Dynamics of the individual pathogens.

We consider two distinct pathogen species, strains, or clones (henceforth ‘pathogens’), which we assume do not interact; i.e., the interaction between the host and one of the pathogens is entirely unaffected by its infection status with respect to the other. Epidemiological properties that are therefore unaffected by the presence or absence of the other pathogen include initial susceptibility; within-host dynamics, including rates of accumulation and/or movement within tissues; host responses to infection; and onward transmission. Assuming a fixed-size host population and susceptible-infected-susceptible (S-I-S) dynamics [52], the proportion of the host population infected by pathogen i∈{1,2} follows (1) in which the dot denotes differentiation with respect to time, β_i is a pathogen-specific infection rate, and μ is the host's natural death rate.

Whereas natural mortality may be negligible for acute infections, it cannot be neglected for chronic (i.e., long-lasting) infections, which are responsible for a large fraction of coinfections in humans and animals [3, 53]. Likewise, plants remain infected over their entire lifetime following infection by most pathogens, including almost all plant viruses, as well as the anther smut fungus, which drives one of our examples here [45].

We assume that the disease-induced death rate (virulence) is zero, as otherwise there would be ecological interactions between pathogens [30]. However, our model can be extended to handle pathogen-specific rates of clearance (S1 Text, Section 4; S2 Text, Section 3).

Tracking coinfection.

Making identical assumptions, but instead distinguishing hosts infected by different combinations of pathogens, leads to an alternate representation of the dynamics. We denote the proportion of hosts infected by only one of the two pathogens by J_i, with J_1,2 representing the proportion coinfected. Pathogen-specific net forces of infection are (2) and so (3) in which J_∅ = 1−J₁−J₂−J_1,2 is the proportion of hosts uninfected by either pathogen (Fig 1).

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Fig 1. Schematic of the model tracking a pair of noninteracting pathogens.

The model is defined in Eqs 1–3: J_∅ denotes uninfected hosts; J₁ and J₂ are hosts singly infected by pathogens 1 and 2, respectively; J_1,2 are coinfected hosts; I₁ = J₁+J_1,2 and I₂ = J₂+J_1,2 are net densities of hosts infected by pathogens 1 and 2, respectively.

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Prevalence of coinfected hosts.

We assume the basic reproduction number, R_0,i = β_i/μ>1, for both pathogens. Solving Eq 3 numerically for arbitrary but representative parameters (Fig 2A) shows the proportion of coinfected hosts (J_1,2) to be larger than the product of the individual prevalences (P = I₁I₂ from Eq 1). That J_1,2(t)≥P(t) for large t (for all parameters) can be proved analytically (S1 Text, Section 1.1). Numerical exploration of the model suggests that J_1,2(t) invariably becomes larger than P(t) relatively rapidly, and well within the lifetime of an average host, over a wide range of initial conditions and plausible sets of parameter values (S1 Text, Section 1.2; S1 Fig).

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Fig 2. Simulations of the two-pathogen model show that net densities of the two pathogens are positively correlated.

J₁. and J₂ are hosts singly infected by pathogens 1 and 2, respectively; J_1,2 is coinfected hosts; I₁=J₁+J_1,2 and I₂=J₂+J_1,2 are net densities of hosts infected by pathogens 1 and 2, respectively. (A) Dynamics of the deterministic model (Eqs 1–3), with β₁ = 5, β₂ = 2.5, and μ = 1 (parameters have units of inverse time). (B) Dynamics of a stochastic version of the model, in a population of size N = 1,000 (see also Methods section ‘Stochastic models’). (C) A single trajectory from the stochastic simulation (black line) in panel B (restricted to the time interval starting from the dashed line at t = 5) in the phase plane (I₁,I₂), and the 90% and 99% confidence ellipses (dashed and dotted curves, respectively) generated from an analytical approximation to the stochastic model. The grey dots represent pairs of individual values of (I₁,I₂) sampled at time T = 10 from 10³ independent replicates of the stochastic model. (D) The probability density of Λ, the relative deviation of the prevalence of coinfection (Eq 5), as estimated at T = 10 from 10⁴ independent replicates of our stochastic model when N = 1,000. The relative deviation was much greater than zero in all individual simulations. The mean value of Λ as estimated from these simulations, , was extremely close to the prediction from the deterministic model, Λ = 1/(5+2.5−1)≈0.1538 (Eq 5; these values are marked by red open and blue filled dots on the x-axis).

