Table 1.
The Markov chain events and their transition rates used to stochastically simulate the epidemiological submodel. Events are shown for a focal variant i. Events and transition rates for all other variants j in the set {1, 2, 3, …, n} are analogous. The population size N is given by Si + Ii + Ri. Several constraints exist in the system and are respected during the simulation. Because Sj + Ij + Rj = N for all variants j, a birth event, occurring at rate μN, increases the number of susceptible hosts for each variant j by 1 (i.e. births do not occur independently for each variant j). Similarly, a death, also occurring at rate μN, decreases the number of hosts S, I or R to each variant j by 1. This decrease is taken from Sj, Ij or Rj with probability Sj/N, Ij/N and Rj/N, respectively. The rate of possible infection with variant j (related to the force of infection) is given by βIj. When i is equal to j, a ‘possible infection’ event results in an infection with probability Si/N. When i is not equal to j, this possible infection event results in a gain of immunity through polarized cross-immunity with probability σij (Si/N). Recovery of an individual infected with variant i and the loss of immunity to variant i occur at rates νIi and γRi, respectively. An antigenic emergence event results in a decrease in the number of individuals infected with variant i and the stochastic appearance of a new antigenic variant, variant n + 1.
| event | change | rate |
|---|---|---|
| birth | (Si,Ii,Ri) → (Si + 1,Ii,Ri) | μN |
| death | (Si,Ii,Ri) → (Si − 1,Ii,Ri) with probability Si/N; (Si,Ii,Ri) → (Si,Ii − 1,Ri) with probability Ii/N; (Si,Ii,Ri) → (Si,Ii,Ri − 1) with probability Ri/N | μN |
| possible infection | for i = j: (Si,Ii,Ri) → (Si − 1,Ii + 1,Ri) with probability Si/N; (Si,Ii,Ri) → (Si,Ii,Ri) with probability (1 − Si/N) for i ≠ j: (Si,Ii,Ri) → (Si − 1,Ii,Ri + 1) with probability σijSi/N; (Si,Ii,Ri) → (Si,Ii,Ri) with probability (1 − σijSi/N) | βIj |
| recovery | (Si,Ii,Ri) → (Si,Ii − 1,Ri + 1) | νIi |
| loss of immunity | (Si,Ii,Ri) → (Si + 1,Ii,Ri − 1) | γRi |
| antigenic emergence | (Si,Ii,Ri) → (Si,Ii − 1,Ri + 1) and (Sn+1, 0,Rn+1) → (Sn+1 − 1, 1,Rn+1) | h(t − tie)Ii |