Question

2(6pts). Suppose a student carrying a flu virus returns to an isolated college campus of 1000 students. If it is assumed that the rate at which the virus spreads is directly proportional not only the number 1 of infected students but also to the number of students not infected, determine the number of infected students after 6 days. Only, after 3 days it is observed that 100 students are infected. sol.

Answer 1

Differential equation

Toral students = 1000. da 1000-).x. I no. of infected person (student) = x y no. of non-infected shudenrs = 1000-2 -) dx = K (1000-2)2 , X(0)=1. dt . .dx = k dt. (1000-x)a Inregearing on both sides Saida Skt C 1000-2 75261000-2; 02 - SK DES ??600 Să tiédoazdx = kt te.' 7 en 1821 + en 11000-41 -1000kt +10000" en 1600-2 =1000kt +C" . ----C":10007 1000kt. 9 a ... ens c. 1000-2 ce

Usin ing initial condition, glo) - 1 - C 100001s ceº C =%999 3) PCE) = 1000 e 4) = 1000 1000kt 1000kt 999 + but X(3) = 100 :-) 100 = 1000 e soook 999 +23000K 1000 - 30ook 100 = 1+999 e -3000K) = 1000 - 100 (1+999 - -3000K = 1000 -100. >) 100x999 e ď 9900 e -3000* 900 -30001 - 900 99900 * - 3000k - en )

K OO (). K = 0.00157 1000kt - P(t) = 1000 e Cookt 999 + 9 x (t) = 1000 (+999 e-TCookt 1000 = alt) - - It 999 261000) (0.0015737 = Q(t) = 1000 1 -1.57 E I + 999 e' 1000 - -1.57X 6. a(6) = ( 6 ) : 3 2 5 1 0 6 1 19 3 25 ... Afree 6 - days. 925 studenis get infection.