9JK21vZHVsZW5hbWVHRiVJL1ZlY3RvckNhbGN1bHVzRzYkRilJKF9zeXNsaWJHRiU2JCIjRC1JIipHRkE2JCIiI0kia0dGJSIjOi9GTTsiIiEiI1g8Iy1GKDYkLUY9NiRGSC1GQDYkRk4tRko2JCIiJkZNL0ZNO0YwIiNdPCMtRig2JC1GPTYkIiQ/Ii1GQDYkRjBGWi9GTTtGUUZpbjwjLUYoNiQtRj02JC1GQDYkIiNxRkkiJF0jL0ZNO0ZRIiM/IiIi Example 2 QyQ+SSJGRzYiLUkmU0lTdG9HRiU2KCQiIiMhIiMkIiImRisiJCsiRilGLEYuISIi The equlibria are QyQtSSZlcXVpMkc2IjYjSSJGR0YlIiIi QyQ+SSNkZkc2Ii1JKWRpcmZpZWxkR0YlNidJIkZHRiUkIiRsKiEiIiQiJCYpKkYsRipGLUYs QyQtSShkaXNwbGF5RzYiNiNJI2RmR0YlIiIi QyQtSShkaXNwbGF5RzYiNiRJI2RmR0YlLUkkc2VxRyUqcHJvdGVjdGVkRzYkLUkqcGhhc2VkaWEyR0YlNiVJIkZHRiVJI1AwR0YlIiM1L0YwNyQtSSQ8LD5HRiU2JCQiJGwqISIiRjctRjU2JEY3JCIkdipGOSIiIg== QyQtSSp0aW1lcGxvdDJHNiI2JUkiRkdGJS1JJDwsPkdGJTYkIiM/IiNJIiNTIiIi QyQtSSp0aW1lcGxvdDJHNiI2JUkiRkdGJS1JJDwsPkdGJTYkIiNOIiNJIiNTIiIi LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkMvJStleGVjdXRhYmxlR0Y0Ri8=