A model for the operations to render epidemic-free a hog farm infected by the Aujeszky disease

The model for this part of the farm can now be constructed mathematically. Each cohort is represented by subscripts 1-3, then 4-6, 7-9, 10-12. Here x₁, x₄, x₇, x₁₀ represent the populations of the immunized animals of each cohort, in which only natural mortality m is present. Further, x₂, x₅, x₈, x₁₁ represent the susceptible animals: individuals leave this class either by natural mortality or by becoming infected, at rate t if the latter is due to biohazards or external factors, and at rate a if this occurs via direct contact with infected individuals. Here, biohazards are represented by all those factors that are not explicitly built as variables in this model, like for instance absence of filters at the windows, lack of attention to cleanliness, rats, leakage of contaminated liquids, contact with external entities to the farm, such as trucks or people entering the farm without proper protection and so on. Note that the disease incidence is proportional to the ratio of total number of infected in the total population. Finally, x₃, x₆, x₉, x₁₂ denote the infected animals, which enter into this class from the corresponding exiting terms of the susceptible class and leave via natural mortality. Letting X=Σi=112xi$\begin{array}{} \displaystyle X = {\rm{\Sigma }}_{i = 1}^{12}{x_i} \end{array}$ represent the total animal population in this part of the farm, the equations for the gestation unit are therefore

dx1dt=−mx1;dx2dt=−mx2−τx2−αx21X[x3+x6+x9+x12]dx3dt=−mx3+τx2+αx21X[x3+x6+x9+x12];dx4dt=−mx4;dx5dt=−mx5−τx5−αx51X[x3+x6+x9+x12];dx6dt=−mx6+τx5+αx51X[x3+x6+x9+x12];dx7dt=−mx7;dx8dt=−mx8−τx8−αx81X[x3+x6+x9+x12];dx9dt=−mx9+τx8+αx81X[x3+x6+x9+x12];dx10dt=−mx10;dx11dt=−mx11−τx11−αx111X[x3+x6+x9+x12];dx12dt=−mx12+τx11+αx111X[x3+x6+x9+x12].$$\begin{array}{} \displaystyle {\frac{{d{x_1}}}{{dt}} = - m{x_1};}\\ {\frac{{d{x_2}}}{{dt}} = - m{x_2} - \tau {x_2} - \alpha {x_2}\frac{1}{X}[{x_3} + {x_6} + {x_9} + {x_{12}}]}\\ {\frac{{d{x_3}}}{{dt}} = - m{x_3} + \tau {x_2} + \alpha {x_2}\frac{1}{X}[{x_3} + {x_6} + {x_9} + {x_{12}}];}\\ {\frac{{d{x_4}}}{{dt}} = - m{x_4};}\\ {\frac{{d{x_5}}}{{dt}} = - m{x_5} - \tau {x_5} - \alpha {x_5}\frac{1}{X}[{x_3} + {x_6} + {x_9} + {x_{12}}];}\\ {\frac{{d{x_6}}}{{dt}} = - m{x_6} + \tau {x_5} + \alpha {x_5}\frac{1}{X}[{x_3} + {x_6} + {x_9} + {x_{12}}];}\\ {\frac{{d{x_7}}}{{dt}} = - m{x_7};}\\ {\frac{{d{x_8}}}{{dt}} = - m{x_8} - \tau {x_8} - \alpha {x_8}\frac{1}{X}[{x_3} + {x_6} + {x_9} + {x_{12}}];}\\ {\frac{{d{x_9}}}{{dt}} = - m{x_9} + \tau {x_8} + \alpha {x_8}\frac{1}{X}[{x_3} + {x_6} + {x_9} + {x_{12}}];}\\ {\frac{{d{x_{10}}}}{{dt}} = - m{x_{10}};}\\ {\frac{{d{x_{11}}}}{{dt}} = - m{x_{11}} - \tau {x_{11}} - \alpha {x_{11}}\frac{1}{X}[{x_3} + {x_6} + {x_9} + {x_{12}}];}\\ {\frac{{d{x_{12}}}}{{dt}} = - m{x_{12}} + \tau {x_{11}} + \alpha {x_{11}}\frac{1}{X}[{x_3} + {x_6} + {x_9} + {x_{12}}].} \end{array}$$

