Keywords: epidemic, incubation, latency, simulation.
CR Classification: G.3.
By devising a model of a natural phenomenon we attempt to analyse and gain a better understanding of its behaviour and ultimately are able to predict and control that behaviour. For an epidemiologist or public health official it is desirable to know how much time is left before a virus epidemic becomes pandemic or whether a virus will propagate at all. To make such predictions accurately a detailed model is needed. However, to make predictions efficiently model implementations must avoid excessive detail. Therefore we need to assess the impact and hence the significance of different epidemic spread parameters for inclusion in such models.
The incubation time of a virus refers to the time between individuals contracting the virus and when they start showing physical signs of infection. During incubation time a host is deemed infective but not sick. Latency time refers to the period when a host is sick, but is not infective; there is only a subtle difference between these two parameters. Hosts can progress from being incubating to latent but not vice versa.
After examining some past works [1,2] I suspect that increases in latency and incubation times accelerate epidemic spread.
Ahmed and Agiza [3] propose an implementation of the SIRS epidemic model using cellular automata (CA) and outline four sets of CA transition rules. Each set of rules incorporates different realism parameters that introduce heterogeneity into an otherwise plain and idealised CA world. Two of these rule sets integrate epidemic latency and incubation. After implementing these test cases I compare their results with those from pure SIRS models and examine the differences. Ahmed and Agiza have shown in their simulations that incubation and latency characteristics promote epidemic spread rather than attenuate it.
The SIRS epidemic model represents individuals as finite 3-state automata: Susceptible, Infective, and Recovered. To measure whether a model's inclusion of latency and incubation times will influence population behaviour at steady state, comparisons are made between susceptibility phase plots. These plots are generated before and after incubation, and latency times are integrated into their respective epidemic model. The intention is to show that even when probabilistic variables are kept constant other non-stochastic parameters such as those relating to time still have an effect on epidemic spread.
The CA implementation proposed by Ahmed and Agiza uses a total population of 500 and runs for 10000 time steps. It is important to note that their model deals with only one spatial dimension, that is, each cellular automaton has only two neighbours. This appears to be an overly restrictive simulation environment because I wish to investigate two dimensional spread. Hence I chose to use interaction neighbourhoods of size 9, that is, each individual can interact with 8 eight neighbours. To offset the four-fold increase in neighbourhood size and allow meaningful comparison with the results of Ahmed and Agiza, I increased the host population size in my simulation by the same factor.
My simulation executes for fewer time steps than that of Ahmed and Agiza, namely 100 rather than 10000. This is primarily to reduce computation time. My assessment of the significance of non-stochastic variables in epidemic spread models focusses on population ratios rather than absolute figures. Therefore, as long as all experiments are carried out over the same number of time steps, qualitative experimental results should not be affected.
Ahmed and Agiza [3] have devised four ``next-state'' automaton algorithms each of which are functions of two probabilities: the probability of infection, p1; and the probability of susceptibility or relapse, p2. These four CA model rule sets will be used in my experiments:
The above case scenarios will be examined separately and their results compared. The first two cases do not have additional non-probabilistic variables and serve as references for comparison. The latter two test my hypothesis that epidemics with latency and incubation characteristics will spread faster.
In my model the world comprises a lattice of 2500 individuals; the lattice dimensions are 50 ×50. It is important to note that the exact lattice dimensions are arbitrary because the world edges ``wrap-around'' creating a toroidal surface. Ahmed and Agiza include in their model spontaneous infection with probability of 0.0001. This probability is an abstraction of probable external infection effects such as foreign or travelling entities; I have mirrored this variable in my simulation. As I am examining epidemic behaviour in the infinite time limit, absolute time intervals are not of interest; therefore I have assumed infections only last one time quantum, that is, individuals who are Infective at this time step will be Recovered by the next time step. Ahmed and Agiza use this same assumption in their model.
I have used Java 2 to implement the four case scenarios and ran the simulations on Celeron 700MHz computers. These machines have 128Mb of RAM and run Linux RedHat 7.2.
For each of the case scenarios outlined by Ahmed and Agiza the two probabilities p1 and p2 are varied and plots are generated to indicate the proportion of the population that is still not infected after 100 time steps. Comparisons will be made with each phase plot and possible conclusions will be discussed. When interpreting the phase plots, it is important to note that their mottled appearance is due to the finite steps I used in selecting probabilities to execute the simulation over. If more samples were taken then the jump discontinuities in the plots would be less apparent. However, the actual shape of the shaded regions is not the focus of this investigation, rather it is their combined areas that is of note.
The pure SIRS epidemic model does not include any heterogeneity in the simulation environment. Similar to the other test cases recovery time is constant, however in this case, susceptibility is uniform amongst hosts also. The transition rules are outlined in Table 1 and the simulation results after 100 time steps displayed in Figure 1.
| Present state | Next state |
| Susceptible with an Infective neighbour | Infective with probability p1 |
| Infective | Recovered |
| Recovered | Susceptible with probability p2 |
| Any | Infective with probability 0.0001 |
|
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The horizontal axis of the phase plot refers to the probability of infection of an individual, given that it has an infective neighbour. The vertical axis denotes the probability that a Recovered individual loses its immunity and reverts back into the Susceptible state. The shading of the graph corresponds to the proportion of the total population that are Infective; the darker the shading, the more Infectives.