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Simulations of a stochastic analogue of the model (Fig 2B) reveal the key driver of this behaviour. The net prevalences of the pathogens considered in isolation, I₁ and I₂, are positively correlated (Fig 2C; Eq 27 in Methods section ‘Stochastic models’), because of simultaneous reductions whenever coinfected hosts die. The full distribution of point estimates of the relative deviation from statistical independence (see Eq 5) indicates the deviation is reliably greater than zero across an ensemble of runs of our stochastic model (Fig 2D). That the deviation is routinely positive is robust to alternative formulations of the stochastic model including environmental as well as demographic noise (S1 Text, Section 6.1; S2 Fig). It also becomes apparent quickly across a wide range of initial conditions; i.e., the sign and magnitude of the relative deviation are relatively robust to transient behaviour of our model (S1 Text, Section 6.2; S3 Fig).

Quantifying the deviation from statistical independence.

For R_0,i>1, the equilibrium prevalence of coinfection in our deterministic model is given by (4)

(See also Methods section ‘Equilibria of the two-pathogen model’). We introduce Λ, the relative deviation of the prevalence of coinfection from that required by statistical independence (), which here is given by (5)

The deviation is zero if, and only if, the host natural death rate is μ = 0. The observed outcome would therefore conform with statistical independence only for noninteracting pathogens when there is no host natural death (at the time scale of an infection). This reiterates the role of host natural death in causing deviation from a statistical association pattern. The relative deviation from statistical independence, Λ, becomes smaller as either or both values of R_0,i become larger. Deviations are therefore more apparent for smaller values of R_0,i. This is unsurprising, since if either pathogen has a very large value of R₀, almost all hosts infected with the other pathogen would be expected to become coinfected, and so both our model and the assumption of statistical independence would lead to very similar predictions.

This result (Eq 5) was first published by Kucharski and Gog [32] in a different context (model reduction in multistrain influenza models). Moreover, using a continuous age-structured model, these authors showed that one may recover statistical independence within infinitesimal age classes. The result in Eq 5 is related to ageing, as individuals acquire more infections as they age. As age increases, so does the probability of being infected with pathogens 1 and/or 2. Therefore, the prevalences of pathogens 1 and 2 are positively correlated [33]. A greater deviation from independence as the mortality rate μ increases is likely due to the fact that prevalence is increasing and concave with respect to age and saturates in older age classes [31].

Modelling coinfection by n noninteracting pathogens.

We denote the proportion of hosts simultaneously coinfected by the (nonempty) set of pathogens Γ to be J_Γ and use Ω_i = Γ\{i} (for i∈Γ) to represent combinations with one fewer pathogen.

The dynamics of the 2ⁿ−1 distinct values of J_Γ follow (6) in which the net force of infection of pathogen i is (7) and ∇_i is the set of all subsets of {1,…,n} containing i as an element. Eq 6 can be interpreted by noting the following:

• the first term tracks inflow due to hosts carrying one fewer pathogen becoming infected;
• the second term tracks the outflows due to hosts becoming infected by an additional pathogen, or death.

If R_0,i = β_i/μ>1 for all i = 1,…,n, the equilibrium prevalence of hosts predicted to be infected by any given combination of pathogens, , can be obtained by (recursively) solving a system of 2ⁿ linear equations (Eq 16 in Methods section ‘Equilibria of the n-pathogen model’).

These equilibrium prevalences are the prediction of our ‘Noninteracting Distinct Pathogens’ (NiDP) model, which in dimensionless form has n parameters (the R_0,i's, i = 1,…,n; Methods section ‘Fitting the models’).

If we simplify the model by assuming that all pathogens are epidemiologically interchangeable and so all pathogen infection rates are equal (i.e., β_i = β for all i), then if R₀ = β/μ>1, the proportion of hosts infected by k distinct pathogens can be obtained by (recursively) solving n+1 linear equations (Eq 22 in Methods section ‘Deriving the NiSP model from the NiDP model’). This constitutes the prediction of our ‘Noninteracting Similar Pathogens’ (NiSP) model, a simplified form of the NiDP model requiring only a single parameter (R₀).

Using the models to test for interactions.

If either the NiSP or NiDP model adequately explains coinfection data, those data are consistent with the underpinning assumption that pathogens do not interact. Which model is fitted depends on the form of the available data, specifically whether only the number of pathogens or instead which particular combination of pathogens infecting each host is known.

Studies often quantify only the number of distinct pathogens carried by individual hosts, without necessarily specifying the combinations involved [22, 45–50]. There are insufficient degrees of freedom in such data to fit the NiDP model, and so we fall back upon the NiSP model. In using the NiSP model, we additionally assume all pathogens within a given study are epidemiologically interchangeable.

