Vector-Borne Disease Models with Active and Inactive Vectors: A Simple Way to Consider Biting Behavior

Abstract

Vector-borne diseases are a serious public health problem, mosquitoes being one of the most important vectors. To analyze the dynamics of this type of disease, Ross–Macdonald models are commonly used. In its simplest formulation and the most common in scientific literature, it is assumed that all mosquitoes are biting at a given rate. To improve this general assumption, we developed a vector-borne disease model with active and inactive vectors as a simple way to incorporate the more general characteristics of mosquito feeding behavior into disease dynamics. Our objective is to obtain an estimate of the Ross–Macdonald biting rate from the feeding parameters that reproduce the same dynamics as the model with active and inactive vectors. Two different cases were analyzed: a SIS–SI model and a SIR–SI model with a single epidemic. Different methods to estimate the biting rate in the Ross–Macdonald model were proposed and analyzed. To compare the results of the models, different epidemiological indicators were considered. When the biting rate is estimated considering that both models have the same basic reproduction number, very similar disease dynamics are obtained. This method is a simple way to incorporate the mosquito feeding behavior into the standard Ross–Macdonald model.

Appendices

Pseudo-code of the Vector Population Dynamics

In this Appendix, we present the pseudo-code used to simulate the vector population dynamics. The computational program developed from this pseudo-code was used to generate the population curves showed in Fig. 1.

As mentioned in the main text, the mortality rate of vectors was considered constant $\mu _v$, thus the probability of a vector dying in a time interval of duration $\Delta t$ is equal to $1 - \exp (-\mu _v \Delta t)$. Considering this, the demographic processes in the vector population were simulated as follow. At each time step $\Delta t$, and for each individual in the vector population, a uniform distributed pseudo-random number u in the interval [0,1] is generated. If $u < 1 - \exp (-\mu _v \Delta t)$, the vector is removed from the vector population.

On the other hand, the birth rate was also considered constant. The number of newborns vectors in a time interval of duration $\Delta t$ was modeled considering a Poisson random variable with parameter $V_{eq}\mu _v \Delta t$, where $V_{eq}$ is the deterministic equilibrium value for the vector population. Thus, at each $\Delta t$ a pseudo-random number according to this distribution is generated, and this amount of vectors is incorporated in the population as inactive vector.

Active and inactive periods are random variables exponentially distributed with parameter $\lambda _a = \mu _a$ and $\lambda _i = \mu _i$ for active and inactive periods, respectively. Periods in each activity class (active and inactive) were simulated as follows. For each new vector, a value of the inactive period ($\tau _i$) is simulated, generating a pseudo-random number from an exponential distribution with parameter $\lambda _i$. This value $\tau _i$, plus the current time t, was stored in the variable $A\_change$. This variable is a future time at which the individual will change the activity state. When the current time t becomes greater or equal to $A\_change$, we change the agent’s activity state from inactive to active. In a similar way, for each newly active vector a value for the active period ($\tau _a$) from an exponential distribution with parameter $\lambda _a$ is generated and the variable $A\_change$ is updated as $\tau _a$ plus the current time t. When the current time reach this value, the activity state of the vector changes from active to inactive, and the process is repeated until the vector dies.

The simulation procedure used is described in the following pseudo-code:

• 1. Initialization of variables and parameters

  • (a) Set the vector (V(0)) population size, and the initial conditions: $V_i(0) = V(0)$ and $V_a(0)=0$.

  • (b) Set the time step $\Delta t$, the simulation duration $t_{sim}$ and the current time t equal to 0.

  • (c) Set the values of parameters $\mu _v$, $\mu _a$, $\mu _i$.

• 2. While $t \le t_{sim}$

  • (a) A random number of inactive vector are added to the population according to a Poisson distribution with parameter $V_{eq} \mu _v \Delta t$

  • (b) For each vector in the population

    • i. A uniform pseudo-random number in the interval (0, 1) is generated.

    • ii. If this number is less than or equal to $1 - \exp (-\mu _v \Delta t)$:

      The vector dies, and it is removed from vector population.

    • iii. Else

      • If vector.A_change is less than or equal to the current time t, the vector changes its activity state

        • If the vector is inactive:

          • Vector becomes ACTIVE, $V_i(t) = V_i(t) - 1$, $V_a(t) = V_a(t) - 1$.

          • Generate an active period $\tau _a$ according to the corresponding exponential distribution.

          • Set an active period $vector.A\_change = t + \tau _a$.

        • If the vector is active:

          • Vector becomes INACTIVE, $V_a(t) = V_a(t) - 1$, $V_i(t) = V_i(t) - 1$.

          • Generate an inactive period $\tau _i$ according to the corresponding exponential distribution.

          • Set an inactive period $vector.A\_change = t + \tau _i$.

  • (c) Increase the current time t in a time step $\Delta t$: $t = t + \Delta t$.

