Text S2: Dynamic system and equilibria for stages of mosquitoes infected with the fungal pathogen

(i) Mosquito population dynamics

Susceptible mosquitoes

We now consider susceptible mosquitoes that have been infected with the fungal pathogen for u days, in host-seeking stages i, . Mosquitoes remain in the first host-seeking stage (i=1) for u days after contracting the fungus if they do not die due to fungal pathogen-induced mortality or other sources of mortality, and they do not find a blood meal in that time. The probability of this occurring, , is defined in Table S2. The density of mosquitoes in the first host-seeking stage, denoted , is thus given by

(S2.1)

Host-seeking stages are denoted , where mosquitoes spend days in stage i. These stages are comprised of mosquitoes that entered via three possible routes. Firstly, fungal pathogen-infected mosquitoes that left the i-1th host-seeking stage at time when their fungal infection age was , having found a host and not contracted the Plasmodium (Table S3). These then survived the i-1th non-host-seeking period, lasting days, and remained in the ith host-seeking period for time. They have fungal infection age . Secondly, mosquitoes that contracted the fungus during the i-1th non-host-seeking stage at time t-u, survived the rest of this non-host-seeking stage and then survived in the ith host-seeking stage for time without finding a blood meal. These mosquitoes have a fungal infection age < u < TM + . Thirdly, mosquitoes in that contracted the fungus whilst in the ith host-seeking stage at time t-u and then survived for u days without finding a blood meal. For these mosquitoes, . Thus

(S2.2)

where and are stage-specific probability functions defined in Table S2.

In calculating the density of non-host-seeking mosquitoes that are infected with the fungus we track both the age of the fungal infection, u, and the time since the most recent blood meal, a, (Table S3). The mosquito density in non-host-seeking stage i, denoted , consists firstly of mosquitoes that were in the ith host-seeking stage at time t-a with fungal infection age u-a, at which point they found a blood meal and did not contract the Plasmodium. They then survived in the ith non-host seeking stage for a days, avoiding fungal-pathogen-induced mortality and other mortality sources, with probability (Table S2). Stages also consist of mosquitoes that contracted the fungus whilst in the ith non-host-seeking stage at time t-u. They then survived with the fungal infection for u days. Thus,

(S2.3)

and

(S2.4)

The total densities of susceptible, fungal pathogen-infected mosquitoes in each stage of the gonotrophic cycle are denoted and for host-seeking and non-host-seeking stages respectively.

(S2.5)

(S2.6)

Exposed mosquitoes

For exposed, host-seeking mosquitoes that are infected with the fungal pathogen, we track both the Plasmodium infection age p and the fungal infection age u. The density of mosquitoes in the first host-seeking stage (i=1), denoted , consists of mosquitoes that entered the stage from three possible sources. Firstly, from susceptible, fungal-pathogen infected mosquitoes that found a blood meal and contracted the Plasmodium at time t-p when their fungal infection age was u-p days. These mosquitoes survived the first non-host-seeking stage of the exposed stage, lasting days, and then avoided mortality due to fungal infection or other mortality sources for p- days in the first host-seeking stage, without finding a blood meal. For these mosquitoes u>p. Secondly, from susceptible mosquitoes uninfected with the fungal pathogen who found a blood meal at time t-p and then contracted the fungal pathogen during the first non-host-seeking stage of the exposed stage, at time t-u. They then survived the remainder of this non-host-seeking stage, and survived in the first host-seeking stage for p- days without finding a bloodmeal. For these mosquitoes Lastly, from exposed mosquitoes who contracted the fungal pathogen in the first host-seeking stage at time p-u then survived without finding a blood meal for u days. For these mosquitoes For all mosquitoes in , . Therefore,

(S2.7)

where the stage-specific survival probabilities and are defined in Table S2.

For host-seeking stages , the density of exposed fungal-pathogen infected mosquitoes is denoted where is the time spent in the ith host-seeking stage (Table S3). Similar to (S2.7), mosquitoes in this stage can also come via three different routes. Firstly, from exposed mosquitoes that found a blood meal in host-seeking stage i-1 at time when their Plasmodium infection age was and their fungal infection age was . They then survived the following non-host-seeking period, lasting days, and then remained in the ith feeding cycle for days without finding a blood meal. For these mosquitoes, . Secondly, from exposed mosquitoes uninfected with the fungal pathogen that found a blood meal in host-seeking stage i-1 at time when their Plasmodium infection age was days. These mosquitoes then contracted the fungal infection during the ith non-host-seeking period at time t-u, survived the rest of the non-host-seeking stage, and then survived for days in the ith host-seeking stage without finding a blood meal. For these mosquitoes, . Lastly, from exposed mosquitoes that became infected with the fungus in the ith host-seeking stage, and then survived for a further u days without finding a blood meal. For these mosquitoes . For all mosquitoes in , . Therefore,

(S2.8)

For the first non-host-seeking stage (i=1), the density of exposed, fungal-pathogen infected mosquitoes, denoted , consists of mosquitoes from two possible sources. Firstly from susceptible, host-seeking mosquitoes that found a blood meal and contracted the Plasmodium at time t-a when their fungal infection age was u-a. They then survived a days in the first non-host-seeking stage. For these mosquitoes,and u>p. Secondly, from exposed mosquitoes that caught the fungus in the first non-host-seeking stage of the exposed stage at time t-u. They then survived in the first non-host-seeking stage for a further u days. For these mosquitoes and .

(S2.9)

Densities of fungal pathogen-infected mosquitoes in non-host-seeking stages i>1, denoted, are calculated in a similar manner. is composed firstly of mosquitoes that found a blood meal when they were exposed and in host-seeking stage i-1 at time t-a with Plasmodium infection age p-a and fungal infection age u-a. They then survived a days in non-host-seeking stage i. For these mosquitoes u>a . Secondly, mosquitoes enter by catching the fungus in the ith non-host-seeking stage and then surviving in this non-host-seeking stage for a further u days. For these mosquitoes . For all mosquitoes in .

