Following the work in36, we construct an epidemiological model which tracks the disease dynamics and population of two species of hosts following the introduction of a pathogen. The native host (hereafter simply referred to as “type 1”) is vulnerable to the disease, but due to being well adapted to the native habitat has high fecundity when uninfected. The invasive host (hereafter referred to as “type 2”), has coevolved defenses to the pathogen that increase both its tolerance of and resistance to the disease, but is not inherently as well-adapted to the habitat in the absence of infection (i.e., its intrinsic rate of growth in the new habitat is lower than that of the native).
Our initial conditions correspond to a population of uninfected type 1 hosts with a small number of both uninfected and infected type 2 hosts, representing an invasion by a novel competitor carrying a novel pathogen into the type 1 population. We consider a vector-borne pathogen, and make the simplifying assumption that there is an already abundant competent vector species in the habitat. (For this initial formulation, we considered a scenario of mosquito-borne infections in birds, such as avian malaria37 or West Nile virus38, to motivate concrete choices.)
The model couples two biological dynamics: the daily vector-borne spread of the disease among hosts, and a yearly host breeding cycle. We simulate in discrete time-steps that represent days using an SIR model taking into account the interactions between the disease, the two species of host, and the vectors. The model also includes a passive death rate for hosts of vectors, which increases for hosts while infected. While the vectors are assumed to breed daily, the hosts reproduce as part of an assumed annual breeding season, every (t_c) time-steps (typically equal to 365). These dynamics were informed by considering an annually breeding bird population in a tropical environment, however, they are not meant to reflect the realism of any one biological system. They are chosen here merely to allow a clean interpretation of modeled scenarios. Future models should explore the impact of greater variety in the dynamics of possible vector and host reproductive patterns.
Epidemiological model
The model tracks eight variables corresponding to combinations of host species and vectors with their infection status. Hosts may be of type 1 or 2, and are either susceptible to the disease ((S_1, S_2)), currently infected ((I_1, I_2)), or recovered ((R_1, R_2)). We assume that recovery is complete and recovered individuals suffer no residual effects from their infection aside from a lifelong immunity to becoming reinfected. (We later set the recovery rate for host type 1 to 0, so (R_1 = 0) at all times, but leave it defined for the sake of generality.) For simplicity, we model using only one stage of infection in which individuals are both infectious and symptomatic. The model also tracks the status of the vector population, which may either be susceptible ((S_v)) or infected ((I_v)). We assume that vectors do not recover from the disease, but also suffer no negative effects from being infected, acting only as carriers.
For convenience of notation, we denote the total number of hosts
$$begin{aligned} H = S_1 + I_1 + R_1 + S_2 + I_2 + R_2 end{aligned}$$
and the relative frequencies of infection within their respective population
$$begin{aligned} F_1 = frac{I_1}{H}, F_2 = frac{I_2}{H},F_v = frac{I_v}{S_v+I_v} end{aligned}$$
which allows some equations to be written more compactly. Table 1 shows a summary of these variables.
The model also has several constant parameters that affect the dynamics. (beta _j) determines the probability that hosts of type j become infected when bitten by a single infected vector. We typically set (beta _1 > beta _2), making type 2 hosts less likely to become infected.
Likewise, (delta _j) determines the probability that a vector becomes infected when biting an infected host of type j.
(b_j) determines the bite rate for vectors on host type j. We assume that each vector bites the same number of hosts per day, so each vector’s probability of becoming infected depends only on the frequency of infection among hosts, while each host will be bitten more if there are more vectors.
(gamma _j) determines the proportion of infected hosts of type j that recover from the disease each day. We typically set (gamma _1 = 0 < gamma _2), meaning infected hosts of type 1 do not recover, while infected type 2 recover after an average of (1/gamma _2) days.
(mu _{j-}) determines the daily death rate for uninfected hosts of type j and (mu _{j+}) determines the death rate for infected host of type j. We typically set (mu _{1-} = mu _{2-}< mu _{2+} < mu _{1+}), meaning uninfected hosts have the same death rate regardless of type, infected type 2 have a higher death rate than uninfected hosts, and infected type 1 have the highest. (Both susceptible and recovered hosts are considered to be uninfected.) Table 2 shows a summary of parameters related to the SIR dynamics.