The initial conditions for the immunized animals are

x1(nT+)=z1(nT−),x4(nT+)=x1(nT−),x7(nT+)=x4(nT−),x10(nT+)=x7(nT−);$$\begin{array}{} \displaystyle {{x_1}(n{T^ + }) = {z_1}(n{T^ - }),\quad {x_4}(n{T^ + }) = {x_1}(n{T^ - }),{\rm{ }}}\\ {{x_7}(n{T^ + }) = {x_4}(n{T^ - }),\quad {x_{10}}(n{T^ + }) = {x_7}(n{T^ - });} \end{array}$$

those for the susceptibles are

x2(nT+)=z2(nT−),x5(nT+)=x2(nT−),x8(nT+)=x5(nT−),x11(nT+)=x8(nT−);$$\begin{array}{} \displaystyle {{x_2}(n{T^ + }) = {z_2}(n{T^ - }),\quad {x_5}(n{T^ + }) = {x_2}(n{T^ - }),}\\ {{x_8}(n{T^ + }) = {x_5}(n{T^ - }),\quad {x_{11}}(n{T^ + }) = {x_8}(n{T^ - });} \end{array}$$

finally for the infected we have

x3(nT+)=z3(nT−),x6(nT+)=x3(nT−),x9(nT+)=x6(nT−),x12(nT+)=x9(nT−).$$\begin{array}{} \displaystyle {{x_3}(n{T^ + }) = {z_3}(n{T^ - }),\quad {x_6}(n{T^ + }) = {x_3}(n{T^ - }),}\\ {{x_9}(n{T^ + }) = {x_6}(n{T^ - }),\quad {x_{12}}(n{T^ + }) = {x_9}(n{T^ - }).} \end{array}$$

Note that the new sows in the first cohort are those discharged from the farrowing unit, z_k, for k = 1, 2, 3.

The parameters have been chosen as follows. For natural mortality, m, the value is taken from data from the particular raising farm considered here. Moreover, since we have the data of the new acquisitions of sows, since these are accurately selected and vaccinated, we added these monthly acquisitions in the set of the immune sows of the cohort in the first stage of the gestation, since they are waiting to be inseminated. In each cohort, the deaths are assumed proportionally subdivided among their 3 subsets, susceptible, infected, immune.

Letting Z=Σi=13zi$\begin{array}{} \displaystyle Z = {\rm{\Sigma }}_{i = 1}^3{z_i} \end{array}$, the farrowing unit equations are the following ones,

dz1dt=−mz1;dz2dt=−mz2−Δz2−βz21Zz3;dz3dt=−mz3+Δz2+βz21Zz3;du1dt=b(z1+z3);du2dt=bz2−Δu2−γu21Z(z3+u3);du3dt=Δu2+γu21Z(z3+u3).$$\begin{array}{} \displaystyle {\frac{{d{z_1}}}{{dt}} = - m{z_1};}\\ {\frac{{d{z_2}}}{{dt}} = - m{z_2} - \Delta {z_2} - \beta {z_2}\frac{1}{Z}{z_3};}\\ {\frac{{d{z_3}}}{{dt}} = - m{z_3} + \Delta {z_2} + \beta {z_2}\frac{1}{Z}{z_3};}\\ {\frac{{d{u_1}}}{{dt}} = b({z_1} + {z_3});}\\ {\frac{{d{u_2}}}{{dt}} = b{z_2} - \Delta {u_2} - \gamma {u_2}\frac{1}{Z}({z_3} + {u_3});}\\ {\frac{{d{u_3}}}{{dt}} = \Delta {u_2} + \gamma {u_2}\frac{1}{Z}({z_3} + {u_3}).} \end{array}$$

The first three equations describe the sows cohort, immunized z₁, susceptible z₂ and infected z₃, the remaining ones are for the newborns again immunized u₁, susceptible u₂ and infected u₃.

Clearly, the initial conditions of the last equations are zero. The initial conditions for the immunized animals are

z1(nT+)=x10(nT−),u1(nT+)=0;$$\begin{array}{} \displaystyle {z_1}(n{T^ + }) = {x_{10}}(n{T^ - }),\quad {u_1}(n{T^ + }) = 0; \end{array}$$

those for the susceptibles are

z2(nT+)=x11(nT−),u2(nT+)=0;$$\begin{array}{} \displaystyle {z_2}(n{T^ + }) = {x_{11}}(n{T^ - }),\quad {u_2}(n{T^ + }) = 0; \end{array}$$

finally for the infected we have

z3(nT+)=x12(nT−),u3(nT+)=0;$$\begin{array}{} \displaystyle {z_3}(n{T^ + }) = {x_{12}}(n{T^ - }),\quad {u_3}(n{T^ + }) = 0; \end{array}$$

Note that the new sows in the first cohort are those discharged from the farrowing unit, z_k, for k = 1, 2, 3.