Figure 1 shows that as the two probabilities approach unity, the proportion of Infectives after 100 time steps grows markedly. When p1 and p2 are small, there appears to be no infection at all. It is important to note that the anomalous grey regions near the centre of the plot are probably due to the spontaneous infection factor as introduced by Ahmed and Agiza [3].
Ahmed and Agiza, though they did not clearly outline why, included a variable susceptibility test case into their experiments. Although I have not hypothesised about the affect of variable susceptibility on epidemic spread I have included it in my series of experiments also. Regardless of the reason for its inclusion, the variable susceptibility phase plot can be used as a secondary point of reference when examining the later cases that incorporate latency and incubation.
The effect of variable susceptibility is realised by including an intermediate state between Susceptible and Infective; this state is called Less Susceptible. The introduction of this new state means that a less susceptible, or higher immunity individual will require 2 consecutive exposures to an Infective neighbour before it too becomes Infective. In Ahmed and Agiza's model, the ratio of normal susceptible hosts to lower susceptible hosts is 7:3. I am unsure where these figures have been take from, perhaps from statistical data, but their value is arbitrary for comparison purposes.
| Present state | Next state |
| Susceptible and has an Infective neighbour | Infective with probability p1 |
| Less Susceptible and has an Infective neighbour | Susceptible |
| Infective | Recovered |
| Recovered | Susceptible with probability 0.7 ×p2 or less susceptible with probability 0.3 ×p2 |
| Any | Infective with probability 0.0001 |
 |
In Figure 2 the dark shaded region has a larger area than the one in Figure 1. This is somewhat counterintuitive as I expected the decrease in susceptibilities to have an attenuating affect on epidemic spread. However, it is important to note that these plots only show Susceptible individuals and do not distinguish between Recovered and Infectives, that is, the dark areas will include some Recovered hosts. Even so, if we attribute the isolated dark regions to spontaneous infections and ignore them, then the area of the shaded region in the top right is not much larger than that in Figure 1. The focus of these experiments is not on whether the area of the shaded region has increased or decreased, but that changes in that area imply the ratio of Susceptibles to Infectives to Recovered has changed.
The magnitude of this ratio change indicates how influential a particular epidemic spread parameter is on the steady state characteristic of an infected population.
Incubation means a host is infecting other hosts, but is not sick itself. Such hosts are unwitting carriers of the infection. An incubation state is added to the basic SIRS model which precedes the Infective state. This is more indicative of real life because the symptoms of a disease are not always apparent. The state transitions for this case are summarised in Table 3.
After examining the phase plot for this case in Figure 3 we can see that the infection pattern has become more dispersed. Andrewes [4] explains this effect by saying that we now have two groups of individuals who can pass on the infection: Incubating and Infective. Incubators behave in the same manner as Infectives, hence increasing the total number of contacts between Susceptibles and Infectives. This also means that any spontaneous infections are more likely to become large scale outbreaks as represented by the numerous black spots in Figure 3.
| Present state | Next state |
| Susceptible with Incubating or Infective neighbours | Incubating with probability p1 |
| Incubating | Infective |
| Infective | Recovered |
| Recovered | Susceptible with probability p2 |
| Any | Spontaneous Infective with probability 0.0001 |
 |
The final case under consideration is the one that incorporates both incubating and latent host states. A latent individual is infected, but is not able to pass the pathogen to adjacent individuals; Latents are a subset of the Infective population whereas incubating individuals are a subset of the Susceptible population. Latency time has the effect of extending the total infection time of a virus.
| Present state | Next state |
| Susceptible with an incubating neighbour | Incubating with probability p1 |
| Incubating | Latent |
| Latent | Recovered |
| Recovered | Susceptible with probability p2 |
| Any | Spontaneous Infective with probability 0.0001 |
 |
The extended length of infection means that there are more contacts between Infective and Susceptible individuals; this should result in faster epidemic spread and this is visible in Figure 4. We can see that not only has the darkened region in the high probability quadrant expanded in area, but spontaneous infections are more likely to become outbreaks.
A more rapid spread of an outbreak is expected because there are more active states: Infective, Incubating and Latent; compared to passive states such as Recovered and Susceptible. Mollison [2] describes Latent individuals as a means for the virus to permeate through a population and flare up simultaneously in many different locations. This has the effect of not only more rapid spread, but more distributed spread.
To make model implementation easier, unnecessary details need to be abstracted out. A question then arises: which variables can be omitted without sacrificing model accuracy? In the case of epidemic models, Ahmed and Agiza [3] implement a probabilistic SIRS model using CA and propose four different rule sets, each rule set incorporating different epidemic spread parameters. Those parameters are variable susceptibility, virus incubation and virus latency. To assess whether any of these factors have a significant affect on an epidemic's spread, they are implemented in isolated models and run over similar idealised environments.
It is shown here that viruses that exhibit incubation characteristics appear to spread faster than those without, and that viruses that exhibit both incubation and latency characteristics will spread even faster. As far as devising an epidemic model is concerned, to maintain realism we should include incubation and latency in our implementations.