Using ideas developed in [14] on multi-type birth-death process (MTBD), we develop a multi-type HIV model that captures the dynamics of the spread of HIV within the context of a generalised HIV epidemic in Africa. In this model, the host population comprises of different demes. A deme is a subdivision of a population that consists of individuals that possess closely related characteristics. The demes are represented by various types. The disease dynamics are modelled using birth-death processes, hence the name multi-type birth-death infection model. Within each deme, HIV dynamics are homogeneous. However, at the population level, the resultant dynamics are heterogeneous. Figure 1 shows a structured host population with two demes and the parameters for the model.

For example, deme 1 can be a general population and deme 2 a sub-population at risk.

We describe the parameters in the first deme. Those for the second deme are described analogously. The following are the parameters for the model in deme 1;

1) Parameter λ 11 is the rate at which an individual in deme 1 transmits or infects an individual of deme 1. Similarly, λ 12 is the rate at which an individual in deme 1 transmits to an individual of deme 2. These are the speciation (transmission) rate parameters in deme 1. There is both within ( λ 11 ) and across ( λ 12 ) deme transmission in this multi-type birth-death model.

2) Parameter μ 1 is the rate at which an individual in deme 1 dies. This is the extinction rate parameter.

3) Parameter ψ 1 is the rate at which an individual of deme 1 is sampled. Sampling in this context is the way HIV sequences are included into the study.

4) Parameter r is the probability that an individual in either of the demes becomes non-infectious upon sampling (being removed from infectious pool). An individual therefore remains infectious with probability 1 − r . Parameter r is the added parameter in our model which is not in the model presented in [14]. In our model framework, an individual either remains infectious or is removed from the infectious pool upon sampling. Therefore, r μ 1 ψ 1 is the rate at which an individual in deme 1 is removed from the infectious pool upon sampling. The rate at which an individual in deme 1 who remains infectious, transmits to an individual in deme 1 upon sampling is ( 1 − r ) λ 11 ψ 1 . The rate, ( 1 − r ) λ 12 ψ 1 is analogously defined.

For the derivation of the likelihood inference for parameter estimation, we still build on ideas used in [14]. Figure 2 illustrates the multi-type birth-death tree that we use in the derivation for likelihood of multi-type birth-death tree.

In Figure 2, black dots are sampled individuals, individuals either belong to deme 1 or 2. E is an edge which represents an individual. For parameters on the

dotted vertical line, 0 is the present, τ is the time at which a certain individual that belongs to deme 2 was sampled, t is the time taken to give raise to the sub-tree (dashed oval) that is subtended by an individual represented by an edge E. T is the time taken for the process or epidemic. This is the time from the origin of the epidemic to present. The model illustrated in Figure 1 results in a multi-type tree as visualized in Figure 2. In our model, the multi-type tree can have a sampled descendant, while a sampled individual in the model presented in [14] had no sampled descendants.

In this derivation, D E 1 ( t ) is the probability density for an individual represented by an edge E in deme 1 giving raise to the sub-tree of age t (dotted oval). For D E 1 ( t ) , t > τ as shown in Figure 2, where τ is the time at sampling of an individual. This implies that time increases backwards from present to the origin. We define another term Z 1 ( t ) , which is the probability that an individual in deme 1 is not sampled and does not have a descendant between t and 0. In order to obtain D E 1 ( t + Δ t ) , we assume that D E 1 ( t ) is known and computed where Δ t is a very small time step. Along an edge E for time step Δ t , either no event happens or birth events happen. This gives:

D E 1 ( t + Δ t ) = ( 1 − ( ∑ j = 1 2     λ 1 , j + μ 1 + ψ 1 ) Δ t ) D E 1 ( t ) + ∑ j = 1 2     λ 1 , j Δ t Z j ( t ) D E 1 ( t )     + ∑ j = 1 2     λ 1 , j Δ t Z 1 ( t ) D E j ( t ) + O ( Δ t 2 ) (1)

From Equation (1), the first term on the right is for no birth, death and sampling events, second term represents birth of an individual of deme j while lineage j produces no samples in time t, third term signifies birth of an individual of deme j while lineage 1 produces no samples in time t. The last term on right is the probability that more than one event occurs in the time interval Δ t .