(S2.10)

Total densities of exposed mosquitoes infected with the fungal pathogen are denoted and for host-seeking and non-host-seeking stages respectively.

(S2.11)

(S2.12)

Infectious mosquitoes

The density of infectious mosquitoes in each host-seeking stage i that are infected with the fungal pathogen, denoted , is calculated for each fungal infection age u and time spent in the ith host seeking stage, . Starting with the first host-seeking stage (i=1), mosquitoes in can come from four possible sources. Firstly, from fungus-infected host-seeking mosquitoes whose Plasmodium infection age reached TE when their fungal infection age was u- at all times t-. These mosquitoes then avoided mortality due to fungal infection and other mortality sources for days without finding a blood meal, with probability (Table S2). Secondly, from infectious mosquitoes that contracted the fungal pathogen during the first host-seeking stage of the infectious stage, then survived for a further u days without finding a blood meal. Thirdly, from non-host-seeking, fungal-pathogen infected mosquitoes whose Plasmodium infection age reaches TE at days through a non-host-seeking stage. They then survived the remaining days in this non-host-seeking stage, with probability (Table S2), and go on to survive days in the first host-seeking stage without finding a blood meal. Lastly, from non-host-seeking mosquitoes uninfected with the fungal pathogen whose Plasmodium infection age reaches TE at days through a non-host-seeking stage. These mosquitoes then survive the remaining days in this non-host-seeking stage, contracting the fungal pathogen at some point through this non-host-seeking stage, where . They then survive days in the first host-seeking stage without finding a blood meal. Thus,

(S2.13)

Mosquitoes in host-seeking stages i>1 come from three possible sources. Firstly, fungal pathogen-infected mosquitoes that found a blood meal in host-seeking stage i-1 at time when their fungal infection age was , survived the ith non-host-seeking period then survived for days in the ith non-host-seeking stage without finding a blood meal. Secondly, mosquitoes uninfected with the fungal pathogen that found a blood meal at time and contracted the fungus during the ith non-host-seeking stage at time t-u. They then survived the rest of the ith non-host-seeking stage, and survived for days in the ith non-host-seeking stage without finding a blood meal. Lastly, from mosquitoes that contracted the fungus in the ith host-seeking stage at time t-u, then survived for u days without finding a blood meal.

(S2.14)

To calculate the density of infectious, non-host-seeking mosquitoes that are fungus infected, keep track of the time since the most recent blood meal, a (Table S3). The density of mosquitoes in the first non-host-seeking stage (i=1), denoted , consists of mosquitoes from two different sources. Firstly, from fungal-pathogen infected, non-host-seeking mosquitoes whose Plasmodium infection age reached TE at point in a non-host-seeking stage (Table S3). These mosquitoes then survived a further days in the first non-host-seeking stage of the infectious stage, with probability (Table S2). Secondly, from non-host-seeking mosquitoes without fungal infection whose Plasmodium infection age reached TE at point in a non-host-seeking stage. These mosquitoes then survived a further days in the first non-host-seeking stage of the infectious stage, contracting the fungal infection at time t-u, where . Thus,

(S2.15)

Mosquito densities in non-host-seeking stages i>1, denoted , consist firstly of fungal pathogen-infected mosquitoes that found a blood meal at time t-a in the i-1th host-seeking stage when their fungal infection age was u-a, and then survived a days in the ith non-host-seeking stage. Secondly, infectious mosquitoes that are without fungal infection and in the ith non-host-seeking stage enter by contracting the fungal pathogen at time t-u and then surviving a further u days, . Thus,

(S2.16)

The total densities of malaria infectious mosquitoes infected with the fungal pathogen are denoted and for host-seeking and non-host-seeking stages respectively.

(S2.17)

(S2.18)

(ii) Equilibrium solution

Expressions for the equilibrium density of susceptible, exposed and infectious host-seeking mosquitoes infected with the fungal pathogen are derived for the limiting case in which , and . For the case where the shape parameter , fungal infection has a constant linear effect on the mortality rate μ, so that . Thus define a constant parameter c, , so that MF(u) + μ=c μ.

Susceptible mosquitoes

Beginning with the equilibrium density of susceptible, fungus-infected mosquitoes in the first host-seeking stage, denoted , substitute MF(u) + μ=c μ into equation (S2.1) to give

(S2.19)

The equilibrium densities in host-seeking stages i>1 consists of all mosquitoes that became infected with the fungal pathogen in either host-seeking stages m=1,…,i or non-host-seeking stages m=1,…,i-1. These mosquitoes then survived to take enough blood meals to enter the ith host-seeking stage. First, denote the equilibrium density of susceptible, host-seeking mosquitoes in host-seeking stage i that caught the fungal infection in host-seeking stage m as . These mosquitoes survived with the fungal infection for days, progressing through host-seeking stages m,…,i and non-host-seeking stages m,…,i-1, taking i-m blood meals without contracting the Plasmodium. Since the probability of surviving host-seeking stages m,…,i is

(S2.20)

and the probability of surviving non-host-seeking stages m,…,i-1 is , we have

(S2.21)

Now denote the equilibrium density of susceptible, host-seeking mosquitoes in host-seeking stage i that caught the fungal infection in non-host-seeking stage m as . For these mosquitoes we need to consider at which point in the non-host-seeking stage the mosquito contracts the fungus (denoted a).

(S2.22)

We now have

(S2.23)

Similar, but more complex, expressions can be derived for and , the equilibrium densities of fungal pathogen-infected exposed and infectious host-seeking mosquitoes respectively.