We identified four suitable studies reporting data concerning strains/clones of a single pathogen and tested whether these data are consistent with no interaction. For all four studies (Fig 3), the best-fitting NiSP model is a better fit to the data than the corresponding binomial model assuming statistical independence (Eq 28 in Methods section ‘Models corresponding to assuming statistical independence’). Application of our model to three additional examples for data sets considering distinct pathogens, which deviate more markedly from the epidemiological equivalence assumption, is described in S2 Text, Section 1 (see also S4 Fig, S1 Table, S2 Table).

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Fig 3. Comparing predictions of the NiSP model with binomial models assuming statistical independence.

In using the NiSP model, pathogens are assumed to be epidemiologically interchangeable: we have therefore restricted attention to data sets concerning strains/clones of a single pathogen species. (A) Strains of human papillomavirus [22]; (B) strains of the anther smut pathogen (Microbotryum violaceum) on the white campion (Silene latifolia) [45]; (C) strains of tick-transmitted bacteria (Borrelia afzelii) on bank voles (Myodes glareolus) [46]; and (D) clones of malaria (Plasmodium vivax) [47]. Insets to each panel show a ‘zoomed-in’ section of the graph corresponding to high multiplicities of clone/strain coinfection, using a logarithmic scale on the y-axis for clarity. Asterisks indicate predicted counts smaller than 0.1. In all four cases, the NiSP model is a better fit to the data than the binomial model (Δ AIC = 572.8,158.6,293.8 and 596.3, respectively). For the data shown in (A), there is no evidence that the NiSP model does not fit the data (lack of goodness of fit p = 0.08), and so our test indicates the human papillomavirus strains do not interact. For the data shown in (B–D), there is evidence of lack of goodness of fit (all have lack of goodness of fit p<0.01). Our test therefore indicates these strains/clones interact (or are epidemiologically different). The underlying data for this figure can be found in S3 Data, S4 Data, S5 Data, and S6 Data. AIC, Akaike information criterion; NiSP, Noninteracting Similar Pathogens.

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In one case—coinfection by different strains of human papillomavirus (HPV) [22] (Fig 3A)—we find no evidence that the reported data cannot be explained by the NiSP model. These data therefore support the hypothesis of no interaction—and indeed no epidemiological differences—between the pathogen strains in question.

In the three other cases we considered—strains of anther smut (M. violaceum) on the white campion (S. latifolia) [45] (Fig 3B), strains of the tick-transmitted bacterium B. afzelii on bank voles (M. glareolus) [46] (Fig 3C), and clones of a single malaria parasite (P. vivax) infecting children [47] (Fig 3D)—despite outperforming the model corresponding to statistical independence, the best-fitting NiSP model does not adequately explain the data. We therefore reject the hypotheses of no interaction in all three cases, noting that our use of the NiSP model means it might be epidemiological differences between pathogen strains/clones—or perhaps simply lack of fit of the underpinning S-I-S model—that have in fact been revealed.

Other studies report the proportion of hosts infected by particular combinations (rather than counts) of pathogens, although many of those concentrate on helminth macroparasites for which our underlying S-I-S model is well known to be inappropriate [54].

However, a methodological article by Howard and colleagues [17] introduces the use of log-linear modelling to test for statistical associations. Conveniently, that article reports the results of that methodology as applied to a large number of studies focusing on Plasmodium spp. causing malaria.

By interrogating the original data sources (Methods section ‘Combinations of pathogens [NiDP model]’), we found a total of 41 studies of malaria reporting the disease status of at least N = 100 individuals, and in which three of P. falciparum, P. malariae, P. ovale, and P. vivax were considered. Data therefore consist of counts of the number of individuals infected with different combinations of three of these four pathogens, a total of eight classes. There were sufficient degrees of freedom to fit the NiDP model, which here has three parameters, each corresponding to the infection rate of a single Plasmodium spp. Fig 4A shows the example of fitting the NiDP model to data from a study of malaria in Nigeria [51].

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Fig 4. Using the NiDP model to reanalyse malaria data sets considered by Howard and colleagues [17].