Equation 1 shows continuous ordinary differential equations approximating the dynamics. Note that the actual model instantiates these in discrete time-steps using the forward Euler method with (h = 1).
$$ begin{aligned}&frac{dS_1}{dt} = – S_1 beta _1 b_1 I_v /H – S_1 mu _{1-} &frac{dI_1}{dt} = S_1 beta _1 b_1 I_v /H – gamma _1 I_1 – I_1 mu _{1+} &frac{dR_1}{dt} = I_1 gamma _1 – R_1 mu _{1-} &frac{dS_2}{dt} = -S_2 beta _2 b_2 I_v /H – S_2 mu _{2-} &frac{dI_2}{dt} = S_2 beta _2 b_2 I_v /H – I_2 gamma _2 – I_2 mu _{2+} &frac{dR_2}{dt} = I_2 gamma _2 – R_2 mu _{2-}&frac{dS_v}{dt} = alpha _v H -S_v delta _1 b_1 F_1 -S_v delta _2 b_2 F_2 -S_v mu _v&frac{dI_v}{dt} = S_v delta _1 b_1 F_1 + S_v delta _2 b_2 F_2 – I_v mu _v end{aligned} $$
(1)
Following a standard SIR model, susceptible hosts can become infected, and infected hosts become recovered, but each equation also contains a negative term corresponding to deaths. Thus, the total population of hosts is strictly decreasing in this time-frame. We assume that the vectors breed on a much shorter timescale than hosts, so we include a term for their births here, while host births are implemented by a yearly breeding event. We assume no vertical disease transmission, so all new vectors begin in the susceptible category. We assume that the daily birthrate for each vector increases with access to hosts, and decreases with competition among other vectors for hosts and breeding sites, so we set it equal to (frac{alpha _v H}{S_v + I_v}), where (alpha _v) is a constant scaling factor. Since the birthrate for each vector contains the total number of vectors in its denominator, the total number of vector births in the population will simply be (alpha _v H).
A population with a larger number of hosts will be able to sustain a larger number of vectors. For a population with a constant number of hosts, the equilibrium vector population will be proportional to the number hosts: aH where (a = frac{alpha _v}{mu _v}) is the equilibrium vector density (number of vectors per host). For instance if (a = 2), then in equilibrium there will be twice as many vectors as hosts. Given a fixed number of hosts, the population of vectors will asymptotically approach the equilibrium value. In practice the total number of hosts is constantly changing, so the population of vectors will chase after this moving equilibrium, though for our standard parameters (alpha _v) and (mu _v) are sufficiently large such that this will occur on a short timescale, and the population of vectors remains close to the current equilibrium value.
Breeding event
Table 3 shows a summary of parameters related to the breeding event. Every (t_c) days (typically 365), a breeding event occurs according to the following process.
Let
$$begin{aligned}&Delta S_1 = t_c alpha _{1-}(S_1+R_1)+t_calpha _{1+} I_1 &Delta S_2 = t_c alpha _{2-}(S_2+R_2)+t_calpha _{2+} I_2 end{aligned}$$
be the number of new host offspring of each type born this generation. In order to maintain consistency of temporal units among the parameters, each birthrate parameter is multiplied by (t_c). Let H be the current total number of hosts. Let
$$begin{aligned} c = {left{ begin{array}{ll} 0 &{} hbox {if } H ge kappa 1 &{} hbox {if } H + Delta S_1 + Delta S_2 le kappa frac{kappa -H}{Delta S_1 + Delta S_2} &{} hbox {otherwise} end{array}right. } end{aligned}$$
be the proportion of offspring that survive to adulthood. (None, if the population is already above carrying capacity. All, if the difference between the reproducing population size and the carrying capacity exceeds the new births. If the population is approaching carrying capacity, juvenile mortality scales proportionally so that the population will hit carrying capacity but not exceed it.)
Then
$$begin{aligned}&S_1 + c Delta S_1 rightarrow S_1 &S_2 + c Delta S_2 rightarrow S_2 end{aligned}$$
We assume there is no vertical disease transmission, so all new hosts begin in the susceptible category. We assume that the host population is iteroparous, such that the new offspring and the existing adult population both carry over to the next generation. If the new population would exceed the carrying capacity, we assume the limited space or supplies reduces the number of successful offspring so that the population exactly reaches the carry capacity by reduction in juvenile survival rather than population-wide competition that could also reduce the adult population.