The first equation describes the immune sows evolution. The second one, the susceptible ones: they leave this set via natural mortality and via contagion due either to external means, at rate Δ < τ, since in general this unit is much more checked and disinfected than all the other parts of the farm, or by contact with an infected, at rate β < α, for the same reasons given above. Note that in this equation we do not take into account the contagion from infected newborns to susceptible sows, i.e. u₃ is not added to z₃ in the last term, because it is well known that this type of contagion cannot biologically occur, [16].

The third equation concerns the infected, which enter this class through terms arising from the infection processes described in the former equation, and leave only via mortality. The fourth equation represents the immune newborns, which are so via the maternal immunity that lasts for about 90-100 days. Only the sows which were exposed to the virus whether wild or coming from the vaccine, can give this maternal immunity protection to their offsprings. Thus this equation has net birthrate b proportional to the sum of the immune and infected sows. Note that we take the same birth rate b for both susceptible and infected sows, because as mentioned earlier, by “infected” sows we mean sows that have been vaccinated, and therefore are immune and do not experience clinical signs. Recall that, as stated in Section 2, exposure to the virus is the only way the “disease” is present in farms nowadays. Thus, in practice, no difference in the birth rates between susceptible and infected sows is observed. The fifth equation contains the susceptible newborns, coming from susceptible sows. The birthrate, b, is proportional to the number of susceptible sows. The equation presents two contagion exits: either via biohazards at rate Δ and via direct contact with infected sows or offsprings at rate γ. Finally the last equation describes the infected newborns, with entries from the exits of the previous equation.

The value of the net birthrate is b = 0.2358 on the basis of the available data. Note that again, the “disease” is in fact only an infection, as stated in Section 2 and thus disease-related mortality does not really arise. Therefore no disease-related mortality appears in the equation for the infected newborns u₃. To validate this choice, in the simulation we compared every four months the estimated number of newborns with the real data, see Table 2.

We now describe the weaning unit. Here the weaned pigs are left for about two months, before entering the fattening unit of the farm. We subdivide the weaning unit into two classes, I and II. The weaned pigs spend about a month in each of them. In fact this subdivision is artificial, it is done for modelling purposes only, while in reality this does not occur. However it should be noted that in class II, i.e. between the 60th and 90th day of life, the first vaccination occurs. The second one will follow after another three to four weeks from the first one. Letting W=Σi=17wi$\begin{array}{} \displaystyle W = {\rm{\Sigma }}_{i = 1}^7{w_i} \end{array}$, the governing equations are:

dw1dt=−nw1;dw2dt=−nw2−Sw2−ηw21W(w3+w6);dw3dt=−nw3+Sw2+ηw21W(w3+w6);dw4dt=−nw4−kw4;dw5dt=−nw5−Sw5−kw5−ηw51W(w3+w6);dw6dt=−nw6+Sw5+ηw51W(w3+w6);dw7dt=−nw7+k(w4+w5).$$\begin{array}{} \displaystyle {\frac{{d{w_1}}}{{dt}} = - n{w_1};}\\ {\frac{{d{w_2}}}{{dt}} = - n{w_2} - S{w_2} - \eta {w_2}\frac{1}{W}({w_3} + {w_6});}\\ {\frac{{d{w_3}}}{{dt}} = - n{w_3} + S{w_2} + \eta {w_2}\frac{1}{W}({w_3} + {w_6});}\\ {\frac{{d{w_4}}}{{dt}} = - n{w_4} - k{w_4};}\\ {\frac{{d{w_5}}}{{dt}} = - n{w_5} - S{w_5} - k{w_5} - \eta {w_5}\frac{1}{W}({w_3} + {w_6});}\\ {\frac{{d{w_6}}}{{dt}} = - n{w_6} + S{w_5} + \eta {w_5}\frac{1}{W}({w_3} + {w_6});}\\ {\frac{{d{w_7}}}{{dt}} = - n{w_7} + k({w_4} + {w_5}).} \end{array}$$

