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← 03ps-06 Malaria Solution

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Showing Revision 2 created 10/24/2012 by Amara Bot.

  1. The first thing that we need to deal in this problem is the issue of the mosquito nets.
  2. Well we know that at first there are no mosquito nets in place at all,
  3. so we're going to create this variable called net factor and initialize it to 1.
  4. We know that after 100 days, net factor is reduced by bite reduction by net,
  5. this constant that we created up here.
  6. We add in this if statement right here saying that after 100 days,
  7. net factor is reduced by 0.9 which is bite reduction by net.
  8. Remember that this is basically the probability that a person is going to be bitten.
  9. It makes sense that if you add in the mosquito net, the probability of you being bitten will go down.
  10. Now for implementing the forward Euler method with the infected population,
  11. both human and mosquito.
  12. For the number of infected humans at the next time step,
  13. we of course, need to start with the number of infected humans at the previous step.
  14. To this, we're going to add on the number of people
  15. that are being added to the infected population.
  16. We are going to subtract the number of people that are moving out
  17. of the infected into the susceptible population.
  18. Let's first think about the term that's being added to the number of infected humans.
  19. We take the probability of a person being bitten which is in that factor
  20. times the number of bites per day and per mosquito that a person gets
  21. also multiply that by the number of infected mosquitoes
  22. and then we multiply this by the fraction of the total human population that is still susceptible
  23. also multiply this by the probability of a mosquito transmitting malaria to a human.
  24. On the other side of the coin, we have factors that move people away
  25. from the infected population and back into a susceptible population.
  26. This is a very simple term.
  27. We simply take the number of people that were infected at the previous time step
  28. and divide this by the time that it takes a person to recover.
  29. Remember that all of that was multiplied by age
  30. and added to the number of people infected at the previous time step.
  31. Now, let's take a look at how the infected mosquito population was changing.
  32. Since we're using the forward Euler method, we started with the number of infected mosquitoes
  33. at the previous step and then add to this age times a bunch of other things.
  34. Now to find out how many mosquitoes are moving into the infected group,
  35. we start with the probability of a mosquito biting a person,
  36. which of course might give the mosquito malaria as well,
  37. multiply this again by the bites per day of mosquito,
  38. multiply this by the number of susceptible mosquitoes and the number of infected humans.
  39. We then divide this by the total number of humans
  40. and multiply the transmission probability from human to mosquito.
  41. If you remember the drawing that I showed earlier on in the video introducing this problem,
  42. I told you that the only way the mosquitoes can no longer be infected is to die.
  43. They don't move back to being susceptible.
  44. This means that the only thing that is going to decrease
  45. the number of infected mosquitoes is death of mosquitoes.
  46. So, we subtract from this entire expression if 1 over the mosquito lifetime
  47. times another infected mosquitoes at the previous step.
  48. I know that that was a lot of constants and variables to keep track of,
  49. but great job if you're able to fill all this up.
  50. Now, let's take a peek at what our final graph for Unit 3 looks like.
  51. Now in this graph, the blue curve shows the fraction of humans that were infected
  52. and the green curve shows the fraction of the number of mosquitoes that are infected.
  53. At first glance, it might look like there are many many more humans infected the mosquitoes.
  54. But remember that because this is fraction, this doesn't actually show us plain numbers.
  55. In fact, we started with 10¹⁰ mosquitoes, but only 10⁸ humans.
  56. So even if a much smaller percent of the mosquito population was infected from the human population,
  57. the mosquitoes are able to infect such a great portion of humans
  58. because there are so many more mosquitoes than there are humans to it.
  59. You can see that the number of both humans and mosquitoes infected,
  60. at first, increases sharply and then levels off and stays at a very high number.
  61. However, as soon as 100 days hits, and we introduce the mosquito nets.,
  62. the number of infected in each population starts to decrease certainly.
  63. We can see that a mosquito net idea was definitely very effective
  64. for helping slow the spread of malaria in the human population as well as the mosquito population.
  65. Great job on all three problems in Unit 3.
  66. I hope you found this discussion of the SIR model and its variations interesting.