After re-arranging the terms in Equation (1) and letting Δ t → 0 , we obtain the differential equation:

d d t D E 1 ( t ) = − ( ∑ j = 1 2     λ 1 , j + μ 1 + ψ 1 ) D E 1 ( t ) + ∑ j = 1 2     λ 1 , j Z j ( t ) D E 1 ( t )     + ∑ j = 1 2     λ 1 , j Z 1 ( t ) D E j ( t ) (2)

The initial condition at τ is given as:

D E 1 ( τ ) = r μ 1 ψ 1 + ( 1 − r ) ∑ j = 1 2     λ 1, j ψ 1 ,   for     j = 1 , and D E 1 ( τ ) = 0, for     j ≠ 1, (3)

The first term on the right of Equation (3) depicts individuals who become non-infectious upon sampling while the last term signifies those who are not removed from the infectious pool, i.e., they remain infectious even after being sampled. In typical African communities, many individuals infected with HIV remain infecting others either knowingly or unknowingly. When r is 1, the initial condition given in Equation (3) reduces to that used in [14].

Equation (2) is used to obtain the quantity D E 1 ( T ) which is the desired probability density for the multi-type birth-death tree. The derivation for Z 1 ( t ) is required to obtain the density of the tree. Z 1 ( t ) is derived using similar steps as those used in the derivation for Equation (2). The differential equation for Z 1 ( t ) is given as:

d d t Z 1 ( t ) = ( 1 − ψ 1 ) μ 1 − ( ∑ j = 1 2     λ 1 , j + μ 1 + ψ 1 ) Z 1 ( t ) + ∑ j = 1 2     λ 1 , j Z 1 ( t ) Z j ( t ) , (4)

with the initial condition,

Z 1 ( 0 ) = Z 2 ( 0 ) = 1. (5)

The first term on the right of Equation (4) represents death without sampling, the second signifies no birth, death and sampling as well as an individual in deme 1 producing no samples. The last term depicts a birth of an individual j but both individuals 1 and j producing no samples in time t. The initial condition given in Equation (5) implies that an individual at time t = 0 is not sampled. This is the case because the model used accounts for sampling through time only. Contemporaneous sampling, which is sampling at only one time point especially the present is not modelled in this inference.

The differential equations (2) and (4) with initial condition given by (3) are for deme 1. The corresponding differential equations for deme 2 were analogously derived. The system of differential equations with their initial conditions that is integrated along a given edge for 2 demes is given as:

d d t Z 1 ( t ) = ( 1 − ψ 1 ) μ 1 − ( λ 1 , 1 + λ 1 , 2 + μ 1 + ψ 1 ) Z 1 ( t ) + λ 1 , 1 Z 1 2 ( t ) + λ 1 , 2 Z 1 ( t ) Z 2 ( t ) d d t Z 2 ( t ) = ( 1 − ψ 2 ) μ 2 − ( λ 2 , 1 + λ 2 , 2 + μ 2 + ψ 2 ) Z 2 ( t ) + λ 2 , 1 Z 2 ( t ) Z 1 ( t ) + λ 2 , 2 Z 2 2 ( t ) d d t D E 1 ( t ) = − ( λ 1 , 1 + λ 1 , 2 + μ 1 + ψ 1 ) D E 1 ( t ) + ( λ 1 , 1 Z 1 ( t ) + λ 1 , 2 Z 2 ( t ) ) D E 1 ( t )                                 + λ 1 , 1 Z 1 ( t ) D E 1 ( t ) + λ 1 , 2 Z 1 ( t ) D E 2 (t)

d d t D E 2 ( t ) = − ( λ 2 , 1 + λ 2 , 2 + μ 2 + ψ 2 ) D E 2 ( t ) + ( λ 2 , 1 Z 1 ( t ) + λ 2 , 2 Z 2 ( t ) ) D E 2 ( t )                                 + λ 2 , 1 Z 2 ( t ) D E 1 ( t ) + λ 2 , 2 Z 2 ( t ) D E 2 ( t ) Z 1 ( 0 ) = Z 2 ( 0 ) = 1 D E 1 ( τ ) = r μ 1 ψ 1 + ( 1 − r ) ( λ 1 , 1 + λ 1 , 2 ) ψ 1 D E 2 ( τ ) = r μ 2 ψ 2 + ( 1 − r ) ( λ 2 , 1 + λ 2 , 2 ) ψ 2 (6)

Solving the system of equations in (6) numerically gives the calculations along an edge of a tree. This is then followed by pruning at the nodes and finally reaching the root of the tree. This results in obtaining either D E 1 ( T ) or D E 2 ( T ) depending on whether the root was in deme 1 or 2.