The initial conditions for the immunized animals are

w1(nT+)=u1(nT−)+u3(nT−),w4(nT+)=w1(nT−);$$\begin{array}{} \displaystyle {w_1}(n{T^ + }) = {u_1}(n{T^ - }) + {u_3}(n{T^ - }),\quad {w_4}(n{T^ + }) = {w_1}(n{T^ - }); \end{array}$$

those for the susceptibles are

w2(nT+)=u2(nT−),w5(nT+)=w2(nT−);$$\begin{array}{} \displaystyle {w_2}(n{T^ + }) = {u_2}(n{T^ - }),\quad {w_5}(n{T^ + }) = {w_2}(n{T^ - }); \end{array}$$

for the infected we have

w3(nT+)=0,w6(nT+)=w3(nT−);$$\begin{array}{} \displaystyle {w_3}(n{T^ + }) = 0,\quad {w_6}(n{T^ + }) = {w_3}(n{T^ - }); \end{array}$$

and finally for the vaccinated ones, since none is so initially, we have

w7(nT+)=0.$$\begin{array}{} \displaystyle {w_7}(n{T^ + }) = 0. \end{array}$$

The first three of them describe class I, the remaining ones class II. The first equation represents the weaned pigs that were born one month earlier either from sound and well vaccinated mother, or from infected mothers, i.e. the weaned pigs who are immune from the disease in view of the action of the maternal immunity. Animals can leave this class only by death, at rate n, here chosen to equal the sows’ mortality, n = m. The second equation contains the dynamics of the weaned pigs that were born from susceptible mothers, so that they are not immune from the AD. In addition to natural mortality, we must here include also an exit from the class due to contagion, either direct of by biohazard, at respective rates η and S. The third equation is for the infected weaned pigs. Animals enter this class from the previous one, via contagion, and exit by natural mortality. Note that the data collected on which the birth rate parameter is calibrated already discount possible deaths. Therefore they represent a net birth rate, so there is no need to add here an extra disease-related mortality.

The fourth equation describes the sound weaned pigs, born two months in advance. Here animals can leave by death, but also by being vaccinated, at rate k = 0.03. This value has been so chosen on the basis of experience, namely the good results it gives. In this last case the weaned pigs enter the class of vaccinated weaned pigs. The latter is described in the seventh equation, into which also enter weaned pigs from the fifth equation, for whom vaccination has been successful. The latter equation describes the evolution of susceptible weaned pigs, which can leave also by contracting either directly or by biohazards the disease, at respective rates η and S. The sixth equation contains the infected weaned pigs dynamics; in it come the animals leaving from the fifth equation via the two ways of contagion. Note that the pigs in this class are vaccinated as all the animals in the farm are. But for the individuals in this class the vaccination has not been effective, for whatever reason. In other words, the vaccination rate k mentioned above is to be understood rather to represent the successful rate of vaccination.

The initial conditions of class I are the final values of the newborns coming from the farrowing unit. Also, the initial condition of class II is represented by the final values of weaned pigs in class I, clearly with the value of vaccinated animals initially set to zero. Furthemore, since at about the end of the third month the maternal immunity ends, when the weaned pigs leave class II, becoming fattening pigs, the subdivision between immune and susceptible animals has no further meaning. Therefore both immune and susceptible pigs at the end of the weaning in class II will merge into susceptible fattening pigs.

We finally model the fattening unit of the farm. It contains pigs from their 4th up to the 9th month of life. Again we subdivided the unit in several classes. After weaning, an animal goes through all the six classes, from I to VI, remaining in each of them for a month. At the end of the sixth fattening month, when the animal is at least 9 months old, it is slaughtered. At this time indeed it reaches the prescribed weight established by the certification procedures for the production of raw ham. In addition, in class II and IV, i.e. at about the fifth and seventh month of age, the animals are vaccinated again, but this renewed vaccinations will be effective only if the former ones had been performed well and they were successful.

In order to better simulate the system, since not all the initial conditions for the fattening class are not known, in what follows these are marked with question marks, in practice they have been chosen as the final outcomes of a previously run one year long simulation, which in turn started from arbitrarily chosen initial conditions. By running the simulation for a whole year, we considered that the arbitrary choice of the initial conditions would in this way be smoothed out. The results of this simulation confirm the goodness of the assumptions and of the general strategy.