The carrying capacity is therefore what drives the interspecific host competition. Because births of both species are summed and then normalized by the total number of births, the higher the birthrate of one host, the larger a fraction of the available space it will capture during the breeding event. Similarly, the lower the death-rate of a host, the less space it frees up for the next breeding event. Even if one host species would be able to sustain a stable population on its own, the presence of a more fit competitor can lead to the extinction of the less fit type by driving its effective birth rate down.
Immune-reproductive trade-offs and boundary conditions
We assume that host type 1 is evolutionarily stable in the absence of the disease; an uninfected monoculture population below the carrying capacity will have at least as many births as deaths each cycle. In a continuous version of this model where births and deaths happened simultaneously, this might be defined by (alpha _{1-} ge mu _{1-}) . However in our model, the population spends many days decreasing due to deaths before the next breeding event occurs. The population exponentially decays throughout the cycle, and then jumps up during the breeding event. The number of new host births is proportional to the number of hosts at the start of the breeding event, which will be the lowest value of any other time during the cycle. Thus, the birth rate needs to be high enough that the surviving hosts can compensate despite their diminished numbers. Taking this into account, we get the condition
$$begin{aligned}&alpha _{1-} ge frac{1-(1- mu _{1-})^{t_c}}{(1-mu _{1-})^{t_c}} end{aligned}$$
Which is a higher bound on (alpha _{1-}) than the simpler one above, but will be close to it if (mu _{1-}) and (t_c) are small.
To implement the scenario in which type 2 has increased resistance and tolerance to the disease at the expense of overall fecundity, we implement the following boundary conditions:
$$begin{aligned}&beta _1> beta _2 &0 = gamma _1< gamma _2 &mu _{1-} = mu _{2-}< mu _{2+} < mu _{1+} &alpha _{1-}> alpha _{2-}> alpha _{2+} > alpha _{1+} end{aligned}$$
Type 2 hosts are less likely to contract the disease, and are able to recover from it, while type 1 lack the immunological strength to eradicate it completely. Additionally, while both types of host are weakened by the disease, type 2 suffer fewer negative effects. However, this stronger immune response comes at the cost of reducing their birth rate when compared to healthy type 1 hosts.
Due to the heterogeneous population, there is ambiguity in defining (R_0) for the disease. The two types of host have different transmission rates and durations of infection, and will therefore be responsible for different amounts of disease spread. To resolve this, we define several related values. Let (R_0^j) be the (R_0) of the disease in a homogeneous population of type j hosts: the average number of hosts infected (indirectly, through vectors) from a single infected host in a population consisting entirely of type j hosts.
$$begin{aligned}&R_0^1 = frac{delta _1 beta _1 a b_1^2}{mu _v mu _{1+}} &R_0^2 = frac{delta _2 beta _2 a b_2^2}{mu _v (mu _{2+}+gamma _2)} end{aligned}$$
We simplify the equation for (R_0^1) since (gamma _1 = 0). We define w to be the frequency of host type 1: (w := (S_1 + I_1)/H). Then (R_0) for the vectors is
$$begin{aligned} R_0^v = R_0^1 w + R_0^2 (1-w) end{aligned}$$
which will also be the effective (R_0) of the disease for the hosts in the mixed population.
For simplicity of results, we restrict to the case where type 1 is more infectious overall than type 2, in particular (R_0^1 > R_0^2). This allows us to avoid edge cases in simulation outcomes which are beyond the scope of this paper. We intend to lift this restriction and study these outcomes in future work.
Note
Although usual epidemiological model formulations can rely on the value 1 as the boundary condition for (R_0) to determine the epidemic potential of an outbreak, in this case we are calculating effective (R_0) in a dynamic host population, such that the decrease in disease spread due to saturation from recovered hosts and already infected hosts increases the actual thresholds. More accurate criteria require a technical and somewhat cumbersome analysis, which we leave for a future paper.
Source: Ecology - nature.com