The probability density of the tree with the first individual at time T conditioned on observing at least an individual who is sampled for a structured population with 2 demes is given as:

p ( T | λ , μ , ψ , T ) = f 1 D E 1 ( T ) 1 − Z 1 ( T ) + f 2 D E 2 ( T ) 1 − Z 2 ( T ) (7)

The probability that an individual at time T is of deme 1 is given by f 1 and its distribution must be specified. Equation (7) is the likelihood of the parameters given the data. This is used for parameter inference. It should be noted that neither Z 1 ( T ) nor Z 2 ( T ) can be 1. This is because if any takes on 1, then the process will not give raise to the inferred tree. The conditioning is necessary since studies have shown that this conditioning gives more accurate parameter values [16]. The parameters that we estimated were:

λ = ( λ 1 , 1 , λ 1 , 2 , λ 2 , 1 , λ 2 , 2 ) μ = ( μ 1 , μ 2 ) , ψ = ( ψ 1 , ψ 2 ) , r and T .

The necessary changes suggested in Equation (3) to reflect HIV dynamics in an African setting were effected in R using an R package, Treesim of [17]. The parameters for simulated trees were then estimated using an R package, TreePar of [18] with changes to suit our model. In the first set of simulations, 50 trees with 100 tips were simulated under multi-type birth-death model with 2 demes (MTBD-2), with sampling probability of 0.2 for each of the two sub-populations ( ψ 1 = ψ 2 = 0.2 ). We first set r = 0.2 and r = 0.8 to investigate the effect of the probability of removal on parameter inference. We then increased the number of simulated trees to 100 but kept r = 0.8. This was to investigate the effect of increasing the number of simulated trees on parameter estimates.

Another set of simulations was done for 100 simulated trees, with 200 sampled tips. We varied the probability of removal, taking on values, r = 0.8 and r = 0.5. The sampling probability remained at 0.2 for both sub-populations. In another set of simulations, the sampling probabilities were set to ψ 1 = ψ 2 = 0.05 for both the sub-populations. This depicted sparse sampling which is highly the case in several HIV epidemics for Africa. Again the probability of removal was varied as it was set at r = 0.2 and r = 0.8. To investigate the effect of varying the sampling proportions, we simulated trees when the sampling proportions were different. We set ψ 1 = 0.2 and ψ 2 = 0.01 for this kind of parameter inference.

For comparison between two models, we used likelihood ratios. From [19], suppose that H 0 is derived from a full alternative ( H 1 ). Suppose further Θ 0 and Θ 1 are defined such that H 0 corresponds to a situation that a true parameter θ is in Θ 0 ⊆ θ , and H 1 corresponds to a case θ ∈ Θ 1 − Θ 0 . The log likelihood ratio statistic is given as:

L R = − 2 [ max θ ∈ Θ 0 log ( L ( θ ) ) − max θ ∈ Θ 1 log ( L ( θ ) ) ] (8)

In [20], they defined the likelihood ratio statistic as 2 ( ln ( X | H 1 ) ( X | H 0 ) ) , where

( X | H ) is the likelihood of the data (X) under hypothesis H. This likelihood statistic simplifies to the one given in Equation (8). Likelihood ratio statistic is asymptotically chi-squared distributed under the null hypothesis according to [19].

For heterogeneous dynamics, we assume that the population is structured into demes, say 2 as in our model. Each of the deme has different birth and death parameters. For homogeneous case, the population is not structured and individuals have the same birth and death rates. We investigated how well our model distinguished between heterogeneous and homogeneous dynamics. For simplicity, we had two sets of simulations, named A and B.