In using the NiDP model, there is no need to assume malaria-causing Plasmodium spp. are epidemiologically interchangeable. (A) Comparing the predictions of the NiDP model with a multinomial model of infection (i.e., statistical independence) for the data set on P. falciparum (‘F’), P. malariae (‘M’), and P. ovale (‘O’) coinfection in Nigeria reported by Molineaux and colleagues [51]. The NiDP model is a better fit to the data than the multinomial model (Δ AIC = 326.2); additionally, there is no evidence of lack of goodness of fit (p = 0.40). This data set is therefore consistent with no interaction between the three Plasmodium species. (B) Comparing the results of fitting the NiDP model and the methodology of Howard and colleagues [17] based on log-linear regression and so statistical independence. For 16 (i.e., 12 + 4) out of the 41 data sets we considered, the conclusions of the two methods differ. The underlying data for this figure can be found in S7 Data and S8 Data. AIC, Akaike information criterion; NiDP, Noninteracting Distinct Pathogens.

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Fitting the NiDP model allows us to test for interactions between Plasmodium spp., without assuming they are epidemiologically interchangeable. In 18 of the 41 cases we considered, our methods suggest the data are consistent with no interaction (Fig 4B). We note that in 12 of these 18 cases, the methodology based on statistical independence of Howard and colleagues [17] instead suggests the Plasmodium spp. interact.

Equilibria of the two-pathogen model.

The two-pathogen model is given by Eqs 1–3. Since the population size is constant, J_∅ = 1−J₁−J₂−J_1,2, and so it follows that (8)

It is well known [52] that if R_0,i = β_i/μ>1 and I_i(0)>0, the prevalence of pathogen i will tend to an equilibrium .

Since F_i = β_iI_i and J_i = I_i−J_1,2, the rate of change of coinfected hosts in Eq 3 can be recast as (9) which leads immediately to the results given in Eqs 4 and 5.

Equilibria of the n-pathogen model.

The n-pathogen model is given by Eqs 1, 6 and 7. Since the host population size is constant, J_∅ = 1−∑_Γ∈∇J_Γ, where ∇ is the set of all 2ⁿ−1 sets with infected or coinfected hosts. It is also true that (10)

At equilibrium, Eq 6 becomes (11) in which and are equilibrium prevalences, and is the force of infection of pathogen i at equilibrium, i.e., (12)

Since these forces of infection are constant and do not depend on the equilibrium prevalences, the set of 2ⁿ−1 equations partially characterising the equilibrium is linear, with (13)

Similarly, Eq 10 is linear (14)

The equilibrium prevalences can be written very conveniently in a recursive form (i.e., using the first equation to fix , using to independently calculate all values of for |Γ| = 1, then using the set of values of when |Γ| = 1 to independently calculate all values of for |Γ| = 2, and so on). A recurrence relation to find all equilibrium prevalences can therefore be initiated with the following expression for the density of uninfected hosts: (15)

Then, one may recursively use the following equation, equivalent to Eq 13: (16)

Since the densities in Eq 16 are entirely in terms of the equilibrium densities of hosts carrying one fewer pathogen (), this allows us to recursively find the densities of all pathogens given pathogen-by-pathogen values of R_0,i.

Deriving the NiSP model from the NiDP model.

If all pathogens are interchangeable and so have identical values of R_0,i = R₀ ∀_i, then for any pair of combinations of infecting pathogens, Γ₁ and Γ₂, it must be the case that whenever |Γ₁| = |Γ₂|. This means the equilibrium prevalences of hosts infected by the same number of distinct pathogens must all be equal, irrespective of the particular combination of pathogens that is carried. In this case, solving the system is much simpler. First, Eq 11 can be rewritten as (17) in which . The net prevalence of hosts infected by k distinct pathogens is (18) in which ∇(k) is the set of combinations of {1,…,n} with k elements. Since the form of Eq 17 depends only on |Γ|, all individual prevalences involved in are identical, and so (19) in which is a combinatorial coefficient, and is any of the individual prevalences for which |Γ| = k. The ratio between successive values of is given by (20)

From Eq 15, it follows that (21) in which R₀ = β/μ. For 1≤k≤n, Eqs 17 and 20 together imply (22) a form that admits a simple recursive solution.

Stochastic models.

Fig 2B and 2C were generated by simulating the stochastic differential equation (SDE) corresponding to Eq 3, simulating a continuous-time Markov chain model using Gillespie's algorithm gave consistent results. Confidence ellipses were obtained from an approximate expression for the covariance matrix at equilibrium (see below).