Let Y=Σi=120yi$\begin{array}{} \displaystyle Y = {\rm{\Sigma }}_{i = 1}^{20}{y_i} \end{array}$. The equations for the three-months old pigs entering this unit are then

dy1dt=−ny1−Qy1−ξy11Y(y2+y5+y9+y12+y16+y19);dy2dt=−ny2+Qy1+ξy11Y(y2+y5+y9+y12+y16+y19);dy3dt=−ny3;$$\begin{array}{} \displaystyle {\frac{{d{y_1}}}{{dt}} = - n{y_1} - Q{y_1} - \xi {y_1}\frac{1}{Y}({y_2} + {y_5} + {y_9} + {y_{12}} + {y_{16}} + {y_{19}});}\\ {\frac{{d{y_2}}}{{dt}} = - n{y_2} + Q{y_1} + \xi {y_1}\frac{1}{Y}({y_2} + {y_5} + {y_9} + {y_{12}} + {y_{16}} + {y_{19}});}\\ {\frac{{d{y_3}}}{{dt}} = - n{y_3};} \end{array}$$

for the four-months old ones

dy4dt=−ny4−Qy4−ξy41Y(y2+y5+y9+y12+y16+y19);dy5dt=−ny5+Qy4+ξy41Y(y2+y5+y9+y12+y16+y19);dy6dt=−ny6−Ky6;dy7dt=−ny7+Ky6;$$\begin{array}{} \displaystyle {\frac{{d{y_4}}}{{dt}} = - n{y_4} - Q{y_4} - \xi {y_4}\frac{1}{Y}({y_2} + {y_5} + {y_9} + {y_{12}} + {y_{16}} + {y_{19}});}\\ {\frac{{d{y_5}}}{{dt}} = - n{y_5} + Q{y_4} + \xi {y_4}\frac{1}{Y}({y_2} + {y_5} + {y_9} + {y_{12}} + {y_{16}} + {y_{19}});}\\ {\frac{{d{y_6}}}{{dt}} = - n{y_6} - K{y_6};}\\ {\frac{{d{y_7}}}{{dt}} = - n{y_7} + K{y_6};} \end{array}$$

for the fifth-months old ones

dy8dt=−ny8−Qy8−ξy81Y(y2+y5+y9+y12+y16+y19);dy9dt=−ny9+Qy8+ξy81Y(y2+y5+y9+y12+y16+y19);dy10dt=−ny10;$$\begin{array}{} \displaystyle {\frac{{d{y_8}}}{{dt}} = - n{y_8} - Q{y_8} - \xi {y_8}\frac{1}{Y}({y_2} + {y_5} + {y_9} + {y_{12}} + {y_{16}} + {y_{19}});}\\ {\frac{{d{y_9}}}{{dt}} = - n{y_9} + Q{y_8} + \xi {y_8}\frac{1}{Y}({y_2} + {y_5} + {y_9} + {y_{12}} + {y_{16}} + {y_{19}});}\\ {\frac{{d{y_{10}}}}{{dt}} = - n{y_{10}};} \end{array}$$

for the six-months old ones

dy11dt=−ny11−Qy11−ξy111Y(y2+y5+y9+y12+y16+y19);dy12dt=−ny12+Qy11+ξy111Y(y2+y5+y9+y12+y16+y19);dy13dt=−ny13−Ky13;dy14dt=−ny14+Ky13;$$\begin{array}{} \displaystyle \begin{array}{*{20}{c}} {\frac{{d{y_{11}}}}{{dt}} = {\rm{ }} - n{y_{11}} - Q{y_{11}} - \xi {y_{11}}\frac{1}{Y}({y_2} + {y_5} + {y_9} + {y_{12}} + {y_{16}} + {y_{19}});} \hfill \\ {\frac{{d{y_{12}}}}{{dt}} = {\rm{ }} - n{y_{12}} + Q{y_{11}} + \xi {y_{11}}\frac{1}{Y}({y_2} + {y_5} + {y_9} + {y_{12}} + {y_{16}} + {y_{19}});} \hfill \\ {\frac{{d{y_{13}}}}{{dt}} = {\rm{ }} - n{y_{13}} - K{y_{13}};} \hfill \\ {\frac{{d{y_{14}}}}{{dt}} = {\rm{ }} - n{y_{14}} + K{y_{13}};} \hfill \\ \end{array} \end{array}$$

