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The dynamics of the simulation process are:

  1. Discrete Markov process.

  2. The simulation has the following parameters:

    a. New variant emergence at rate X. b. For each variant k:

    • Unvaccinated individuals become sick rate C(k),
    • Mortality rate D(k),
    • Recovery rate H(k),
    • Vaccines have an efficaccy rate E(v,k) and pseudo vaccines (recovered) have efficacy rate E(r,k) < E(v,k). In general, the probability of i acquiring the disease k from j will be equal to
    P(i gets the disease from j | their states) = C(k) * (1 - E(i,k)) * (1 - E(j, k))

    where (i,j) in (u,v,r). Efficacy rate for unvaccinated is zero.

    • Vaccinated individuals have a reduced mortality rate D(k,v) > D(k), and recovered individuals D(k,r) in (D(k,v), D(k)]
    • Vaccinated individuals have an increased recovery rate H(k,v) > H(k), whereas recovered's rate H(k,r) in [H(k), H(k,v)).

    The sum of mortality and recovery rates is less than one since the difference represents no change.

    c. Each country vaccinates citizens at rate V function of A (availability) and B (citizens' acceptance rate.) d. In each country i, the entire population N(i) distributes between the following states:

    • Healthy unvaccinated (N(i,t,u)),
    • Healthy vaccinated (N(i,t,v)),
    • Deceased (N(i,t,d)),
    • Recovered (N(i,t,r)),
    • Unvaccinated and sick with variant (N(i,t,s,k|u)) k., and
    • Vaccinated and sick with variant (N(i,t,s,k|v)) k.

    Total sick are N(i,t,k,s) = sum(g in {u,v}) N(i,t,k,s|g)

    Globally, we keep track of the prevalence of new variants. Variants can disappear if no more individuals port the variant, i.e., the prevalence rate P(k,t)=sum(i) N(i,s,k) equals zero.

    d. Vaccines are manufactured at each country at rates M(i) and uniformly shared with other countries at rate S(i). c. Population flows between each country pair (i,j) at a rate F(i,j). Flows between countries do not change Population and are symmetric.

  3. The simulation process is as follows:

    1. Countries are initialized with a total population N(i).

    2. Variant zero initializes at a random location i, with an initial prevalence P(k,t) = N(i,t,k).

    3. For time t in (0,T) do:

      a. Unvaccinated individuals can become sick of variant k with probability:

      Pr(h->s|i,t,k,u) ~ sum(g in {u,v}) (N(i,t - 1,s,k|g) + sum(j != i) F(i,j) * N(j,t-1,s,k|g)) * C(k) / (N(i) + sum(j != i) N(j))

      b. Vaccinated individuals can become sick of variant k with probability: Pr(v->s|i,t,k,v) ~ Pr(h->s|i,t,k) * (1 - E(v,k)).

      b. Recovered individuals can become sick of variant k with probability: Pr(v->s|i,t,k,r) ~ Pr(h->s|i,t,k) * (1 - E(r,k)).

      c. Sick individuals with variant k die with probability D(k) or recover with probability H(k), otherwise they stay infected; with the rates depending on their vaccination status v or n.

      d. Unvaccinated individuals vaccinate in country i with probability P(u->v) ~ V(A(i,t), B(i)).

      e. The country vaccine supply changes.