Contact Networks: When Will the COVID-19 Pandemic End?
Sources:
https://journals.plos.org/plosone/article?id=10.1371/journal.pone.0237832
In class we have discussed the spread of epidemic diseases through contact networks using several models. The one with richer properties, the SIR model, is a proper simplified representation of the COVID-19 pandemic we are all currently experiencing. The SIR model embodies nodes that can be susceptible (S), infectious (I), and removed (R), which are the three stages that a person can be among a pandemic. Susceptible means the individual does not carry the disease yet and is not contagious, but he or she has the probability of being infected. Infectious means the person carries pathogens that can be spread to others. Removed indicates immunity that the individual develops after he or she has recovered from the disease and is no longer vulnerable to the disease. Each node in state I in this contact network can infect its neighbors in states S with the infection rate being probability p. After attempting to spread the disease, the node in state I switches to R and is no longer actively involved in the spreading of the epidemic.
In the first article “Simulating the progression of the COVID-19 disease in Cameroon using SIR models,” the authors analyzed the spread of pandemic using the concepts I explained above. The initial model considers all individuals in the network to be susceptible. If they are infected by the virus, they immediately switch to infectious and remain so until recovery, assuming immunity during the rest of the outbreak. Supposing a closed network of size N, an individual who switches to state I remains so for a time determined exponentially with a decay rate of γ (the removal rate). Thus, the inverse of γ refers to the average number of days an infectious node has to transmit the virus before switching to state R (placed in quarantine, hospitalized, recovered, or died). During the infectious period, an individual has close contact with the rest of the network at a rate β (the effective contact rate), which is the product of the average number of exposures per unit of time (τ) and the likelihood of infection at each occasion of exposure (μ): β = τ * μ.
If assuming that at the beginning of the epidemic, where S(0), I(0), R(0) = N-1, 1, 0, then the number of susceptible individuals decreases symmetrically as shown in this equation:
dS/dt = –β( S(t)I(t)/N ).
The variation in the number of nodes infected is thus given by:
dI/dt = β( S(t)I(t)/N ) – γI(t).
The result of this model is the following dynamic system:
The simplicity of this dynamic system reflecting the SIR model gives us rapid information on the spreading rate of the epidemic. Indeed, an epidemic occurs if the number of infected individuals increases continuously:
β/γ * S(t)/N > 1.
Since nearly everyone is susceptive at the beginning, S(t)/N can be approximated to 1 and the above equation can be written as:
β/γ = R0 > 1.
Based on this model, the evolution of COVID-19 in Cameroon is presented below, and it should be a plausible scenario if no action is taken to reduce the spread of the virus. Specifically, about 7.7% of the Cameroonian population might have ended up being infected, which is close to 2,015,200 individuals.
This is only a simplified model of the pandemic in one area of the world, and there are more aspects to it. For example, the nodes in state I might switch from R back to S after some time, or they might not switch to state R at all but directly to S. This is due to the fact that the coronavirus have developed different strings as a potent pathogen, and people immune to one string are still vulnerable to another. Therefore the most appropriate solution to end this pandemic once and for all is to get individuals in the network vaccinated.
In the article from Mckinsey, final data from the Pfizer/BioNTech1 vaccine trial and interim data from the Moderna trial both show efficacy of approximately 95%, which is higher than many dared to hope for. With the vaccines mass produced and distributed, the States will most likely reach an epidemiological end to the pandemic (achieving herd immunity) around Q3 or Q4 2021. The reason that vaccines can reduce the spread of the epidemic is by reducing the probability p of infection significantly when two individuals are in physical contact. A decline in p leads towards the basic reproductive number R0 decreasing, and when p becomes close to 0 (as the vaccines are tested to be 95% effective), the value of R0 becomes close to 0 and thus less than 1. The disease would die out in a finite number of steps because it can no longer maintain a larger than 100% reproduction rate (i.e. 1 infectious node will guarantee at least 1 neighbor to also get infected). However, this scenario would only be the ideal case, and there are many more economic, political, and social factors beyond simply getting everyone vaccinated and thus immuned. Hopefully the pandemic will soon come to an end as scientists and politicians are working together towards an optimistic result for all.