In A, both trees simulated and parameter inference were made under MTBD-2. For B, parameter inference was made for multi-type birth-death model with 1 deme (MTBD-1), yet trees were obtained under MTBD-2. We then performed likelihood tests to find out the number of trees that were rejected in support of the MTBD-2 (heterogeneous) dynamics. For homogeneous dynamics, we simulated trees under MTBD-1 and made parameter inference under both MTBD-1 and MTBD-2. Likelihood ratio tests were used to find out how many trees were rejected whose inference was made under MTBD-2.

Since the use of likelihood ratios requires that H 0 is derived from a full alternative H 1 , in A, we tested H 0 : Tree simulation under MTBD-2 and parameter inference under MTBD-1 against H 1 : Tree simulation under MTBD-2 and parameter inference under MTBD-2. In B, we had H 0 : Tree simulation under MTBD-1 and parameter inference under MTBD-1 against H 1 : Tree simulation under MTBD-1 and parameter inference under MTBD-2. H 0 and H 1 are defined as null and alternative models, respectively.

In the absence of real DNA sequence data for a structured host population, DNA sequences were simulated using a stochastic agent based model called, the Discrete Spatial Phylo Simulator (DSPS). DSPS was also used in [21]. The simulated DNA sequences were for a structured population which was composed of homosexual men (MSM), heterosexuals and bisexual men. The homosexual men were further split into LowMSM, MedMSM and HighMSM that were of increasing risk behaviour. Heterosexuals consisted of Low Females, High Females, Low Males and High Males. Bisexual men contacted both heterosexuals and homosexual men. During the simulations, half of bisexual men were diagnosed as if they were homosexual men (this was more likely early in infection) and the other half like heterosexual men (this was more likely late in infection). It was assumed that all bisexual men contacted men and women at the same rate on average. This group was not sub-divided further. The sampling was at 100% for 10 years (25 - 35, equivalent to 1995-2005). The simulated DNA sequences were from pol gene.

In order to make analysis faster, we down sampled the set of sequences which we used in the analysis. In this set, there were 21532 sequences and we sampled 200 sequences uniformly at random. We further reduced the sub-groups to homosexuals, heterosexuals and bisexuals. Out of 200 sequences, 157 were for homosexuals, 38 for bisexuals and only 5 for heterosexuals. Since we were interested in MTBD-2 model, we discarded 5 sequences for heterosexuals. This resulted in analysis of 195 sequences in our set of simulated HIV sequences.

We made a Bayesian study for a sampled set of sequences using Bayesian Evolution Analysis by Sampling Trees (BEAST) developed by [22]. This was done using the multi-type birth-death model package version 0.2.0 of BEAST. The inferred parameters included reproductive number for each deme, sampling proportions, becoming non-infectious rate, among others. For this study, we used the HKY substitution model with 4 gamma categories and estimated base frequencies. We set 0.867 as a proportion of invariant sites. Since our alignment contained sequences sampled at different times and the times were measured in years, we used a strict clock model with starting value set to 0.005. We set sampling change time to 10.0 which was slightly greater than time for the oldest sample. The sampling proportions were set to 0 before the time of the first sample.

We used the mean estimates for the parameters obtained from this Bayesian study to simulate trees with 200 sampled tips. The death rate parameters used in the simulation were equivalent to becoming non-infectious parameters from the Bayesian study. The birth-rate parameters were not inferred directly from this study. The birth-rate parameters were however computed from the reproductive numbers for each deme. This was possible since the BEAST study inferred the reproductive numbers of each deme. Since the reproductive number of each was

obtained using R 0 i = ∑ j = 1 2     λ i , j μ i , and yet we obtained R 0 i and μ i from the BEAST study, we therefore set ∑ j = 1 2     λ i , j = μ i R 0 i during the simulations with the

mean parameter estimates from the BEAST study of the simulated sequences. From the simulated trees, we estimated the parameters to evaluate the performance of our model with parameters that were obtained from the simulated DNA sequences.

We investigated whether there was a relationship between some tree statistics and the structuring of the host population. This was done using tree statistics that measure balance and imbalance of a phylogeny. In [23], they state that measuring the degree of imbalance or asymmetry of a tree topology may provide support for the hypothesis that species have different potential for speciation. It also serves as an indication of the patterns of speciation events for organisms under study. We used three tree statistics that measure tree balance and imbalance. We employed gamma-statistic, Sackin and Colles indices.