The continuous-time Markov chain model corresponding to the unscaled version of Eqs 3–8 tracks a vector of integer-valued random variables X(t) = (J_∅(t),J₁(t),J₂(t),J_1,2(t)). Defining ΔX = X(t+Δt)−X(t) = (ΔJ_∅,ΔJ₁,ΔJ₂,ΔJ_1,2), changes of ±1 to each element of X(t) occur in small periods of time Δt at the rates given in Table 1. Stochastic trajectories from this model can conveniently be simulated via the Gillespie algorithm [72]. Note that the numeric values of the infection rates and the host birth rate must be altered to account for the scaling by population size.

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Table 1. Transitions in the two-pathogen stochastic models.

The prevalence of uninfected host is J_∅, the prevalence of each class of singly infected hosts is J_i (for i∈[1,2]), and the prevalence of coinfected host is J_1,2. The net force of infection of pathogen i is F_i = β_iI_i/N = β_i(J_i+J_1,2)/N (note the scaling by the population size N relative to the forces of infection as used in the deterministic version of the model). To ensure a constant host population size, we have made the simplifying assumption that removal and replacement occur simultaneously; this has no effect on our qualitative results.

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The model can also be written as a system of SDEs, an approximation to the continuous-time Markov chain that is valid for sufficiently large N [73] and that is particularly well suited for simulation of the stochastic model when the population size is large. This form of the model again tracks the seven events in Table 1, although in the SDE formulation the random variables in X(t) are continuous-valued. A heuristic derivation is based on a normal approximation described below. Alternately, the forward Kolmogorov differential equations in the continuous-time Markov chain model are closely related to the Fokker-Planck equation for the probability density function of the SDE model [74].

The expected change and covariance of the changes can be computed from Table 1 to order Δt via (23) where is the unscaled version of the deterministic model as specified in Eqs 3–8 with N = J_∅+J₁+J₂+J_1,2 (a constant) and F_i = β_i(J_i+J_1,2)/N. In addition, the matrix Σ is given by (24)

The changes in a small time interval Δt are approximated by a normal distribution via the central limit theorem: , where 0 = zero vector. The covariance matrix Σ can be written as Σ = GG^T. Letting Δt→0, the SDE model can therefore be expressed as (25)

The matrix G is not unique, but a simple form with dimension 4×7 accounts for each event in Table 1 [74]. Each entry in matrix G involves a square root, and W is a vector of seven independent standard Wiener processes, where dW_i≈ΔW_i(t) = W_i(t+Δt)−W_i(t)~Normal(0,Δt). An explicit form for the SDE model in Eq 25 is (26)

In S1 Text, Section 2, we show that the covariance between the prevalences of pathogen 1 and pathogen 2 as they fluctuate in the vicinity of their equilibrium values is approximately (27) with equality if, and only if, μ = 0 (assuming β_i>μ, i = 1,2). Only in the specific case μ = 0 is the deviation from statistical independence equal to zero (Eq 5).

Models corresponding to assuming statistical independence.

If data are observations of numbers of individuals infected with k distinct pathogens, O_k, for k∈[0,n], statistical independence corresponds to assuming the infection load of a single individual follows the one-parameter, binomial model Bin(n,p), in which p is the pathogen prevalence (assumed identical for each pathogen and fitted appropriately to the data), and n is the maximum number of infections that is possible (i.e., the total number of distinct pathogens under consideration). Model predictions are then simply N samples from this binomial distribution, where N = ∑_kO_k is the total number of individuals observed in the data. One interpretation is as a multinomial model in which (28)

For the data for malaria corresponding to numbers of individuals, O_Γ, infected by different sets of pathogens, Γ, statistical independence corresponds to an n-parameter multinomial model, parameterised by the prevalences of the individual pathogens p_i (again fitted to the data), i.e., (29)

Fitting the models.

The host natural death rate, μ, can be scaled out of the equilibrium prevalences by rescaling time. Fitting the models therefore corresponds to finding value(s) for scaled infection rate(s) β_i, i.e., R_0,i = β_i/μ (all are equal for the NiSP model).

The method used to fit the model does not depend on whether the data are numbers of hosts infected by a particular combination of pathogens or numbers of hosts carrying particular numbers of distinct pathogens, since both can be viewed as N samples drawn from a multinomial distribution, with q_j observations of the j^th class. If the corresponding probabilities generated by the model being fitted are p_j, then the log-likelihood is (30)

The models were fitted by maximising L via optim() in R [75]. Convergence to a plausible global maximum was checked by repeatedly refitting the model from randomly chosen starting sets of parameters. All models were fitted in a transformed form to allow only biologically meaningful values of parameters; i.e., the basic reproduction numbers were estimated after transformation with log(R_0,i−1) to ensure R_0,i>1.

Model comparison.