ABM SIS–SI Pseudo-code

In this Appendix, we present the pseudo-code used to simulate SIS–SI agent-based model. The computational program developed from this pseudo-code was used to generate the curves showed in Fig. 2. The procedure is similar to that presented in Simoy and Aparicio (2020), but considering active and inactive vectors, and without considering latent periods.

For the vector population, the process is the same as explained in Appendix A. In the case of human population, we disregard the demographic processes (births and deaths).

The biting process is simulated as follows. For each active vector in each time step $\Delta t$, a pseudo-random number $u_b$ from an uniform distribution in the interval [0,1] is generated. If $u_b < b \Delta t$, the vector bites a host selected at random. If the vector is susceptible and the host is infected, the vector can be infected with probability $p_v$. In turn, if the vector is infected and the host is susceptible, the vector can transmit the infection with probability $p_h$.

The infectious period in the host population was considered exponentially distributed with parameter $r_h$. The procedure is the same as that was considered for the active and inactive periods in the vector population. When a host becomes infected, a pseudo-random number is generated from an exponential distribution with parameter $r_h$. This simulated infectious period ($\tau _{h,i}$), plus the current time t, is stored in a variable $T\_change$. As before, $T\_change$ is a future time at which the host will change the epidemiological state. When the current time t is grater than or equal to $T\_change$, the infected host becomes susceptible again.

The simulation procedure used is described in the following pseudo-code:

• 1. Initialization of variables and parameters

  • (a) Set the host (H(0)) and vector (V(0)) population sizes, and the initial conditions $H_s(0)$, $H_i(0)$, $H_r(0)$, $V_{s,a}(0)$, $V_{s,i}(0)$, $V_{i,a}(0)$, $V_{i,i}(0)$, $V_{eq}$.

  • (b) Set the time step $\Delta t$, the simulation duration $t_{sim}$ and the present time $t=0$.

  • (c) Set the values of parameters $\mu _v$, $\mu _i$, $\mu _a$, $p_v$, $p_h$, $r_h$, b.

• 2. While $t \le t_{sim}$ and $0 \le H_i(t) + V_{i,a}(t) + V_{i,i}(t)$ /* this last sentence interrupts the program when infections cannot takes place anymore */

  • (a) The number of vectors that will be added to the population is generated according to a Poisson distribution with parameter $V_{eq} \mu _v \Delta t$.

  • (b) For each of these new vectors

    • i. An inactive period $\tau _i$ is generated from an exponential distribution with parameter $\mu _i$.

    • ii. This inactive time is stored in the variable $vector.A\_change = t + \tau _i$

    • iii. The vector is added to the population as susceptible inactive vector.

  • (c) For each vector in the population

    • i. If the vector is active:

      • A. A uniform random number in the interval (0, 1) is generated.

      • B. If the number is less than or equal to $b \Delta t$, the vector bites.

        • The host bitten is chosen at random.

        • If the vector is susceptible and the host bitten is infected:

          • A uniform random number in the interval (0, 1) is generated.

          • If the number is less than or equal to $p_v$, the vector becomes INFECTED

          •    $V_{s,a} = V_{s,a} - 1$, $V_{i,a} = V_{i,a} + 1$

        • If the vector is infected and the host bitten is susceptible:

          • A uniform random number in the interval (0, 1) is generated.

          • If the number is less than or equal to $p_h$, the host becomes INFECTED

          •    $H_{s} = H_{s} - 1$, $H_{i} = H_{i} + 1$

          •    Generate an infectious period $\tau _{h,i}$ according to the corresponding exponential distribution.

          •    Set an infectious time $host.T\_change = t + \tau _{h,i}$

    • ii. A uniform random number in the interval (0, 1) is generated.

    • iii. If the number is less than or equal to $1 - \exp (-\mu _v \Delta t)$:

      The vector dies, and it is removed from the population.

    • iv. Else

      • If vector.A_change is less than or equal to the current time t, the vector changes its activity state

        • If the vector is inactive:

          • Vector becomes ACTIVE; $V_{i,i} = V_{i,i} - 1$, $V_{i,a} = V_{i,a} + 1$, or $V_{s,i} = V_{s,i} - 1$, $V_{s,a} = V_{s,a} + 1$; depending on the vector epidemiological state.

          • Generate an active period $\tau _a$ according to the corresponding exponential distribution.

          • Set an active period $vector.A\_change = t + \tau _a$.

        • If the vector is active:

          • Vector becomes INACTIVE; $V_{i,a} = V_{i,a} - 1$, $V_{i,i} = V_{i,i} + 1$, or $V_{s,a} = V_{s,a} - 1$, $V_{s,i} = V_{s,i} + 1$; depending on the vector epidemiological state.

          • Generate an inactive period $\tau _i$ according to the corresponding exponential distribution.

          • Set an inactive period $vector.A\_change = t + \tau _i$.

  • (d) For each host

    • i. If the host is infected and host.T_change is equal to t:

      Host becomes SUSCEPTIBLE, $H_i(t)=H_i(t)-1$, ${H_s(t)=H_s(t)+1}$

  • (e) Increase the current time t in a time step $\Delta t$: $t = t + \Delta t$.