for the seven-months old ones

dy15dt=−ny15−Qy15−ξy151Y(y2+y5+y9+y12+y16+y19);dy16dt=−ny16+Qy15+ξy151Y(y2+y5+y9+y12+y16+y19);dy17dt=−ny17;$$\begin{array}{} \displaystyle {\frac{{d{y_{15}}}}{{dt}} = - n{y_{15}} - Q{y_{15}} - \xi {y_{15}}\frac{1}{Y}({y_2} + {y_5} + {y_9} + {y_{12}} + {y_{16}} + {y_{19}});}\\ {\frac{{d{y_{16}}}}{{dt}} = - n{y_{16}} + Q{y_{15}} + \xi {y_{15}}\frac{1}{Y}({y_2} + {y_5} + {y_9} + {y_{12}} + {y_{16}} + {y_{19}});}\\ {\frac{{d{y_{17}}}}{{dt}} = - n{y_{17}};} \end{array}$$

and finally for the eigth-months old ones

dy18dt=−ny18−Qy18−ξy181Y(y2+y5+y9+y12+y16+y19);dy19dt=−ny19+Qy18+ξy181Y(y2+y5+y9+y12+y16+y19);dy20dt=−ny20;$$\begin{array}{} \displaystyle {\frac{{d{y_{18}}}}{{dt}} = - n{y_{18}} - Q{y_{18}} - \xi {y_{18}}\frac{1}{Y}({y_2} + {y_5} + {y_9} + {y_{12}} + {y_{16}} + {y_{19}});}\\ {\frac{{d{y_{19}}}}{{dt}} = - n{y_{19}} + Q{y_{18}} + \xi {y_{18}}\frac{1}{Y}({y_2} + {y_5} + {y_9} + {y_{12}} + {y_{16}} + {y_{19}});}\\ {\frac{{d{y_{20}}}}{{dt}} = - n{y_{20}};} \end{array}$$

The initial conditions for the susceptible animals are for k ∊ {4, 11, 18} and for i ∊ {8, 15}

y1(nT+)=w4(nT−)+w5(nT−),yk(nT+)=yk−3(nT−),yi(nT+)=yi−4(nT−);$$\begin{array}{} \displaystyle {y_1}(n{T^ + }) = {w_4}(n{T^ - }) + {w_5}(n{T^ - }),\quad {y_k}(n{T^ + }) = {y_{k - 3}}(n{T^ - }),\quad {y_i}(n{T^ + }) = {y_{i - 4}}(n{T^ - }); \end{array}$$

those for the infected are for k ∊ {5, 12, 19} and for i ∊ {9, 16}

y2(nT+)=w6(nT−),yk(nT+)=yk−3(nT−),yi(nT+)=yi−4(nT−);$$\begin{array}{} \displaystyle {y_2}(n{T^ + }) = {w_6}(n{T^ - }),\quad {y_k}(n{T^ + }) = {y_{k - 3}}(n{T^ - }),\quad {y_i}(n{T^ + }) = {y_{i - 4}}(n{T^ - }); \end{array}$$

for the properly vaccinated animals we have for k ∊ {10, 17, 20}

y3(nT+)=w7(nT−),yk(nT+)=yk−3(nT−);$$\begin{array}{} \displaystyle {y_3}(n{T^ + }) = {w_7}(n{T^ - }),\quad {y_k}(n{T^ + }) = {y_{k - 3}}(n{T^ - }); \end{array}$$

and finally for the vaccinated ones for which renewed vaccinations occur, we have

y6(nT+)=y3(nT−),y13(nT+)=y10(nT−),y7(nT+)=0,y14(nT+)=0.$$\begin{array}{} \displaystyle {y_6}(n{T^ + }) = {y_3}(n{T^ - }),\quad {y_{13}}(n{T^ + }) = {y_{10}}(n{T^ - }),\quad {\rm{ \;}}{y_7}\left( {n{T^ + }} \right) = 0,{\rm{ \;}}{y_{14}}\left( {n{T^ + }} \right) = 0. \end{array}$$

The equations for y₁, y₄, y₈, y₁₁, y₁₅ and y₁₈ describe the susceptible fattening pigs. They exit these classes via mortality at rate n, and by becoming infected through biohazard at rate Q or via direct contact, at rate ξ . The variables y₂, y₅, y₉, y₁₂, y₁₆ and y₁₉ denote the infected animals, which can die at rate n, and enter these classes by becoming infected, i.e. from the corresponding former classes. The equations for y₃, y₁₀, y₁₇ and y₂₀ stand for the properly vaccinated animals, in which the vaccination has been effective; y₆ and y₁₃ are as the former ones, but in addition since they describe the classes in which the renewed vaccinations occur, they are characterized by terms containing the vaccination rate K that we take to be the same as for the weaning unit, K = k. If the vaccination is successful, the animal migrates into the pigs subjected to booster vaccination class, y₇ and y₁₄, simply denoted by booster in what follows.

