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States (sum up to the whole population) S(t) = # of susceptible people at time t E(t) = # of exposed people at time t (incubation state) I(t) = # of infectious people at time t Q1(t) = # of non-symptomatic people in quarantine at time t Q2(t) = # of symptomatic (infectious) people in quarantine at time t Q(t) = # of people in quarantine at time t = Q1(t) + Q2(t) D(t) = # of people dead at time t M(t) = # of people disabled/maimed at time t How beneficial is a “disabled” state? Will we be able to get decent data on the expected occurrence of a disability from the disease we model? Will the errors in the model actually be larger than the number of people that end up moving into this state? R(t) = # of people recovered at time t s(t) = fraction of total population susceptible at time t e(t) = fraction of total population exposed at time t (incubation state) i(t) = fraction of total population infectious at time t q1(t) = fraction of total population who are non-symptomatic & in quarantine at time t q2(t) = fraction of total population e who are symptomatic (infectious) & in quarantine at time t q(t) = fraction of total population in quarantine at time t = q1(t) + q2(t) d(t) = fraction of total population dead at time t m(t) = fraction of total population disabled/maimed at time t r(t) = fraction of total population recovered at time t Transition Rates (instantaneous rates … # per unit time) B(t) = # of susceptible people who become infected (exposed) C(t) = # of exposed people who develop symptoms QS(t) = # of susceptible people who are quarantined QE(t) = # of exposed people who are quarantined QI(t) = # of infectious people who are quarantined QQ(t) = # of quarantined exposed people who develop symptoms DI(t) = # of infectious people who die MI(t) = # of infectious people who become disabled/maimed DQ(t) = # of quarantined people who die MQ(t) = # of quarantined people who become disabled/maimed RI(t) = # of infectious people who recover RS(t) = # of susceptible people who recover RQ1(t) = # of quarantined non-symptomatic people who recover RQ2(t) = # of quarantined infectious people who recover Input Parameters (all are instantaneous rates) Do all of these definitions make sense? β = transmission rate (# of new people exposed per infectious person per time period) α = close contacts identification rate (# of non-symptomatic persons that have been identified as a close contact of an infectious person who was quarantined within the last time period) d = mortality rate of the disease (fraction of the infected people who will die) m=disability rate of the disease (fraction of the infected people who end up disabled/maimed) φ = treatment rate (fraction of non-symptomatic people treated/vaccinated) γ = quarantine rate (fraction of infectious people who get quarantined during each time period) μ1 = incubation period (mean length of time) μ2 = infectious period (mean length of time) ODEs dS QS (t ) B(t ) RS (t ) dt dE B(t ) C (t ) QE (t ) dt dI C (t ) DI (t ) M I (t ) RI (t ) QI (t ) dt dQ1 QS (t ) QE (t ) RQ1 (t ) QQ (t ) dt dQ2 QI (t ) DQ (t ) M Q (t ) RQ 2 (t ) QQ (t ) dt dD DI (t ) DQ (t ) dt dM M I (t ) M Q(t ) dt dR RI (t ) RQ1 (t ) R Q 2 (t ) RS (t ) dt (1) (2) (3) (4) (5) (6) (7) (8) Note: I need to add the transition equations to this document so that we have them all laid out individually – that should make it easier to make updates, since we will likely not need to change all of them. 1 1 i (t ) s (t ) i (t ) s (t ) 1 i (t ) i (t ) s (t ) 1 (1a) 2 2 S to Q1 S to E S to R The S to Q1 transition is α * (I to Q2) * s(t) which is also α * QI(t) * s(t) The S to R transition was originally just φ * s(t) but we decided to assume that the S to Q1 and S to E transitions occur prior to the treatment of the S population, so those rates were subtracted from φ ds dt de 1 1 i(t ) e(t ) i(t ) s(t ) e(t ) 1 dt 1 2 S to E E to I E to Q1 The E to Q1 transition is α * (I to Q2) * e(t) which is also α * QI(t) * e(t) (2a) di 1 1 1 1 1 i(t ) e(t ) d i(t ) m i(t ) (1 d m) i(t ) 1 dt 1 2 2 2 2 E to I I to D I to M I to R I to Q2 For transitions out of I, assume that I to D, M, & R occur first (before I to Q2) with 1/μ2 capturing the delay associated with the symptomatic period. The transition from I to Q2 then requires 1-1/μ2 to ensure that the only people going to quarantine are those that have not already been designated to die, become “maimed”, or recover. di 1 1 e(t ) i(t ) dt 1 2 1 i(t ) 1 2 (3a) 1 1 1 s(t ) 1 e(t ) i(t ) s(t ) 1 i(t ) e(t ) q1 (t ) 1 q1 (t ) 1 s(t ) e(t ) 1 s(t ) e(t ) 2 2 S to Q1 E to Q1 Q1 to R Q1 to Q2 People moving out of Q1 can go to either Q2 or to R. We use 1/μ2 to capture the delay between states (will not know that an individual has actually recovered until they make it through an incubation period without getting sick). The ratios with s(t)+e(t) in the denominator use used to differentiate between the people in Q1 who have the disease and those who don’t (all are non-symptomatic). The assumption is that the fraction of people that fall into the two categories (exposed, but not yet showing symptoms, and nonexposed) are the same in Q1 as they are in the general population. dq1 dt dq1 dt 1 i(t ) s(t ) 1 2 1 1 i(t ) e(t ) q1 (t ) 1 1 2 (4a) dq2 dt 1 1 1 1 1 e(t ) i(t ) d q2 (t ) 1 m q2 (t ) (1 d m) q2 (t ) q1 (t ) 2 2 2 1 s(t ) e(t ) 2 I to Q2 Q2 to D Q2 to M Q2 to R Q1 to Q2 dq2 dt 1 1 1 e(t ) i(t ) q2 (t ) 1 q1 (t ) 2 1 s(t ) e(t ) 2 (5a) dd 1 1 d i (t ) d q 2 (t ) dt 2 2 I to D Q2 to D (6a) dm 1 1 m i (t ) m q 2 (t ) dt 2 2 I to M Q2 to M (7a) dr 1 1 s(t ) 1 (1 d m) i (t ) q1 (t ) (1 d m) q 2 (t ) s(t ) dt 2 1 s(t ) e(t ) 2 Q2 to R Q1 to R Q2 to R S to R B(t) E (exposed) S (susceptible) (8a) RS(t) QS(t) QE(t) C(t) Q=Q1+Q2 Q1 (quarantined non-symptomatic) I (infectious) QI(t) DI(t) MI(t) RI(t) DQ(t) D (dead) M (maimed) QQ(t) Q2 (quarantined symptomatic) MQ(t) RQ1(t) RQ2(t) R (recovered)