Gamma ( γ ) statistic was defined in [24]. Let g 2 , g 3 , ⋯ , g n be the inter-node distances of the reconstructed phylogeny with n taxa, the γ -statistic is defined as:

γ = ( 1 n − 2 ∑ i = 2 n − 1 ( ∑ k = 2 i     k g k ) ) − ( T 2 ) T 1 12 ( n − 2 ) (9)

where

T = ∑ j = 2 n     j g j

We analysed whether the values for γ -statistic are affected by the structuring of a host population. We computed gamma-statistic for both structured and non-structured host population. In [25], it is stated that under pure birth process, γ -statistic values of complete reconstructed phylogenies follow a standard normal distribution. If gamma-statistic values are greater than zero, then the internal nodes are closer to the leaves than expected under a pure birth process. On the other hand, if gamma-statistic values are less than zero, then internal nodes are closer to the origin (root) than expected under a pure birth model.

Another tree statistic which we used is called Sackin index which adds the number of internal nodes between each leaf and the root. It is defined as:

I s n = ∑ i = 1 n     N i (10)

where n is the number of leaves of the tree and N i is the number of internal nodes crossed in the path from leaf i to the root. It is well known in systematic biology that the expectation of I s n under Yule model is of order 2 n ln ( n ) . The normalized Sackin index converges in distribution as the number of leaves, n grows

to infinity. The normalized Sackin index is defined in [25] as I ^ s n = I s n − E ( I s n ) n .

Colless index is another tree statistic that assesses tree imbalance. It is defined as:

I c = 2 ( n − 1 ) ( n − 2 ) ∑ i = 1 n − 1 | T R − T L | (11)

where i is an internal node, n is the total number of leaves. For each internal node, T R is the number of terminal taxa subtended by the right hand branch and T L is the number of terminal taxa subtended by left hand branch. For a normalised I c , it ranges from 0 (perfect balance) to 1 (incomplete) balance.

Tree balance is an important consideration for phylogenies because the balance of the true phylogeny affects the accuracy of its estimates as stated in [26]. For Sackin and Colless indices, the higher the values for these two indices, the more unbalanced the trees are [26].

We simulated phylogenetic trees in R using TreeSim package of [17]. For a structured host population, we set our parameters as: λ 11 = 17 , λ 12 = 4 , λ 21 = 6 , λ 22 = 8 , , and For a non-structured host population, the parameters were set to, , , , , and Parameter is the rate at which an individual in sub-population i infects an individual in sub-population j, is the rate at which an individual in sub-population i dies and is the rate at which an individual is sub-population i is sampled, for and 2.

In the first set of simulations, we simulated 5000 trees with 200 sampled tips under both structured and non-structured host populations (A1 and A2). In the second set of simulations, we increased the number of sampled tips to 500 while other parameters remained the same as those for the first set of simulations under both structured and non-structured host populations (B1 and B2). In the third set of simulations, parameters were the same as those in first set of simulations but increased the number of sampled trees to 15,000 for both structured and non-structured populations (C1 and C2). Under each of the simulations, we computed gamma-statistic values for the simulated trees. Using an R package apTreeshape of [27], we computed Colless and Sackin indices for the simulated trees. We additionally computed their mean, median and 95% confidence intervals for the three tree statistics in each of the simulation sets.

Figure 3 and Figure 4 display the estimates for some of the selected parameters graphically. Parameter estimates for and when and are depicted in Figure 3. Figure 4 shows parameter estimates for and when and Apart from which is shown in lower part of Figure 4, other parameter estimates are uniformly scattered across their true values (solid red line).

For fixed sampling probabilities (), parameter estimates for, yielded slightly better estimates compared to estimates obtained for as shown in Table 1, upper and middle parts. There was an overall poor estimation for the second death parameter (). The lower part of Table 1 shows parameter estimates when the number of simulated trees was increased from 50 to 100. Better estimates were realized when the number of simulated trees was increased. Parameter estimation for greatly improved when the number of sampled trees was increased to 100. A close look at the middle and lower parts of Table 1, reveals that 95% confidence interval for narrows by close to half of the interval for the same parameter when the number of simulated trees was increased from 50 to 100.