From the four classes, susceptible, infected, well vaccinated and booster pigs, the animals migrate, in the following month, to only three possible classes of susceptible, infected and well vaccinated pigs, since the susceptible pigs will contain the previous susceptible animals and the leftovers of the well vaccinated ones, i.e. those animals for which the third vaccination has not been successful, so that their immunization is being lost as time goes by. The new well vaccinated class at this time will contain the former booster.

Finally the seventh and fourteenth equations describe the booster, and contain the populations leaving from equations 6 and 13 at rate K.

Fix α = 0.006, β = 0.002, γ = 0.004.

Fig. 4

Fig. 5

Fig. 6

Fig. 7

Fig. 8

Fig. 9

To better analyze the results, for each simulation we report in the following table the total number of infected newborns and the percentage of infected sows, in the whole year, for the two reproductive units, gestation and farrowing units: these are the averages of the percentage of infected sows in each month of the year.

In the absence of external contamination and biohazards, we have respectively the average number of infected weaned pigs as 2.244, and the percentage of infected shows at 14.44.

We now analyze the relationship between biohazard values in the two units, i.e. τ in the gestation unit and Δ in the farrowing unit, and the percentage of infected sows.

We take simulation A as reference because it is the common simulation to both cases that includes both biohazards and external contamination.

Taking a varying value for Δ from 0.002, for simulations C, F, G and H, we have for Δ = 0 the value of infected sows 23.86%, for Δ = 0.003 the value 25.39%, for Δ = 0.006 the value 26.69% and for Δ = 0.01 the value 28.15%.

We can therefore conclude that in order to decrease the number of infected sows, it is not convenient to act excessively on containing the biohazard Δ: since to ensure total biosafety has very high costs, the benefit one obtains in such case is not so much relevant. The ease with which one goes from 24.91% up to almost 29% for Δ = 0.01 shows however that it is not wise to disregard Δ. Its increase in fact leads to observable changes.

When τ varies from 0.003, in simulations D, E and I, we find for τ = 0.006 the value 29.51%, for τ = 0.01 the value 32.98% and for τ = 0 the value 16.70%. Thus, τ is much more relevant. Its growth leads to large percentage increases of the infected sows, but if we can minimize it, also a considerable reduction in the infected animals will be achieved, in some cases they drop even below 17%.

Finally, by minimizing both biohazard parameters, simulation B, we reach the value 14.44%, which is not so far from the threshold one can achieve by acting just upon τ.

We now analyze the relationship between the biohazard parameters and the number of infected weaned pigs which are introduced into the weaning unit and later on into the fattening unit, in the whole year, taking always as reference simulation A.

As Δ varies from the value 0.002, simulations C, F, G and H, for Δ = 0 the value 2.3667 is obtained, for Δ = 0.003 the value 39.947, for Δ = 0.006 the value 70.1768 and for Δ = 0.01 the value 101.6042.

Thus, the role played by Δ is relevant: by minimizing it, we can practically extinguish the disease from the weaned pigs, while if it is left unchecked, the infected animals grow unboundedly.

As τ varies from 0.003 in simulations D, E and I, we find for τ = 0.006 the value 21.1276, for τ = 0.01 the value 15.5476 and for τ = 0 the value 40.4711.

If both biohazard parameters are minimized, simulation B, we find the minimum value 2.244%. This is smaller, even though just by a little, to the threshold found for Δ = 0. Thus biosafety in the farrowing unit is absolute, the contagion probability for newborns is so small that to get the disease for the newborns without maternal immunity, i.e. coming from susceptible mothers, when also τ = 0 is smaller than the advantage we have given by the smaller number of infected sows coming from the gestation unit. Since however to reach total biosafety in at least one of the two environments in practice it is impossible, and in view of the fact that to have τ = Δ = 0 entails little improvement over the situation for Δ = 0, it seems therefore preferable not to exceed in biosafety in the gestation unit, until we are sure to have achieved a very high biosafety standard in the farrowing unit, and to maintain it in time.