To compare the best-fitting NiSP or NiDP model and an appropriate model assuming statistical independence (binomial or multinomial), we use the Akaike information criterion , in which is the log-likelihood of the best-fitting version of each model and k is the number of model parameters. This is necessary because these comparisons involve pairs of models that are not nested.

Goodness of fit.

We use a Monte-Carlo technique to estimate p-values for model goodness of fit, generating 1,000,000 independent sets of samples of total size N from the multinomial distribution corresponding to the best-fitting model, calculating the likelihood (Eq 30) of each of these synthetic data sets, and recording the proportion with a smaller value of L than the value calculated for the data [76]. This was done using the function xmonte() in the R package XNomial [77].

Numbers of distinct pathogens (NiSP model).

Results of fitting the NiSP model to data from four publications for strains of a single pathogen are presented in Fig 3. Error bars are 95% confidence intervals using exact methods for binomial proportions via binconf() in the R package Hmisc [78]. Results for three further data sets concerning different pathogens of a single host [46, 48, 50] are provided in Text S2 Section 1 (see also S4 Fig).

For convenience, the raw data as extracted for use in model fitting are retabulated in Table 2. Results of model fitting are summarised in Table 3. We used the value n = 102 for the number of distinct strains by López-Villavicencio and colleagues [45] following personal communication with the authors; there might be undetected genetic differences due to missing data—which would require a larger value of n in our model-fitting procedure—but we confirmed that our inferences are unaffected by taking any value of n∈[100,200].

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Table 2. Sources of data for fitting the NiSP model in which pathogen types, clones, or strains are assumed to be epidemiologically interchangeable.

The data sets include human papillomavirus [22], anther smut (M. violaceum) [45], B. afzelii on bank voles [46], and malaria (P. vivax) [47]. The underlying data for this table can be found in S1 Data.

https://doi.org/10.1371/journal.pbio.3000551.t002

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Table 3. Fitting the NiSP model.

The NiSP model was highly supported over the binomial model (ΔAIC≫10) in all cases tested. The final column of the table corresponds to the GoF test of the NiSP model; the value p>0.05 is highlighted in bold and corresponds to lack of evidence for failure to fit the data, and so the NiSP model is adequate for the data concerning human papillomavirus [22].

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Combinations of pathogens (NiDP model).

Howard and colleagues [17] report results of analysing 73 data sets concerning multiple Plasmodium spp. causing malaria (rows 68–140 of Table 1 in that paper). We reanalysed the subset of these studies satisfying certain additional constraints as detailed in the main text (see S2 Text, Section 2, for a full description of how the studies were filtered). This left a final total of 41 data sets taken from 35 distinct papers: 24 data sets considering the three-way interaction between P. falciparum, P. malariae, and P. vivax and 17 data sets considering the three-way interaction between P. falciparum, P. malariae, and P. ovale.

We used our method based on the NiDP model to test whether any of these data sets were consistent with no interaction between the Plasmodium spp. considered (Table 4). We found 15 data sets for which the NiDP model was (1) a better fit than the multinomial model as indicated by ΔAkaike information criterion (AIC) ≥ 2 and (2) sufficient to explain the data as revealed by our goodness of fit test. In these 15 cases, our methods therefore support the hypothesis of no interaction. For 11 of these 15 data sets (76, 109, 118, 130, 132, 68, 69, 70, 79, 95, 97, 98, 99, 100, 102), the results as reported by Howard and colleagues [17] instead suggest the strains interact.

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Table 4. Fitting the NiDP model.

Data sets that are consistent with no interaction between the Plasmodium spp. considered are highlighted in grey (and have a row number marked with bold font in the first column). Such data sets have both p-values for the GoF test of the NiDP model p(GoF)>0.05 (marked in bold in the sixth column), and ΔAIC≥2 (for the comparison between the NiDP model and the multinomial model; marked in bold in the 12th column), meaning the NiDP model is adequate. The multinomial model corresponds to the statistical independence hypothesis. Parameters R_0,1 and R_0,2 are associated with P. falciparum and P. malariae, respectively. Parameter R_0,3 corresponds either to P. vivax (upper part of the table, data sets 74–137) or to P. ovale (lower part of the table, data sets 68–103). The final column contains a “Y” whenever at least one association between a pair of pathogens was assessed to be significant by Howard and colleagues [17] (and “N” when not significant). A “Y” in cells shaded pink correspond to possible statistical associations that are consistent with our no-interaction model (NiDP), i.e., cases in which our methods lead to results diverging from those reported in [17].

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