Having observed that increasing the number of simulated trees to 100 improved parameter estimates, we increased the number of sampled tips from 100 to 200 in the next set of simulations. This is shown in Table 2. On making comparisons between the lower part of Table 1 and the upper part of Table 2, we note that increasing the number of sampled tips to 200 and keeping the number of simulated trees constant (100), registers a slight improvement in the parameter

estimation. The 95% confidence intervals for parameters in the upper part of Table 2 are narrower than those in the lower part of Table 1. When we varied the probability of removal (r) from 0.8 to 0.5, there was no significant difference in the parameter estimates as seen in the lower part of Table 2.

To investigate whether sampling intensity had an effect on parameter estimates, we varied the sampling proportion from 0.2 to 0.05. Parameter estimates when the sampling intensity was fixed at are shown in Table 3. Slightly wider intervals were obtained for the parameter estimates for this setting. Parameter was poorly estimated.

When parameter inference was made with different sampling proportions, i.e, for and, there was no difference noted in parameter inference for the simulated trees, results not shown here. Parameter was poorly estimated again.

For the error bars shown in Figure 5 and Figure 6, the intervals for parameter estimates obtained when each simulated tree had 200 sampled tips were narrower than those when the simulated trees had 100 sampled tips. The error bars for two parameters (and) are shown. The two selected parameters gave a representation for the rest of the parameters that were estimated.

To establish whether our model distinguished between heterogeneous population (MTBD-2) from homogeneous population (MTBD-1), we performed likelihood ratio tests. In the likelihood ratio inference, we used a decision rule of rejecting the null-hypothesis if the likelihood ratio (test statistic) is greater than

(9.236) for a significance level () of 0.1. We used since in MTBD-2 inference, 7 parameters were estimated including the likelihood value of the tree, while 4 parameters were estimated for MTBD-1. This gave an average of 6 parameters during the inference and since we needed a theoretical value of, we used for the inference. In the first scenario, 41 out of 50 trees rejected MTBD-1 model in favour of MTBD-2 using likelihood ratio tests at 0.9 level (). All the 50 trees supported MTBD-1 while rejecting MTBD-2. Some of the results are shown in Table 4 and Table 5.

However, according to [28], he suggests using a with 1 degree of freedom. So at of 0.1, we reject the null hypothesis if the likelihood ratio is greater than 2.706. With this rule, instead of 41 out of 50, it is 49 trees that rejected MTBD-1 model in favour of MTBD-2 in first scenario and results for the second scenario remain the same when we use the idea of [28].

We analysed 195 sequences from the sample. These sequences were structured into two groups, namely MSM and Bisexuals. For this sample, we run MCMC in BEAST which was of 12,000,000 length. The mixing was not very good as some parameters had effective sample sizes (ESS) which were low although some had high ESS of above 400. Our interest was mainly on three parameters, namely, becoming non-infectious rate and sampling proportion.

Figure 7 shows one of the trees sampled in the MCMC runs from BEAST. We also simulated 100 trees with 200 tips using the mean values of the parameters obtained from the analysis of the simulated sequences. These results are shown in Table 6.

From results shown in Table 7, generally the gamma-statistic values for simulated trees between structured and non-structured populations were all negative values apart from upper limits of the confidence intervals. The mean values for gamma-statistic values were generally more negative under non-structured population when we considered simulation cases A1, A2, B1 and B2. This was not true for simulation sets C1 and C2. When the number of tips was increased from 200 to 500, the mean and median values for gamma-statistic became more

negative. However, the confidence intervals became narrower. When the sampled trees were increased from 5000 to 15,000 trees, while kept number of sampled tips to 200, the mean values for gamma-statistic became more negative under structured population though remained almost constant under non structured population. The median values remained almost constant.

For Colless index, the mean and median values were higher under structured population compared to non-structured population. Increasing the sampled tips from 200 to 500 resulted in higher mean and median Colless index values. Increasing simulated trees from 5000 to 15,000 resulted in no big difference in the mean and median values. Sackin index values have the same pattern as the Colless index values.

The densities for the three tree statistics are shown in Figure 8. From these densities, there was no significant difference between structured and non-structured populations. The only difference is in the densities for Colless index as the median value for a structured population is higher than that for a non-structured one.