We want now to analyze the situation in the fattening unit. We observe how the number of infected animals to be sent to slaughter changes as a function of the biosafety in the two units, gestation and farrowing units. Indeed these are the two parameters on which it is easier to operate in practice. All the remaining parameters have the same values in all the simulations reported in the next tables.

Fig. 10

Fig. 11

Fig. 12

The previous data show how by improving biosafety in gestation and farrowing units it is not possible to reduce the average percentage of infected animals in the fattening unit, below a threshold of about 35%. When τ = 0 in simulation I, we find 35.85%, for Δ = 0, simulation C, we have 34.80%. Even by decreasing biosafety things do not improve much, since by increasing D to 0.006, simulation G, we find the value 35.61% and for 0.01, simulation H we have 36.01%. By increasing τ to 0.006, simulation D, we arrive at 34.77% and for τ = 0.01, simulation E, we get the value 34.45%.

By setting finally τ = Δ = 0, simulation B, we find the value 35.48%.

From these data, we find that the best way of modifying biosafety is to allow a slight increase in the biohazard in the farrowing unit followed by a reduction of Δ. Improving both parameters does not lead to satisfactory results.

Looking at the percentages, in the various simulations we find once again the previous considerations, but we also remark the large increase of infected percentages in simulation A. In it we arrive at 64.75% for τ = 390. From this, the importance of keeping well below certain thresholds both biohazard rates is clear, in order to avoid a sudden increase in the infected animals in the fattening unit.

In conclusion it does not seem possible to eradicate the disease from the entire farm, just by acting on the gestation and farrowing units. However by improving their biosafety it is possible to reduce significantly the number of infected in the same units, and to contain them in the fattening unit. It remains to assess how to reduce biohazard in the first two units when the disease is present in the fattening unit, since the epidemic spreads not only by direct contact but also via other means, and the closer one animal is to an affected area, the higher the possibility of getting the infection. But these considerations are beyond the scope of this model, since they require to work with biohazard parameters which are directly related to the number of infect animals present in the vicinity of each unit in the farm.

Since one of the main problems arisen during the implementation of the “Piano Nazionale di Controllo della Malattia di Aujeszky” (National Control Plan for the AD) was the omission or the bad execution of the third vaccination, which should be done between the sixth and the seventh month of age, we finally ran further simulations to assess its impact on the disease eradication plan.

We summarize in the tables some of the simulations already discussed above for the analysis of the infected percentages in the fattening unit. Aside of the “infected percentage” column, under the heading “new infected percentage” are the values of the same simulations, performed however under the assumption that the third vaccination is not performed.

Fig. 13

Fig. 14

Fig. 15

From all the simulations it is clear that if the third vaccination is not performed, the percentage of infected animals increases of about 5%.

Finally, we have run a sensitivity analysis on the model to assess what are the parameters which changing affect the system more. The results are shown in Figures 16, 17 and 18 respectively for the variables x, z and u. To allow a better comparison, the vertical scale has been kept the same in all figures, namely the interval [−600, 600]. Clearly the most sensitive quantity is represented by the newborns u, which heavily depend on the birth rate b and on the horizontal disease transmission rate Δ. Secondly, we find that the sow populations are influenced in a very similar way by a couple of parameters: namely, the sows in the gestation unit x by the external contamination, modeled by parameter τ and those in the farrowing unit z once again by the parameter D; in a similar way to this one, they are also both influenced by mortality m. Next, the three populations depend less markedly on the last parameter, the three horizontal transmission rates, namely α for x, β for z and γ for u.

Fig. 16

Fig. 17

Fig. 18

We also calculated the correlation among the parameters relative to the gestation and farrowing units and their populations, Figure 19. The parameters considered are, in the bottom-up order of the figure, m, α, τ, Δ, β , b, γ. The infected sows in both units appear to be negatively correlated with all these parameters. Instead, the newborns appear to be positively correlated with Δ, b and γ, respectively the contamination due to external factors, the net birthrate and the contact rate, all these apparently at the same level. All the immunized animals are negatively correlated with all the parameters with the exception of the newborns, that appear to be influenced by their birthrate, a fact which is quite plausible. The same behavior is exhibited by the three susceptible populations. In fact the correlation coefficients in the two cases differ, but only the least significant digits.

Fig. 19